Best mathematics books according to redditors
We found 7,687 Reddit comments discussing the best mathematics books. We ranked the 2,734 resulting products by number of redditors who mentioned them. Here are the top 20.
We found 7,687 Reddit comments discussing the best mathematics books. We ranked the 2,734 resulting products by number of redditors who mentioned them. Here are the top 20.
Copying my answer from another post:
I was literally in the bottom 14th percentile in math ability when i was 12.
One day, by pure chance, i stumbled across this (free and open) book written by Carl Stitz and Jeff Zeager, of Lakeland Community College
Precalculus
It covers everything from elementary algebra (think grade 5), all the way up to concepts used in Calculus and Linear Algebra (Partial fractions and matrix algebra, respectively.) The book is extremely well organized. Every sections starts with a dozen or so pages of proofs and derivations that show you the logic of why and how the formulas you'll be using work. This book, more than any other resource (and i've tried a lot of them), helped me build my math intuition from basically nothing.
Math is really, really intimidating when you've spent your whole life sucking at it. This book addresses that very well. The proofs are all really well explained, and are very long. You'll basically never go from one step to the next and be completely confused as to how they got there.
Also, there is a metric shitload of exercises, ranging from trivial, to pretty difficult, to "it will literally take your entire class working together to solve this". Many of the questions follow sort of an "arc" through the chapters, where you revisit a previous problem in a new context, and solve it with different means (Also, Sasquatches. You'll understand when you read it.)
I spent 8 months reading this book an hour a day when i got home from work, and by the end of it i was ready for college. I'm now in my second year of computer science and holding my own (although it's hard as fuck) against Calculus II. I credit Stitz and Zeager entirely. Without this book, i would never have made it to college.
Edit: other resources
Khan Academy is good, and it definitely complements Stitz/Zeager, but Khan also lacks depth. Like, a lot of depth. Khan Academy is best used for the practice problems and the videos do a good job of walking you through application of math, but it doesn't teach you enough to really build off of it. I know this from experience, as i completed all of Khan's precalculus content. Trust me, Rely on the Stitz book, and use Khan to fill in the gaps.
Paul's Online Math Notes
This website is so good it's ridiculous. It has a ton of depth, and amazing reference sheets. Use this for when you need that little extra detail to understand a concept. It's still saving my ass even today (Damned integral trig substitutions...)
Stuff that's more important than you think (if you're interested in higher math after your GED)
Trigonometric functions: very basic in Algebra, but you gotta know the common values of all 6 trig functions, their domains and ranges, and all of their identities for calculus. This one bit me in the ass.
Matrix algebra: Linear algebra is p. cool. It's used extensively in computer science, particularly in graphics programming. It's relatively "easy", but there's more conceptual stuff to understand.
Edit 2: Electric Boogaloo
Other good, cheap math textbooks
/u/ismann has pointed out to me that Dover Publications has a metric shitload of good, cheap texts (~$25CAD on Amazon, as low as a few bucks USD from what i hear).
Search up Dover Mathematics on Amazon for a deluge of good, cheap math textbooks. Many are quite old, but i'm sure most will agree that math is a fairly mature discipline, so it's not like it makes a huge difference at the intro level. Here is a Math
OverflowExchange list of the creme de la creme of Dover math texts, all of which can be had for under $30, often much less. I just bought ~1,000 pages of Linear Algebra, Graph Theory, and Discrete Math text for $50. If you prefer paper to .pdf, this is probably a good route to go.Also, How to Prove it is a very highly rated (and easy to read!) introduction to mathematical proofs. It introduces the basic logical constructs that mathematicians use to write rigorous proofs. It's very approachable, fairly short, and ~$30 new.
Here's my list of the classics:
General Computing
Computer Science
Software Development
Case Studies
Employment
Language-Specific
C
Python
C#
C++
Java
Linux Shell Scripts
Web Development
Ruby and Rails
Assembly
Simon Singh explains.
edit: Hey, I didn't expect this to become the top comment. Neat. Might as well abuse it, by providing bonus material:
This is the same Simon Singh discussed in this recent and popular Reddit post; he is a superhero of science popularization. He has written some excellent and highly rated books:
(full disclosure: I have nothing to disclose, other than that I'm a fan of his work.)
The answer is "virtually all of mathematics." :D
Although lots of math degrees are fairly linear, calculus is really the first big branch point for your learning. Broadly speaking, the three main pillars of contemporary mathematics are:
You might also think of these as the three main "mathematical mindsets" — mathematicians often talk about "thinking like an algebraist" and so on.
Calculus is the first tiny sliver of analysis and Spivak's Calculus is IMO the best introduction to calculus-as-analysis out there. If you thought Spivak's textbook was amazing, well, that's bread-n-butter analysis. I always thought of Spivak as "one-dimensional analysis" rather than calculus.
Spivak also introduces a bit of algebra, BTW. The first few chapters are really about abstract algebra and you might notice they feel very different from the latter chapters, especially after he introduces the least-upper-bound property. Spivak's "properties of numbers" (P1-P9) are actually the 9 axioms which define an algebraic object called a field. So if you thought those first few chapters were a lot of fun, well, that's algebra!
There isn't that much topology in Spivak, although I'm sure he hides some topology exercises throughout the book. Topology is sometimes called the study of "shape" and is where our most general notions of "continuous function" and "open set" live.
Here are my recommendations.
Analysis If you want to keep learning analysis, check out Introductory Real Analysis by Kolmogorov & Fomin, Principles of Mathematical Analysis by Rudin, and/or Advanced Calculus of Several Variables by Edwards.
Algebra If you want to check out abstract algebra, check out Dummit & Foote's Abstract Algebra and/or Pinter's A Book of Abstract Algebra.
Topology There's really only one thing to recommend here and that's Topology by Munkres.
If you're a high-school student who has read through Spivak in your own, you should be fine with any of these books. These are exactly the books you'd get in a more advanced undergraduate mathematics degree.
I might also check out the Chicago undergraduate mathematics bibliography, which contains all my recommendations above and more. I disagree with their elementary/intermediate/advanced categorization in many cases, e.g., Rudin's Principles of Mathematical Analysis is categorized as "elementary" but it's only "elementary" if your idea of doing math is pursuing a PhD. Baby Rudin (as it's called) is to first-year graduate analysis as Spivak is to first-year undergraduate calculus — Rudin says as much right in the introduction.
A Book of Abstract Algebra by Charles C. Pinter
Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.
Amazon search for Dover Books on mathematics
There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm
Pre-Calculus / Problem-Solving
Calculus
Linear Algebra
Differential Equations
Number Theory
Proof-Writing
Analysis
Complex Analysis
Functional Analysis
Partial Differential Equations
Higher-dimensional Calculus and Differential Geometry
Abstract Algebra
Geometry
Topology
Set Theory and Logic
Combinatorics / Discrete Math
Graph Theory
P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
I started from scratch on the formal CS side, with an emphasis on program analysis, and taught myself the following starting from 2007. If you're in the United States, I recommend BookFinder to save money buying these things used.
On the CS side:
On the math side, I was advantaged in that I did my undergraduate degree in the subject. Here's what I can recommend, given five years' worth of hindsight studying program analysis:
Final bit of advice: you'll notice that I heavily stuck to textbooks and Ph.D. theses in the above list. I find that jumping straight into the research literature without a foundational grounding is perhaps the most ill-advised mistake one can make intellectually. To whatever extent that what you're interested in is systematized -- that is, covered in a textbook or thesis already, you should read it before digging into the research literature. Otherwise, you'll be the proverbial blind man with the elephant, groping around in the dark, getting bits and pieces of the picture without understanding how it all forms a cohesive whole. I made that mistake and it cost me a lot of time; don't do the same.
Mathematics: Its Content, Methods and Meaning by A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrentev.
Personally read only the first chapter, but the book is praised by lots of people. I bet Mr. Nikolaevich has read it.
You can find it on Amazon https://www.amazon.com/Mathematics-Content-Methods-Meaning-Volumes/dp/0486409163
For real analysis I really enjoyed Understanding Analysis for how clear the material was presented for a first course. For abstract algebra I found A book of abstract algebra to be very concise and easy to read for a first course. Those two textbooks were a lifesaver for me since I had a hard time with those two courses using the notes and textbook for the class. We were taught out of rudin and dummit and foote as mainly a reference book and had to rely on notes primarily but those two texts were incredibly helpful to understand the material.
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If any undergrads are struggling with those two courses I would highly recommend you check out those two textbooks. They are by far the easiest introduction to those two fields I have found. I also like that you can find solutions to all the exercises so it makes them very valuable for self study also. Both books also have a reasonable amount of excises so that you can in theory do nearly every problem in the book which is also nice compared to standard texts with way too many exercises to realistically go through.
You need to develop an "intuition" for proofs, in a crude sense.
I would suggest these books to do that:
Proof, Logic, and Conjecture: The Mathematician's Toolbox by Robert Wolf. This was the book I used for my own proof class at Stony Brook - (edit: when I was a student.) This book goes down to the logic level. It is superbly well written and was of an immense use to me. It's one of those books I've actually re-read entirely, in a very Wax-on Wax-off Mr. Miyagi type way.
How to Read and Do Proofs by Daniel Slow. I bought this little book for my own self study. Slow wrote a really excellent, really concise, "this is how you do a proof" book. Teaching you when to look to try a certain technique of proof before another. This little book is a quick way to answer your TL:DR.
How to Solve it by G. Polya is a classic text in mathematical thinking. Another one I bought for personal collection.
Mathematics and Plausible Reasoning, Vol 1 and Mathematics and Plausible Reasoning, Vol 2 also by G. Polya, and equally classic, are two other books on my shelf of "proof and mathematical thinking."
It just comes from the way we define sums of infinite sums, aka series. .999... is just shorthand for (.9+.09+.09+.009...), which is an infinite sum. We define the sum of a series to be equal to the limit of the partial sums. The limit is rigorously defined, and you can read the definition on wikipedia if you google "epsilon delta". The limit of an infinite sum, if it exists, is unique. For this infinite sum, that limit is exactly 1. By the way we define infinite sums, .999... is therefore exactly equal to 1.
It's not so bad when you remember that all real numbers have a representation as a non-terminating decimal. 0.5 can be written as 0.4999... and 1/3 can be written as 0.333... and pi can be written as 3.14159... for example.
And lastly, if .999... and 1 are different real numbers, then there must exist a number between them. This is because of an axiom we have called trichotomy: for any two real numbers a and b, exactly one of the following is true: a<b, a=b, a>b. If a=/=b, then there exists a real number between them, because the real numbers have a property called "dense". It is easy to prove that here is no such number between .999... and 1, real or otherwise. Therefore .999... is exactly equal to 1.
e: The sum (.9+.09+.009...) is bigger than every real number less than 1. You can check if you want. The smallest number that is greater than every real number less than 1 is 1 itself. We get this from an axiom called the "least upper bound property". Therefore .999... is at least 1. Using our rigorous definition of a limit, we find that it is exactly 1.
e2: Apostol's Calculus vol 1 is a fantastic place to start learning about rigorous math shit. Chapter one starts you out with axioms for real numbers, and about half way through chapter 1 you prove the whole thing about repeating decimals corresponding to rational numbers. It is slow and easy to follow. Other people recommend Spivak but I haven't seen it so idk.
maybe you mean What is Mathematics? by Courant and Robbins.
Stay away from Numberphile. Numberphile oversimplifies mathematical concepts to the point where they will give you misconceptions about common mathematical notions that will greatly impact your learning later on. I'm noticing this happening a lot with the "1+2+... = -1/12" video because it doesn't explain that they aren't using the standard partial sum definition of series convergence.
Not sure how "mathematical" it is, but Secrets of Mental Math is a great, useful book that will help you do really fast calculations in your head.
He's my major advisor, and he loves occasionally showing off (who wouldn't?). I find it very entertaining. As far as I can tell, it's just a lot of practice plus some pattern recognition. For multiplying large numbers he just uses the distributive property combined with a certain method of remembering numbers in his head he uses.
I also read his book Secrets of Mental Math back in high school. He outlines some of the techniques there although its more basic.
It should also be noted that simpler subjects and introductory texts tend to be common knowledge to the point that citation is often not needed. You don't need to cite that water is wet, not even on Wikipedia.
Journals and modern texts about modern subjects tend to be very well cited, because they're building heavily on other sources of information.
When you don't do this, you have to have an enormous amount of backgrounding. For instance, check out this algebra text. It's 944 pages because it doesn't tend to cite much of exposition and instead states it all directly. It includes an enormous amount of information -- it's meant to be used as fundamental education material. It's not just high level conclusions that could fit in 20-50 pages.
So the amount of citation depends a great deal on the purpose of the text and how close it is to common knowledge. However, Anita's criticism is clearly not common knowledge because nobody but her sees it the way she does. Therefore, she should be explaining how she comes to her conclusions, and citing information. She should also be citing the direct quotes she uses, because it's plagiarism otherwise (and we have huge volumes of evidence that she outright plagiarizes a great deal). Plagiarism in academia is something that ends your career.
The problem you are having is that math education is shitty.
> What I want is to have a concrete understanding [...]
If you want to actually understand anything you learn in class, you'll have to seek it out yourself. Actual mathematics isn't taught until you get to college, and even then, only to students majoring in the subject.
"Why the fuck calculus works" typically goes under the name "analysis." You can look up a popular textbook, Baby Rudin, although I've never used it. I had this cheap-o Dover book. You can't beat it for $12. There's also this nice video series from Harvey Mudd.
The general pattern you see in actual, real mathematics isn't method-problem-problem-problem-problem, but rather definition-theorem-proof. The definitions tell you what you're working with. The theorems tell you what is true. The proofs give a strong technical reason to believe it.
> I know that to grasp mathematical concepts, it is advisable to do lots of problems from your textbook.
For some reason, schools are notorious for drilling exercises until you're just about to bleed from the fucking skull. Once you understand how an exercise is done, don't waste your time with another exercise of the same type. If you can correctly take the derivative of three different polynomials, then you probably understand it.
Just a heads up, analysis is built on the foundations of set theory and the real numbers. What you work with in high school are an intuitive notion of what a real number is. However, to do proper mathematics with them, it's better to have a proper understanding of how they are defined. Any good book on analysis will start off by giving a full, rigorous definition of what a real number is. This is typically done either in terms of cauchy sequences (sequences that seem like they deserve to converge), in terms of dedekind cuts (splitting the rational numbers up into two sets), or axiomatically (giving you a characterization involving least upper bounds of bounded sets). (No good mathematical book would ever talk about decimals. Decimals are a powerful tool, but pure mathematicians avoid them whenever possible).
Calculus and analysis can both be summed up shortly as "the cool things you can do with limits". Limits are the primary way we work with infinities in analysis. Their technical definition is often confusing the first time you see it, but the idea behind them is straightforward. Imagining a world where you can't measure things exactly, you have to rely on approximations. You want accuracy, though, and so you only have so much room for error. Suppose you want to make a measurement with a very small error. (We use ε for denoting the maximum allowable error). If the equipment you're using to make the measurement is calibrated well enough, then you can do this just fine. (The calibration of your machine is denoted δ, and so, these definitions commonly go by the name of "ε-δ definitions").
I doubt that you're going to find everything you're looking for in a single book.
I suggest that you start with Axler's Linear Algegra Done Right. Despite the pretentious name it does a good job of introducing linear algebra in a general form.
But Axler doesn't do any applications and almost completely ignores determinants (which I like, but it sounds like you want more of that) so I would supplement with Strang's MIT Lectures and any one of his books.
My main hobby is reading textbooks, so I decided to go beyond the scope of the question posed. I took a look at what I have on my shelves in order to recommend particularly good or standard books that I think could characterize large portions of an undergraduate degree and perhaps the beginnings of a graduate degree in the main fields that interest me, plus some personal favorites.
Neuroscience: Theoretical Neuroscience is a good book for the field of that name, though it does require background knowledge in neuroscience (for which, as others mentioned, Kandel's text is excellent, not to mention that it alone can cover the majority of an undergraduate degree in neuroscience if corequisite classes such as biology and chemistry are momentarily ignored) and in differential equations. Neurobiology of Learning and Memory and Cognitive Neuroscience and Neuropsychology were used in my classes on cognition and learning/memory and I enjoyed both; though they tend to choose breadth over depth, all references are research papers and thus one can easily choose to go more in depth in any relevant topics by consulting these books' bibliographies.
General chemistry, organic chemistry/synthesis: I liked Linus Pauling's General Chemistry more than whatever my school gave us for general chemistry. I liked this undergraduate organic chemistry book, though I should say that I have little exposure to other organic chemistry books, and I found Protective Groups in Organic Synthesis to be very informative and useful. Unfortunately, I didn't have time to take instrumental/analytical/inorganic/physical chemistry and so have no idea what to recommend there.
Biochemistry: Lehninger is the standard text, though it's rather expensive. I have limited exposure here.
Mathematics: When I was younger (i.e. before having learned calculus), I found the four-volume The World of Mathematics great for introducing me to a lot of new concepts and branches of mathematics and for inspiring interest; I would strongly recommend this collection to anyone interested in mathematics and especially to people considering choosing to major in math as an undergrad. I found the trio of Spivak's Calculus (which Amazon says is now unfortunately out of print), Stewart's Calculus (standard text), and Kline's Calculus: An Intuitive and Physical Approach to be a good combination of rigor, practical application, and physical intuition, respectively, for calculus. My school used Marsden and Hoffman's Elementary Classical Analysis for introductory analysis (which is the field that develops and proves the calculus taught in high school), but I liked Rudin's Principles of Mathematical Analysis (nicknamed "Baby Rudin") better. I haven't worked my way though Munkres' Topology yet, but it's great so far and is often recommended as a standard beginning toplogy text. I haven't found books on differential equations or on linear algebra that I've really liked. I randomly came across Quine's Set Theory and its Logic, which I thought was an excellent introduction to set theory. Russell and Whitehead's Principia Mathematica is a very famous text, but I haven't gotten hold of a copy yet. Lang's Algebra is an excellent abstract algebra textbook, though it's rather sophisticated and I've gotten through only a small portion of it as I don't plan on getting a PhD in that subject.
Computer Science: For artificial intelligence and related areas, Russell and Norvig's Artificial Intelligence: A Modern Approach's text is a standard and good text, and I also liked Introduction to Information Retrieval (which is available online by chapter and entirely). For processor design, I found Computer Organization and Design to be a good introduction. I don't have any recommendations for specific programming languages as I find self-teaching to be most important there, nor do I know of any data structures books that I found to be memorable (not that I've really looked, given the wealth of information online). Knuth's The Art of Computer Programming is considered to be a gold standard text for algorithms, but I haven't secured a copy yet.
Physics: For basic undergraduate physics (mechanics, e&m, and a smattering of other subjects), I liked Fundamentals of Physics. I liked Rindler's Essential Relativity and Messiah's Quantum Mechanics much better than whatever books my school used. I appreciated the exposition and style of Rindler's text. I understand that some of the later chapters of Messiah's text are now obsolete, but the rest of the book is good enough for you to not need to reference many other books. I have little exposure to books on other areas of physics and am sure that there are many others in this subreddit that can give excellent recommendations.
Other: I liked Early Theories of the Universe to be good light historical reading. I also think that everyone should read Kuhn's The Structure of Scientific Revolutions.
I enjoyed this one by the same author: Fermat's Enigma. Maybe 1/3 to 1/2 of the book tells the story of Andrew Wiles trying to prove Fermat's Last Theorem (and the significance of it), and mixed in throughout is information about all sorts of mathematical history.
This is not a highly advanced or hard-to-read book. Anybody with an interest in mathematics could enjoy it. If you're looking for some higher-level mathematical knowledge, this is not the book to read. I haven't read The Code Book, so I don't know how similar it is.
EDIT: The first review starts with "After enjoying Singh's "The Code Book"..." The reviewer gave it 5 stars.
Seems a waste not to link to this fantastic book about how he solved it.
> Are the deep mathematical answers to things usually very complex or insanely elegant and simple when you get down to it?
I would say that the deep mathematical answers to questions tend to be very complex and insanely elegant at the same time. The best questions that mathematicians ask tend to be the ones that are very hard but still within reach (in terms of solving them). The solutions to these types of questions often have beautiful answers, but they will generally require lots of theory, technical detail, and/or very clever solutions all of which can be very complex. If they didn't require something tricky, technical, or the development of new theory, they wouldn't be difficult to solve and would be uninteresting.
For any experts that happen to stumble by, my favorite example of this is the classification of semi-stable vector bundles on the complex projective plane by LePotier and Drezet. At the top of page 7 of this paper you'll see a picture representing the fractal structure that arises in this classification. Of course, this required a lot of hard math and complex technical detail to come up with this, but the answer is beautiful and elegant.
> How hard would it be for a non mathematician to go to a pro? Is there just some brain bending that cannot be handled by some? How hard are the concepts to grasp?
I would say that it's difficult to become a professional mathematician. I don't think it has anything to do with an individual's ability to think about it. The concepts are difficult, certainly, but given time and resources (someone to talk to, good books, etc) you can certainly overcome that issue. The majority of the difficulty is that there is so much math! If you're an average person, you've probably taken at most Calculus. The average mathematics PhD (i.e., someone who is just getting their mathematical career going) has probably taken two years of undergraduate mathematics courses, another two years of graduate mathematics courses, and two to three years of research level study beyond calculus to begin to be able tackle the current theory and solve the problems we are interested in today. That's a lot of knowledge to acquire, and it takes a very long time. That doesn't mean you can't start solving problems earlier, however. If you're interested in this type of thing, you might want to consider picking up this book and see if you like it.
Nagel's book 'Gödel's Proof' is a good, intelligible summary of Gödel. I suggest reading that, even if you suck at math.
Besides the Napkin Project I mentioned, which is a genuinely good resource? I got a coordinate-free treatment of linear algebra in my school's prelim. abstract algebra course. We used Dummit and Foote, which must be prescribed by law somewhere because I haven't yet seen a single department not use it. However, in reviewing abstract algebra I instead used Hungerford, which I definitely prefer for its brevity. But really, you can pick any graduate intro algebra text and it should teach this stuff.
> Mathematical Logic
It's not exactly Math Logic, just a bunch of techniques mathematicians use. Math Logic is an actual area of study. Similarly, actual Set Theory and Proof Theory are different from the small set of techniques that most mathematicians use.
Also, looks like you have chosen mostly old, but very popular books. While studying out of these books, keep looking for other books. Just because the book was once popular at a school, doesn't mean it is appropriate for your situation. Every year there are new (and quite frankly) pedagogically better books published. Look through them.
Here's how you find newer books. Go to Amazon. In the search field, choose "Books" and enter whatever term that interests you. Say, "mathematical proofs". Amazon will come up with a bunch of books. First, sort by relevance. That will give you an idea of what's currently popular. Check every single one of them. You'll find hidden jewels no one talks about. Then sort by publication date. That way you'll find newer books - some that haven't even been published yet. If you change the search term even slightly Amazon will come up with completely different batch of books. Also, search for books on Springer, Cambridge Press, MIT Press, MAA and the like. They usually house really cool new titles. Here are a couple of upcoming titles that might be of interest to you: An Illustrative Introduction to Modern Analysis by Katzourakis/Varvarouka, Understanding Topology by Shaun Ault. I bet these books will be far more pedagogically sound as compared to the dry-ass, boring compendium of facts like the books by Rudin.
If you want to learn how to do routine proofs, there are about one million titles out there. Also, note books titled Discrete Math are the best for learning how to do proofs. You get to learn techniques that are not covered in, say, How to Prove It by Velleman. My favorites are the books by Susanna Epp, Edward Scheinerman and Ralph Grimaldi. Also, note a lot of intro to proofs books cover much more than the bare minimum of How to Prove It by Velleman. For example, Math Proofs by Chartrand et al has sections about doing Analysis, Group Theory, Topology, Number Theory proofs. A lot of proof books do not cover proofs from Analysis, so lately a glut of new books that cover that area hit the market. For example, Intro to Proof Through Real Analysis by Madden/Aubrey, Analysis Lifesaver by Grinberg(Some of the reviewers are complaining that this book doesn't have enough material which is ridiculous because this book tackles some ugly topological stuff like compactness in the most general way head-on as opposed to most into Real Analysis books that simply shy away from it), Writing Proofs in Analysis by Kane, How to Think About Analysis by Alcock etc.
Here is a list of extremely gentle titles: Discovering Group Theory by Barnard/Neil, A Friendly Introduction to Group Theory by Nash, Abstract Algebra: A Student-Friendly Approach by the Dos Reis, Elementary Number Theory by Koshy, Undergraduate Topology: A Working Textbook by McClusckey/McMaster, Linear Algebra: Step by Step by Singh (This one is every bit as good as Axler, just a bit less pretentious, contains more examples and much more accessible), Analysis: With an Introduction to Proof by Lay, Vector Calculus, Linear Algebra, and Differential Forms by Hubbard & Hubbard, etc
This only scratches the surface of what's out there. For example, there are books dedicated to doing proofs in Computer Science(for example, Fundamental Proof Methods in Computer Science by Arkoudas/Musser, Practical Analysis of Algorithms by Vrajitorou/Knight, Probability and Computing by Mizenmacher/Upfal), Category Theory etc. The point is to keep looking. There's always something better just around the corner. You don't have to confine yourself to books someone(some people) declared the "it" book at some point in time.
Last, but not least, if you are poor, peruse Libgen.
You could read Timothy Gowers' welcome to the math students at Oxford, which is filled with great advice and helpful links at the bottom.
You could read this collection of links on efficient study habits.
You could read this thread about what it takes to succeed at MIT (which really should apply everywhere). Tons of great discussion in the lower comments.
You could read How to Solve It and/or How to Prove It.
If you can work your way through these two books over the summer, you'll be better prepared than 90% of the incoming math majors (conservatively). They'll make your foundation rock solid.
I'll be that guy. There are two types of Calculus: the Micky Mouse calculus and Real Analysis. If you go to Khan Academy you're gonna study the first version. It's by far the most popular one and has nothing to do with higher math.
The foundations of higher math are Linear Algebra(again, different from what's on Khan Academy), Abstract Algebra, Real Analysis etc.
You could, probably, skip all the micky mouse classes and start immediately with rigorous(proof-based) Linear Algebra.
But it's probably best to get a good foundation before embarking on Real Analysis and the like:
Discrete Mathematics with Applications by Susanna Epp
How to Prove It: A Structured Approach Daniel Velleman
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Book of Proof by Richard Hammock
That way you get to skip all the plug-and-chug courses and start from the very beginning in a rigorous way.
I would guess that career prospects are a little worse than CS for undergrad degrees, but since my main concern is where a phd in math will take me, you should get a second opinion on that.
Something to keep in mind is that "higher" math (the kind most students start to see around junior level) is in many ways very different from the stuff before. I hated calculus and doing calculations in general, and was pursuing a math minor because I thought it might help with job prospects, but when I got to the more abstract stuff, I loved it. It's easily possible that you'll enjoy both, I'm just pointing out that enjoying one doesn't necessarily imply enjoying the other. It's also worth noting that making the transition is not easy for most of us, and that if you struggle a lot when you first have to focus a lot of time on proving things, it shouldn't be taken as a signal to give up if you enjoy the material.
This wouldn't be necessary, but if you like, here are some books on abstract math topics that are aimed towards beginners you could look into to get a basic idea of what more abstract math is like:
Different mathematicians gravitate towards different subjects, so it's not easy to predict which you would enjoy more. I'm recommending these five because they were personally helpful to me a few years ago and I've read them in full, not because I don't think anyone can suggest better. And of course, you could just jump right into coursework like how most of us start. Best of luck!
(edit: can't count and thought five was four)
If you want to learn how calculus actually works (rather than just how to do computations), I highly recommend working through Spivak's Calculus. Spivak builds up calculus from the foundations with mathematical rigor and actual proofs, explaining (and proving) what's really going on. (That includes properly developing sequences and limits.) The exercises are also excellent; many of them require real thought and insight, instead of the usual "repeat the steps you were just told fifty times" exercises that fill up mainstream calculus textbooks.
Also, from a more sophisticated perspective, dx is a differential form.
I had to deal with the no internet thing for some time.
Find some place with free wi-fi(you are using phone?).
Download ebook/pdf reader, FBreader + PDF plugin is good (Assuming that you are using Android phone).
Install Firefox and this add-on Save Page WE, it also work for phones (tested with Android).
Then you can save pages from some of these web sites or Wikipedia:
Poor man's library:
http://gen.lib.rus.ec/
http://b-ok.org/
https://archive.org/
Some books:
[Harold Jacobs - Elementary Algebra] (https://www.amazon.com/Elementary-Algebra-Harold-R-Jacobs/dp/0716710471) (that's the most friendly and still not bullshit introduction I was able to find)
[Basic Mathematics by Serge Lang] (https://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/0387967877) - Make sure that you understand the first 100-150 pages from the first book if you want to start this one.
Precalculus by David Cohen - Lang's Basic Mathematics is harder (and much more interesting) than this...
Use the three web sites written above to download them. PDF for better performance (and some times quality) on cheap phone, epub/djvu for the smaller size.
I think that free wi-fi will be easy to find. Reading books wouldn't be like watching videos but you will actually learn more. Compass, straightedge and protractor would be really useful.
Traditionally, a mathematical proof has one and only one job: convince other people that your proof is correct. (In this day and age, there is such a thing as a computer proof, but if you don't understand traditional proofs, you can't handle computer proofs either.)
Notice what I just said: "convince other people that your proof is correct." A proof is, in some sense, always an interactive undertaking, even if the interaction takes place across gulfs of space and time.
Because interaction is so central to the notion of a proof, it is rare for students to successfully self-study how to write proofs. That seems like what you're asking. Don't get me wrong. Self-study helps. But it is not the only thing you need. You need, at some point, to go through the process of presenting your proofs to others, answering questions about your proof, adjusting your proof to take into account new feedback, and using this experience to anticipate likely issues in future proofs.
What you're proposing to do, in most cases, is the wrong strategy. You need more interactive experience, not less. You should be beating down the doors of your professor or TA in your class during their office hours, asking for feedback on your proofs. (This implies that you should be preparing your proofs in advance for them to read before going to their office hours.) If your school has a tutorial center, that's a wonderful resource as well. A math tutor who knows math proofs is a viable source of help, but if you don't know how to do proofs, it's hard for you to judge whether or not your tutor knows how to do proofs.
If you do self-study anything, you should not be self-studying calculus, linear algebra, real analysis, or abstract algebra. You should be self-studying how to do proofs. Some people here say that How to Prove It is a useful resource. My own position is that while self-studying can be helpful, it needs to be balanced with some amount of external interactive feedback in order to really stick.
The heart of conceptual mathematics (i.e., mathematics that isn't just computation and carrying out algorithms) is mathematical proof. I suggest you work through the book How to Prove It. This will give you the tools to self work through other textbooks (not that it will suddenly be easy).
It's available free online, but I've def got a hard cover copy on my bookshelf. I can't really deal with digital versions of things, I need physical books.
You are in a very special position right now where many interesing fields of mathematics are suddenly accessible to you. There are many directions you could head. If your experience is limited to calculus, some of these may look very strange indeed, and perhaps that is enticing. That was certainly the case for me.
Here are a few subject areas in which you may be interested. I'll link you to Dover books on the topics, which are always cheap and generally good.
Basically, don't limit yourself to the track you see before you. Explore and enjoy.
I take it you want something small enough to fit inside a hollowed-out bible or romance novel, so you can hide your secrets from nosy neighbors?
More seriously, this book is great https://amzn.com/0201558025 but might not be quite what you’re looking for at an introductory level.
I’ve seen recommendations of https://amzn.com/0495391328 (I haven’t looked at this book myself.)
I doubt this can be answered for a five year old, I read an excellent book on the subject and still don't really get it. I will try to recount the jist of what I remember.
Fermat left a small note scribbled in the margins of a book: a^n + b^n = c^n has no solution for positive integers greater then 2.
What fascinated everyone is that if n=2 you have the Pythagorean theorem which every knows, loves, and uses all the time. But to say that there is no solution for a^3 + b^3 = c^3 well that seems a bit crazy. You can sit down and try to plug in the first few values yourself, and low and behold you cant find any solution. Fermat had claimed that he had a proof that showed that this was true from 3 > infinity. (personally I don't think he had an actual proof, more of a very strong gut instinct and if anyone in his lifetime proved him wrong he would have laughed at them and said that he trolled them hard.)
That's the background, now to your questions, what are mathematical proofs? They show that a given formula is true in all cases, any two positive integers plugged into the Pythagorean theorem will result in a real solution for C.
Why is it hard to make them? because you have to show that the theorem works to infinity, you can plug in billions of numbers into a theorem, and prove nothing because the billionth + 1 may not be true
What was so special about Fermat's? Not much, except that it drove people insane with its simplicity, but it took hundreds of years to prove that a^3 + b^3 = c^3 had no real solutions and hundreds of years more for Andrew Wiles and Richard Taylor to discover the general proof.
From wikipedia as to whether Fermat actually had a general formula:
>Taylor and Wiles's proof relies on mathematical techniques developed in the twentieth century, which would be alien to mathematicians who had worked on Fermat's Last Theorem even a century earlier. Fermat's alleged "marvellous proof", by comparison, would have had to be elementary, given mathematical knowledge of the time, and so could not have been the same as Wiles' proof. Most mathematicians and science historians doubt that Fermat had a valid proof of his theorem for all exponents n.
and finally my attempt at EILI5:
You know how you ask me a million questions every day, and some times I don't have the answer. Now imagine going to your teacher and asking them, and they don't know, and ten years from now you ask another teacher and they still don't know, you grow up and go to college and ask your professors and they don't know either. Your question sounds like it should be easy to answer, why doesn't anyone know the answer, then you try to answer it for yourself, and you can't figure it out. You try for thirty years to answer the question, and talk to other people who have tried to answer the question for the last 400 years and still no answer. Some people might give up, but the fact that you could be the first person in the world to know something makes you work even harder to find the answer to this simple question.
Found it.
https://www.amazon.com/Introduction-Error-Analysis-Uncertainties-Measurements/dp/093570275X
I think the advice given in the rest of the thread is pretty good, though some of it a little naive. The suggestion that differential equations or applied math somehow should not be of interest is silly. A lot of it builds the motivation for some of the abstract stuff which is pretty cool, and a lot of it has very pure problems associated with it. In addition I think after (or rather alongside) your initial calculus education is a good time to look at some other things before moving onto more difficult topics like abstract algebra, topology, analysis etc.
The first course I took in undergrad was a course that introduced logic, writing proofs, as well as basic number theory. The latter was surprisingly useful as it built modular arithmetic which gave us a lot of groups and rings to play with in subsequent algebra courses. Unfortunately the textbook was god awful. I've heard good things about the following two sources and together they seem to cover the content:
How to prove it
Number theory
After this I would take a look at linear algebra. This a field with a large amount of uses in both pure and applied math. It is useful as it will get you used to doing algebraic proofs, it takes a look at some common themes in algebra, matrices (one of the objects studied) are also used thoroughly in physics and applied mathematics and the knowledge is useful for numerical approximations of ordinary and partial differential equations. The book I used Linear Algebra by Friedberg, Insel and Spence, but I've heard there are better.
At this point I think it would be good to move onto Abstract Algebra, Analysis and Topology. I think Farmerje gave a good list.
There's many more topics that you could possibly cover, ODEs and PDEs are very applicable and have a rich theory associated with them, Complex Analysis is a beautiful subject, but I think there's plenty to keep you busy for the time being.
Learn math at a more "fundamental" level, and that will test if you love it. For me, I didn't love math until I took a class on proofs and real analysis. One of the books we used was "How to Prove it", and to this day it's my favorite textbook ever. How do we know anything in mathematics? Which rules do we follow and how do we know they are true? This starts from basic logic and truth tables, and works its way up to some really complicated stuff. It's not as fancy as complex integrals and PDE's, but I would say it's a more fundamental form of mathematics and the basis for all other subjects in the field.
You should start with some gentler introduction to real analysis (e.g. the "baby" Rudin )that does the basic topology of the real line and Riemann integration rigorously.
I'll share my reading list for the next 12 months as it's how I plan to become a better learner:
&nbsp;
Learning
Improving Maths
Improving Reading Speed
Algorithmic thinking
Understanding the Mind
Productivity
It's an ambitious reading list for the next 12 months as there is some heavy reading in there but hopefully you can get one or two useful suggestions from it!
One of the most fun things I did when I was first learning about proofs was proving the basic facts about algebra from axioms. Where I first read about these ideas was the first chapter of Spivak's Calculus. This would be a very high level book for an 18 year old, but if you decide to look at it, don't be afraid to take your time a little.
Another option is just picking up an introduction to proof, like Velleman's How to Prove It. This wil lteach you the basics for proving anything, really, and is a great start if you want to do more math.
If you want a free alternative to that last one, you can look at The Book of Proof by Richard Hammack. It's well-written although I think it's shorter than How to Prove It.
I double majored in math and CS as an undergrad and I enjoyed math more than CS. I'm a graduate student right now planning on doing research in a mathy area of CS. Everything I write below comes from that perspective.
But if you're interested in really digging in and understanding some math at an advanced undergraduate level (analysis, abstract algebra, topology, etc.) then I don't think there is any substitute for books.
Ordinary Differential Equations from the Dover Books on Mathematics series. I Just took my final for Diff Eq a few days ago and the book was miles better than the one my school suggested and is the best written math textbook I have encountered during my math minor. My Diff Eq course only covered about the first 40% of the book so there's still a TON of info that you can learn or reference later. It is currently $14 USD on amazon and my copy is almost 3" thick so it really is a great deal. A lot of the reviewers are engineering and science students that said the book helped them learn the subject and pass their classes no problem. Highly Highly recommend. ISBN-10: 9780486649405
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https://www.amazon.com/gp/product/0486649407/ref=ppx_yo_dt_b_asin_title_o08_s00?ie=UTF8&psc=1
Pick up mathematics. Now if you have never done math past the high school and are an "average person" you probably cringed.
Math (an "actual kind") is nothing like the kind of shit you've seen back in grade school. To break into this incredible world all you need is to know math at the level of, say, 6th grade.
Intro to Math:
These books only serve as samplers because they don't even begin to scratch the surface of math. After you familiarized yourself with the basics of writing proofs you can get started with intro to the largest subsets of math like:
Intro to Abstract Algebra:
There are tons more books on abstract/modern algebra. Just search them on Amazon. Some of the famous, but less accessible ones are
Intro to Real Analysis:
Again, there are tons of more famous and less accessible books on this subject. There are books by Rudin, Royden, Kolmogorov etc.
Ideally, after this you would follow it up with a nice course on rigorous multivariable calculus. Easiest and most approachable and totally doable one at this point is
At this point it's clear there are tons of more famous and less accessible books on this subject :) I won't list them because if you are at this point of math development you can definitely find them yourself :)
From here you can graduate to studying category theory, differential geometry, algebraic geometry, more advanced texts on combinatorics, graph theory, number theory, complex analysis, probability, topology, algorithms, functional analysis etc
Most listed books and more can be found on libgen if you can't afford to buy them. If you are stuck on homework, you'll find help on [MathStackexchange] (https://math.stackexchange.com/questions).
Good luck.
Of course efforts like this won't fly because there will be people who sincerely want to can them because it's "computerized racial profiling," completely missing the point that, if race does correlate with criminal behavior, you will see that conclusion from an unbiased system. What an unbiased system will also do is not overweight the extent to which race is a factor in the analysis.
Of course, the legitimate concern some have is about the construction of prior probabilities for these kinds of systems, and there seems to be a great deal of skepticism about the possibility of unbiased priors. But over the last decade or two, the means of constructing unbiased priors have become rather well understood, and form the central subject matter of Part II of E.T. Jaynes' Probability Theory: The Logic of Science, which I highly recommend.
For the very basics (and more), I can highly recommend you Professor Leonard on YouTube.
>What books would you recommend?
How about doing your own research?
Google.com -> book site:reddit.com/r/learnmath
Anyways, take a look at Basic Mathematics by Serge Lang. This is what I'm learning with right now, it's really great.
Mathematics, a learning map
Edit:
Ehm, or take a look at your own thread from a year ago.
https://www.reddit.com/r/learnmath/comments/46xdpp/learning_math_from_scratch_all_by_myself/
No matter what his interests may be, this wonderful survey will cover it, Mathematics: Its Contents, Methods, and Meaning. It was written by a team of prominent Russian mathemations, and became a classic. It's now a single Dover edition, but if possible, find it used in the original MIT 3-volume hardcover edition -- it demands that kind of respect!
Or How to Prove It by Velleman.
There would have been a time that I would have suggested getting a curriculum
text book and going through that, but if you're doing this for independent work
I wouldn't really suggest that as the odds are you're not going to be using a
very good source.
Going on the typical
Arithmetic > Algebra > Calculus
****
Arithmetic
Arithmetic refresher. Lots of stuff in here - not easy.
I think you'd be set after this really. It's a pretty terse text in general.
*****
Algebra
Algebra by Chrystal Part I
Algebra by Chrystal Part II
You can get both of these algebra texts online easily and freely from the search
chrystal algebra part I filetype:pdfchrystal algebra part II filetype:pdfI think that you could get the first (arithmetic) text as well, personally I
prefer having actual books for working. They're also valuable for future
reference. This
filetype:pdfsearch should be remembered and used liberallyfor finding things such as worksheets etc (eg
trigonometry worksheet<br /> filetype:pdffor a search...).Algebra by Gelfland
No where near as comprehensive as chrystals algebra, but interesting and well
written questions (search for 'correspondence series' by Gelfand).
Calculus
Calculus made easy - Thompson
This text is really good imo, there's little rigor in it but for getting a
handle on things and bashing through a few practical problems it's pretty
decent. It's all single variable. If you've done the algebra and stuff before
this then this book would be easy.
Pauls Online Notes (Calculus)
These are just a solid set of Calculus notes, there're lots of examples to work
through which is good. These go through calc I, II, III... So a bit further than
you've asked (I'm not sure why you state up to calc II but ok).
Spivak - Calculus
If you've gone through Chrystals algebra then you'll be used to a formal
approach. This text is only single variable calculus (so that might be calc I
and II in most places I think, ? ) but it's extremely well written and often
touted as one of the best Calculus books written. It's very pure, where as
something like Stewart has a more applied emphasis.
**
Geometry
I've got given any geometry sources, I'm not too sure of the best source for
this or (to be honest) if you really need it for the above. If someone has
good geometry then they're certainly better off, many proofs are given
gemetrically as well and having an intuition for these things is only going to
be good. But I think you can get through without a formal course on it.... I'm
not confident suggesting things on it though, so I'll leave it to others. Just
thought I'd mention it.
****
There are essentially "two types" of math: that for mathematicians and everyone else. When you see the sequence Calculus(1, 2, 3) -> Linear Algebra -> DiffEq (in that order) thrown around, you can be sure they are talking about non-rigorous, non-proof based kind that's good for nothing, imo of course. Calculus in this sequence is Analysis with all its important bits chopped off, so that everyone not into math can get that outta way quick and concentrate on where their passion lies. The same goes for Linear Algebra. LA in the sequence above is absolutely butchered so that non-math majors can pass and move on. Besides, you don't take LA or Calculus or other math subjects just once as a math major and move on: you take a rigorous/proof-based intro as an undergrad, then more advanced kind as a grad student etc.
To illustrate my point:
Linear Algebra:
Linear Algebra Through Geometry by Banchoff and Wermer
3. Here's more rigorous/abstract Linear Algebra for undergrads:
Linear Algebra Done Right by Axler
4. Here's more advanced grad level Linear Algebra:
Advanced Linear Algebra by Steven Roman
-----------------------------------------------------------
Calculus:
Calulus by Spivak
3. Full-blown undergrad level Analysis(proof-based):
Analysis by Rudin
4. More advanced Calculus for advance undergrads and grad students:
Advanced Calculus by Sternberg and Loomis
The same holds true for just about any subject in math. Btw, I am not saying you should study these books. The point and truth is you can start learning math right now, right this moment instead of reading lame and useless books designed to extract money out of students. Besides, there are so many more math subjects that are so much more interesting than the tired old Calculus: combinatorics, number theory, probability etc. Each of those have intros you can get started with right this moment.
Here's how you start studying real math NOW:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers. Essentially, this book is about the language that you need to be able to understand mathematicians, read and write proofs. It's not terribly comprehensive, but the amount of info it packs beats the usual first two years of math undergrad 1000x over. Books like this should be taught in high school. For alternatives, look into
Discrete Math by Susanna Epp
How To prove It by Velleman
Intro To Category Theory by Lawvere and Schnauel
There are TONS great, quality books out there, you just need to get yourself a liitle familiar with what real math looks like, so that you can explore further on your own instead of reading garbage and never getting even one step closer to mathematics.
If you want to consolidate your knowledge you get from books like those of Rodgers and Velleman and take it many, many steps further:
Basic Language of Math by Schaffer. It's a much more advanced book than those listed above, but contains all the basic tools of math you'll need.
I'd like to say soooooooooo much more, but I am sue you're bored by now, so I'll stop here.
Good Luck, buddyroo.
I used Principles of Mathematical Analysis by Walter Rudin. It's very thorough, and covers all the topics you mentioned.
$9.36 and free shipping.
Honestly. You'll be improving yourself while being able to amaze others at your "magic".
Computer scientist here... I'm not a "real" mathematician but I do have a good bit of education and practical experience with some specific fields of like probability, information theory, statistics, logic, combinatorics, and set theory. The vast majority of mathematics, though, I'm only interested in as a hobby. I've never gone much beyond calculus in the standard track of math education, so I to enjoy reading "layman's terms" material about math. Here's some stuff I've enjoyed.
Fermat's Enigma This book covers the history of a famous problem that looks very simple, yet it took several hundred years to resolve. In so doing it gives layman's terms overviews of many mathematical concepts in a manner very similar to jfredett here. It's very readable, and for me at least, it also made the study of mathematics feel even more like an exciting search for beautiful, profound truth.
Logicomix: An Epic Search for Truth I've been told this book contains some inaccuracies, but I'm including it because I think it's such a cool idea. It's a graphic novelization (seriously, a graphic novel about a logician) of the life of Bertrand Russell, who was deeply involved in some of the last great ideas before Godel's Incompleteness Theorem came along and changed everything. This isn't as much about the math as it is about the people, but I still found it enjoyable when I read it a few years ago, and it helped spark my own interest in mathematics.
Lots of people also love Godel Escher Bach. I haven't read it yet so I can't really comment on it, but it seems to be a common element of everybody's favorite books about math.
http://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328 is one of my math books. The bookstore wants $350 for it.
At the moment, psychology is largely ad-hoc, and not a modicum of progress would've been made without the extensive utilization of statistical methods. To consider the human condition does not require us to simply extrapolate from our severely limited experiences, if not from the biases of limited datasets, datasets for which we can't even be certain of their various e.g. parameters etc..
For example, depending on the culture, the set of phenotypical traits with which increases the sexual attraction of an organism may be different - to state this is meaningless and ad-hoc, and we can only attempt to consider the validity of what was stated with statistical methods. Still, there comes along social scientists who would proclaim arbitrary sets of phenotypical features as being universal for all humans in all conditions simply because they were convinced by limited and biased datasets (e.g. making extreme generalizations based on the United States' demographic while ignoring the entire world etc.).
In fact, the author(s) of "Probability Theory: The Logic of Science" will let you know what they think of the shaky sciences of the 20th and 21st century, social science and econometrics included, the shaky sciences for which their only justifications are statistical methods.
_
With increasing mathematical depth and the increasing quality of applied mathematicians into such fields of science, we will begin to gradually see a significant improvement in the validity of said respective fields. Otherwise, currently, psychology, for example, holds no depth, but the field itself is very entertaining to me; doesn't stop me from enjoying Michael's "Mind Field" series.
For mathematicians, physics itself lacks rigour, let alone psychology.
_
Note, the founder of "psychoanalysis", Sigmund Freud, is essentially a pseudo-scientist. Like many social scientists, he made the major error of extreme extrapolation based on his very limited and personal life experiences, and that of extremely limited, biased datasets. Sigmund Freud "proclaimed" a lot of truths about the human condition, for example, Sigmund Fraud is the genius responsible for the notion of "Penis Envy".
In the same century, Einstein would change the face of physics forever after having published the four papers in his miracle year before producing the masterpiece of General Relativity. And, in that same century, incredible progress such that of Gödel's Incompleteness Theorem, Quantum Electrodynamics, the discovery of various biological reaction pathways (e.g. citric acid cycle etc.), and so on and so on would be produced while Sigmund Fraud can be proud of his Penis Envy hypothesis.
Is it really such a big step from du Sautoy's explanation to the formal proof? I don't think so, but maybe I'm biased. I bet there are books on elementary number theory that don't assume much of any background that you could understand. If you're interested in proofs in general, you might enjoy Velleman's How to Prove It.
I picked up a book a couple years ago called How to Prove It.
It has helped me develop a greater appreciation for logic and proofs. I wish I took this stuff more seriously when I started programming. A little bit of knowledge of boolean algebra can help tremendously.
Mathematics, its Content, Methods, and Meaning - an amazing survey of analytic geometry, algebra, ordinary and partial differential equations, the calculus of variations, functions of a complex variable, prime numbers, theories of probability and functions, linear and non-Euclidean geometry, topology, functional analysis, and more.
https://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192
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https://www.amazon.com/Mathematics-Form-Function-Saunders-MacLane/dp/1461293405/ref=sr_1_3?keywords=MacLane%2C+Saunders&qid=1555006726&s=books&sr=1-3 (unfortunately, very expensive)
Seconding /u/khanable_ -- most of statistical theory is built on matrix algebra, especially regression. Entry-level textbooks usually use simulations to explain concepts because it's really the only way to get around assuming your audience knows linear algebra.
My Ph.D. program uses Casella and Berger as the main text for all intro classes. It's incredibly thorough, beginning with probability and providing rigorous proofs throughout, but you would need to be comfortable with linear algebra and at least the basic principles of real analysis. That said, this is THE book that I refer to whenever I have a question about statistical theory-- it's always on my desk.
I really enjoyed Godel's Proof by Nagel + Newman. It's a layman's guide to Godel incompleteness theorem. It avoids some of the more finnicky details, while still giving the overall impression.
https://www.amazon.com/Gödels-Proof-Ernest-Nagel/dp/0814758371/
If you like that, it's edited by Hofstadter, who wrote Godel-Escher-Bach, a famous book about recurrence.
Finally, I would recommend Nonzero: The Logic of Human Destiny by Robert Wright. It's a life-changing book that dives into the relevance of game theory, evolutionary biology and information technology. (Warning that the first 80 pages are very dry.)
https://www.amazon.com/Nonzero-Logic-Destiny-Robert-Wright/dp/0679758941/
If you are getting your degree in math or computer science, you will probably have to take a course on "Discrete math" (or maybe an "introduction to proofs") in your first year or two (it should be by your 3rd semester). Unfortunately, this will probably be the first time you will take a course that is more about the why than the how. (On the bright side, almost everything after this will focus on why instead of how.) Depending on how linear algebra is taught at your university, and the order you take classes in, linear algebra may be also be such a class.
If your degree is anything else, you may have no formal requirement to learn the why.
For the math you are learning right now, analysis is the "why". I'm not sure of a good analysis book, but there are two calculus books which treat the subject more like a gentle introduction to analysis-- Apostol's and Spivak's. Your library might have a copy you can check out. If not, you can probably find pdfs (which are probably[?] legal) online.
As you wish to get into applied statistics (i.e. actually analyzing data), you'll need software. I'd strongly recommend learning and using R because it's completely free and incredibly powerful.
Here are some resources for learning statistics using R:
Then, these websites provide very valuable resources for doing statistics with R:
Hope that helps.
math 271 is easy if you can think logically
just pirate this book: https://www.amazon.ca/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328
and browse the first 10 chapters - thats whats taught at math 271. i had thi dinh or something, some asian dude. he's a hardass but one of my fav profs of all time
ill be honest though, none of this math is plug and chug like calc 1 or even calc 2. i thought 271 required more thinking than linear or calc 1 or calc 2. once you get into counting and probability and set theory and graphs/relations and shits, it gets pretty intense. the first half of the course is easy but a lot of people fail the final exam and then fail the course lmao.
eng319 is hard but u gotta take it bro.
TAKING BOTH??? are you ready to stay indoors/at school all day for 60 days? then you can do it. if u slack off ur gonna fail 271 or 319.
gljuck bro
Tenenbaum and Pollard's ODE book made the subject come quite easily when all my $150 textbook did was confuse me.
The pure mechanics component consists of multivariable differential calculus, a little bit of multivariable integral calculus, and a bit of linear algebra; plus substantial comfort what might be called "systems of equations differential calculus." The fastest way to cover this material is to work through the first five or so chapters of Kaplan's advanced calculus book or something similar. Do the exercises. Your basic Stewart Calculus doesn't adequately cover the systems-of-equations part and Kreyszig's Advanced Engineering Mathematics book is at the right technical level but has all the wrong emphasis and coverage for economists. Kaplan's book isn't ideal, but it's about as close as you're going to get. (This is a hole in the textbook market...)
The theoretical portion mainly consists of basic point-set topology and elementary real analysis. The fastest way to cover this material is to chop through the first eight chapters of Rudin's undergraduate book.
Yale has a lovely set of Math Camp notes that you should also work through side-by-side with Kaplan and Rudin.
To see economic applications, read those two books side-by-side with Simon and Blume's book.
The first chapter of Debreu's Theory of Value covers all the math you need to know and is super slick, but is also far too terse and technical to realistically serve as your only resource. Similarly you should peek at the mathematical appendices in MWG but they will likely not be sufficient on their own.
Hey! This comment ended up being a lot longer than I anticipated, oops.
My all-time favs of these kinds of books definitely has to be Prime Obsession and Unknown Quantity by John Derbyshire - Prime Obsession covers the history behind one of the most famous unsolved problems in all of math - the Riemann hypothesis, and does it while actually diving into some of the actual theory behind it. Unknown Quantity is quite similar to Prime Obsession, except it's a more general overview of the history of algebra. They're also filled with lots of interesting footnotes. (Ignore his other, more questionable political books.)
In a similar vein, Fermat's Enigma by Simon Singh also does this really well with Fermat's last theorem, an infamously hard problem that remained unsolved until 1995. The rest of his books are also excellent.
All of Ian Stewart's books are great too - my favs from him are Cabinet, Hoard, and Casebook which are each filled with lots of fun mathematical vignettes, stories, and problems, which you can pick or choose at your leisure.
When it comes to fiction, Edwin Abbott's Flatland is a classic parody of Victorian England and a visualization of what a 4th dimension would look like. (This one's in the public domain, too.) Strictly speaking, this doesn't have any equations in it, but you should definitely still read it for a good mental workout!
Lastly, the Math Girls series is a Japanese YA series all about interesting topics like Taylor series, recursive relations, Fermat's last theorem, and Godel's incompleteness theorems. (Yes, really!) Although the 3rd book actually has a pretty decent plot, they're not really that story or character driven. As an interesting and unique mathematical resource though, they're unmatched!
I'm sure there are lots of other great books I've missed, but as a high school student myself, I can say that these were the books that really introduced me to how crazy and interesting upper-level math could be, without getting too over my head. They're all highly recommended.
Good luck in your mathematical adventures, and have fun!
You're not really doing higher math right now as much as you're learning tricks to solve problems. Once you start proving stuff that'll be a big jump. Usually people start doing that around Real Analysis like your father said. Higher math classes almost entirely consist of proofs. It's a lot of fun once you get the hang of it, but if you've never done it much before it can be jarring to learn how. The goal is to develop mathematical maturity.
Start learning some geometry proofs or pick up a book called "Calculus" by Spivak if you want to start proving stuff now. The Spivak book will give you a massive head start if you read it before college. Differential equations will feel like a joke after this book. It's called calculus but it's really more like real analysis for beginners with a lot of the harder stuff cut out. If you can get through the first 8 chapters or so, which are the hardest ones, you'll understand a lot of mathematics much more deeply than you do now. I'd also look into a book called Linear Algebra done right. This one might be harder to jump into at first but it's overall easier than the other book.
Not sure what sort of thing you're trying to prove, but there are a few good books on techniques for proof that you'll end up using if you go into higher math. I like How to Prove It by Velleman. It's geared towards students finishing high school math who are planning to do math at the university level, so it might be the sort of thing you're looking for.
https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/
A good book on Gödel's proof is Gödel's Proof.
Talk about a lack of substance!
A book I read way back when that was excellent was Gödel’s Proof by Nagel and Newman. http://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371
If she's bright and interested enough you might want to consider getting her an entry level college calculus book such as Spivak's.
It won't pose a replacement to the technical approach of high school, but it will illuminate a lot.
I think this is a better approach than trying to tie connections between calculus and other areas of math, because calculus has an inherent beauty of its own which could be very compelling when taught with the right philosophical approach.
I think the most important part of being able to see beauty in mathematics is transitioning to texts which are based on proofs rather than application. A side effect of gaining the ability to read and write proofs is that you're forced to deeply understand the theory of the math you're learning, as well as actively using your intuition to solve problems, rather than dry route calculations found in most application based textbooks. Based on what you've written, I feel you may enjoy taking this path.
Along these lines, you could start of with Book of Proof (free) or How to Prove It. From there, I would recommend starting off with a lighter proof based text, like Calculus by Spivak, Linear Algebra Done Right by Axler, or Pinter's book as you mentioned. Doing any intro proofs book plus another book at the level I mentioned here would have you well prepared to read any standard book at the undergraduate level (Analysis, Algebra, Topology, etc).
If you're looking for other texts, I would suggest Spivak's Calculus and Calculus on Manifolds. At first the text may seem terse, and the exercises difficult, but it will give you a huge advantage for later (intermediate-advanced) undergraduate college math.
It may be a bit obtuse to recommend you start with these texts, so maybe your regular calculus texts, supplemented with linear algebra and differential equations, should be approached first. When you start taking analysis and beyond, though, these books are probably the best way to return to basics.
A relatively compact (excuse the pun) rundown of the basic definitions and theorems behind real analysis can be found in a book called "Baby Rudin"
https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X
But beware, this is definitely not ELIF. Math isn't really an ELIF type of thing, but I guess it depends on how deep you need to go to get where you're going.
I wish you luck!
There are a few options. Firstly, if you are more familiar using infinity in the context of Calculus, then you might want to look into Real Analysis. These subjects view infinity in the context of limits on the real line and this is probably the treatment you are probably most familiar with. For an introductory book on the subject, check out Baby Rudin (Warning: Proofs! But who doesn't like proofs, that's what math is!)
Secondly, you might want to look at Projective Geometry. This is essentially the type of geometry you get when you add a single point "at infinity". Many things benefit from a projective treatment, the most obvious being Complex Analysis, one of its main objects of study is the Riemann Sphere, which is just the Projective Complex Plane. This treatment is related to the treatment given in Real Analysis, but with a different flavor. I don't have any particular introductory book to recommend, but searching "Introductory Projective Geometry" in Amazon will give you some books, but I have no idea if they're good. Also, look in your university library. Again: Many Proofs!
The previous two treatments of infinity give a geometric treatment of the thing, it's nothing but a point that seems far away when we are looking at things locally, but globally it changes the geometry of an object (it turns the real line into a circle, or a closed line depending on what you're doing, and the complex plane into a sphere, it gets more complicated after that). But you could also look at infinity as a quantitative thing, look at how many things it takes to get an infinite number of things. This is the treatment of it in Set Theory. Here things get really wild, so wild Set Theory is mostly just the study of infinite sets. For example, there is more than one type of infinity. Intuitively we have countable infinity (like the integers) and we have uncountable infinity (like the reals), but there are even more than that. In fact, there are more types of infinities than any of the infinities can count! The collection of all infinities is "too big" to even be a set! For an introduction into this treatment I recommend Suppes and Halmos. Set Theory, when you actually study it, is a very abstract subject, so there will be more proofs here than in the previous ones and it may be over your head if you haven't taken any proof-based courses (I don't know your background, so I'm just assuming you've taken Calc 1-3, Diff Eq and maybe some kind of Matrix Algebra course), so patience will be a major virtue if you wish to tackle Set Theory. Maybe ask some professors for help!
You would probably like these two books:
Neither of those are "popular math" books; the authors are famous mathematicians, and they explore various fields of mathematics without requiring too much advanced knowledge.
I read a good portion of this book: http://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401/ref=sr_1_1?ie=UTF8&amp;qid=1341547865&amp;sr=8-1&amp;keywords=secrets+of+mental+math
It has many tricks like that. I enjoyed it.
Fermat's Last Theorem by Simon Singh.
(I'm reasonably sure the linked book is Fermat's Last Theorm, just with a different title. It was the closest I could find on US Amazon)
Basic Mathematics by Serge Lang.
https://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/0387967877
Do not enroll in a precalculus class until you have a solid grasp on the foundations of precalculus. Precalculus is generally considered to be the fundamentals required for calculus and beyond (obviously), and a strong understanding of precalculus will serve you well, but in order to do well in precalculus you still need a solid understanding of what comes before, and there is quite a bit.
I do not mean to sound discouraging, but I was tutoring a guy in an adult learning program from about December 2017-July 2018...I helped him with his homework and answered any questions that he had, but when he asked me to really get into the meat of algebra (he needed it for chemistry to become a nurse) I found a precalculus book at the library and asked him to go over the prerequisite chapter and it went completely over his head. Perhaps this is my fault as a tutor, but I do not believe so.
What I am saying is that you need a good foundation in the absolute basics before doing precalculus and I do not believe that you should enroll in a precalculus course ASAP because you may end up being let down and then give up completely. I would recommend pairing Basic Mathematics by Serge Lang with The Humongous Book of Algebra Problems (though any book with emphasis on practice will suffice) and using websites like khanacademy for additional practice problems and instructions. Once you have a good handle on this, start looking at what math courses are offered at your nearest CC and then use your best judgment to decide which course(s) to take.
I do not know how old you are, but if you are anything like me, you probably feel like you are running out of time and need to rush. Take your time and practice as much as possible. Do practice problems until it hurts to hold the pencil.
I'd like to give you my two cents as well on how to proceed here. If nothing else, this will be a second opinion. If I could redo my physics education, this is how I'd want it done.
If you are truly wanting to learn these fields in depth I cannot stress how important it is to actually work problems out of these books, not just read them. There is a certain understanding that comes from struggling with problems that you just can't get by reading the material. On that note, I would recommend getting the Schaum's outline to whatever subject you are studying if you can find one. They are great books with hundreds of solved problems and sample problems for you to try with the answers in the back. When you get to the point you can't find Schaums anymore, I would recommend getting as many solutions manuals as possible. The problems will get very tough, and it's nice to verify that you did the problem correctly or are on the right track, or even just look over solutions to problems you decide not to try.
Basics
I second Stewart's Calculus cover to cover (except the final chapter on differential equations) and Halliday, Resnick and Walker's Fundamentals of Physics. Not all sections from HRW are necessary, but be sure you have the fundamentals of mechanics, electromagnetism, optics, and thermal physics down at the level of HRW.
Once you're done with this move on to studying differential equations. Many physics theorems are stated in terms of differential equations so really getting the hang of these is key to moving on. Differential equations are often taught as two separate classes, one covering ordinary differential equations and one covering partial differential equations. In my opinion, a good introductory textbook to ODEs is one by Morris Tenenbaum and Harry Pollard. That said, there is another book by V. I. Arnold that I would recommend you get as well. The Arnold book may be a bit more mathematical than you are looking for, but it was written as an introductory text to ODEs and you will have a deeper understanding of ODEs after reading it than your typical introductory textbook. This deeper understanding will be useful if you delve into the nitty-gritty parts of classical mechanics. For partial differential equations I recommend the book by Haberman. It will give you a good understanding of different methods you can use to solve PDEs, and is very much geared towards problem-solving.
From there, I would get a decent book on Linear Algebra. I used the one by Leon. I can't guarantee that it's the best book out there, but I think it will get the job done.
This should cover most of the mathematical training you need to move onto the intermediate level physics textbooks. There will be some things that are missing, but those are usually covered explicitly in the intermediate texts that use them (i.e. the Delta function). Still, if you're looking for a good mathematical reference, my recommendation is Lua. It may be a good idea to go over some basic complex analysis from this book, though it is not necessary to move on.
Intermediate
At this stage you need to do intermediate level classical mechanics, electromagnetism, quantum mechanics, and thermal physics at the very least. For electromagnetism, Griffiths hands down. In my opinion, the best pedagogical book for intermediate classical mechanics is Fowles and Cassidy. Once you've read these two books you will have a much deeper understanding of the stuff you learned in HRW. When you're going through the mechanics book pay particular attention to generalized coordinates and Lagrangians. Those become pretty central later on. There is also a very old book by Robert Becker that I think is great. It's problems are tough, and it goes into concepts that aren't typically covered much in depth in other intermediate mechanics books such as statics. I don't think you'll find a torrent for this, but it is 5 bucks on Amazon. That said, I don't think Becker is necessary. For quantum, I cannot recommend Zettili highly enough. Get this book. Tons of worked out examples. In my opinion, Zettili is the best quantum book out there at this level. Finally for thermal physics I would use Mandl. This book is merely sufficient, but I don't know of a book that I liked better.
This is the bare minimum. However, if you find a particular subject interesting, delve into it at this point. If you want to learn Solid State physics there's Kittel. Want to do more Optics? How about Hecht. General relativity? Even that should be accessible with Schutz. Play around here before moving on. A lot of very fascinating things should be accessible to you, at least to a degree, at this point.
Advanced
Before moving on to physics, it is once again time to take up the mathematics. Pick up Arfken and Weber. It covers a great many topics. However, at times it is not the best pedagogical book so you may need some supplemental material on whatever it is you are studying. I would at least read the sections on coordinate transformations, vector analysis, tensors, complex analysis, Green's functions, and the various special functions. Some of this may be a bit of a review, but there are some things Arfken and Weber go into that I didn't see during my undergraduate education even with the topics that I was reviewing. Hell, it may be a good idea to go through the differential equations material in there as well. Again, you may need some supplemental material while doing this. For special functions, a great little book to go along with this is Lebedev.
Beyond this, I think every physicist at the bare minimum needs to take graduate level quantum mechanics, classical mechanics, electromagnetism, and statistical mechanics. For quantum, I recommend Cohen-Tannoudji. This is a great book. It's easy to understand, has many supplemental sections to help further your understanding, is pretty comprehensive, and has more worked examples than a vast majority of graduate text-books. That said, the problems in this book are LONG. Not horrendously hard, mind you, but they do take a long time.
Unfortunately, Cohen-Tannoudji is the only great graduate-level text I can think of. The textbooks in other subjects just don't measure up in my opinion. When you take Classical mechanics I would get Goldstein as a reference but a better book in my opinion is Jose/Saletan as it takes a geometrical approach to the subject from the very beginning. At some point I also think it's worth going through Arnold's treatise on Classical. It's very mathematical and very difficult, but I think once you make it through you will have as deep an understanding as you could hope for in the subject.
If you like logic and the scientific method, I recommend E. T. Jaynes' Probability Theory: The Logic of Science. You can buy it here:
http://www.amazon.com/Probability-Theory-The-Logic-Science/dp/0521592712/
or read a PDF here:
http://shawnslayton.com/open/Probability%2520book/book.pdf
Here is the ooh page on Statisticians:
http://www.bls.gov/oco/ocos045.htm
A job straight out of college might see you as a research assistant. I could see you getting a job at Mathematica perhaps. Try to get a SAS certificate before you graduate, a working knowledge of R, and if you feel like tackling it a programming language good for numerical analysis.
Have you taken a course on Regression? I'd consider that, and perhaps even trying to take a Mathematical Statistics Course, if it is offered. You can try to see if you university would allow you to take a class online, or try a Semester Abroad at a university that has that class.
My background: I am an Economist that uses Statistics heavily, and works with Statistical methods often (ie: econometrics). I love it.
Your plans on studying Calc 2 and Linear Algebra are great. That is perfect.
My pay after 10 years is likely to be 100k-150k.
Before you start your first semester at the graduate level know the following things really well: Set theory, integration, matrix algebra, and proofs.
Get this book: http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521675995 -- read it before you study linear algebra, and maybe even some Calculus. It doesn't require a heavy Math background and will save you a lot of frustration later on.
I'd echo what /u/Odnahc has said.
Struggling in Intro the Proofs isn't he end of the world. I struggled in proofs and still ended up with a BS and MS in Math, however, I bought this book and self studied proofs over the Summer and made sure I had a stronger foundation.
The courses normally taken after proofs (Advanced Calculus and Modern Algebra) usually spend the first class reviewing proofs to make sure students have a handle of the material. After that though, you're expected to know the stuff. And honestly, you'll be doing lot of work trying to understand the new material and you're really going to struggle if you're fighting proof writing instead of the new ideas.
Proceed with caution. Definitely speak to your advisor.
Alternately, any introductory book on mathematical analysis will have a section on sentential logic. 'How to Prove It' by Velleman is a great intro, and comes with a link to a web tool to practice!
I know the answer to this.
First, though: arithmetic and all that, through calculus, is not math.
True math is the discovery of properties of ideas. One interesting example is the fact that there is a hypothetical machine that is proven to be able to do everything a (real) computer can do, but that there are many things that it can never do. Therefore, there are questions that can never be answered by a computer, no matter how powerful.
If you actually want to know about the beauty, you need to see it for yourself. As I recall, How to Prove it is pretty decent.
Indeed; you may feel that you are at a disadvantage compared to your peers, and that the amount of work you need to pull off is insurmountable.
However, you have an edge. You realize you need help, and you want to catch up. Motivation and incentive is a powerful thing.
Indeed, being passionate about something makes you much more likely to remember it. Interestingly, the passion does not need to be a loving one.
A common pitfall when learning math is thinking it is like learning history, philosophy, or languages, where it doesn't matter if you miss out a bit; you will still understand everything later, and the missing bits will fall into place eventually. Math is nothing like that. Math is like building a house. A first step for you should therefore be to identify how much of the foundation of math you have, to know where to start from.
Khan Academy is a good resource for this, as it has a good overview of math, and how the different topics in math relate (what requires understanding of what). Khan Academy also has good exercises to solve, and ways to get help. There are also many great books on mathematics, and going through a book cover-to-cover is a satisfying experience. I have heard people speak highly of Serge Lang's "Basic Mathematics".
Finding sparetime activities to train your analytic and critical thinking skills will also help you immeasurably. Here I recommend puzzle books, puzzle games (I recommend Portal, Lolo, Lemmings, and The Incredible Machine), board/card games (try Eclipse, MtG, and Go), and programming (Scheme or Haskell).
It takes effort. But I think you will find your journey through maths to be a truly rewarding experience.
Get a copy of Div, Grad, Curl. It will walk you through the math you need.
Intro Calculus, in American sense, could as well be renamed "Physics 101" or some such since it's not a very mathematical course. Since Intro Calculus won't teach you how to think you're gonna need a book like How to Solve Word Problems in Calculus by Eugene Don and Benay Don pretty soon.
Aside from that, try these:
Excursions In Calculus by Robert Young.
Calculus:A Liberal Art by William McGowen Priestley.
Calculus for the Ambitious by T. W. KORNER.
Calculus: Concepts and Methods by Ken Binmore and Joan Davies
You can also start with "Calculus proper" = Analysis. The Bible of not-quite-analysis is:
[Calculus by Michael Spivak] (http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1413311074&amp;sr=1-1&amp;keywords=spivak+calculus).
Also, Analysis is all about inequalities as opposed to Algebra(identities), so you want to be familiar with them:
Introduction to Inequalities by Edwin F. Beckenbach, R. Bellman.
Analytic Inequalities by Nicholas D. Kazarinoff.
As for Linear Algebra, this subject is all over the place. There is about a million books of all levels written every year on this subject, many of which is trash.
My plan would go like this:
1. Learn the geometry of LA and how to prove things in LA:
Linear Algebra Through Geometry by Thomas Banchoff and John Wermer.
Linear Algebra, Third Edition: Algorithms, Applications, and Techniques
by Richard Bronson and Gabriel B. Costa.
2. Getting a bit more sophisticated:
Linear Algebra Done Right by Sheldon Axler.
Linear Algebra: An Introduction to Abstract Mathematics by Robert J. Valenza.
Linear Algebra Done Wrong by Sergei Treil.
3. Turn into the LinAl's 1% :)
Advanced Linear Algebra by Steven Roman.
Good Luck.
Hrrumph. Determinants are a capstone, not a cornerstone, of Linear Algebra.
https://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/0387982582
>When university starts, what can I do to ensure that I can compete and am just as good as the best mathematics students?
Read textbooks for mathematics students.
For example for Linear Algebra I heard that Axler's book is very good (I studied Linear Algebra in another language, so I can't really suggest anything from personal experience). For Calculus I personally suggest Spivak's book.
There are many books that I could suggest, but one of the greatest books I've ever read is The Art and Craft of Problem Solving.
I studied with this book on abstract. It's authoritative and brutal.
It's hard to give an objective answer, because any sufficiently advanced book will be bound to not appeal to everyone.
You probably want Daddy Rudin for real analysis and Dummit & Foote for general abstract algebra.
Mac Lane for category theory, of course.
I think people would agree on Hartshorne as the algebraic geometry reference.
Spanier used to be the definitive algebraic topology reference. It's hard to actually use it as a reference because of the density and generality with which it's written.
Spivak for differential geometry.
Rotman is the group theory book for people who like group theory.
As a physics person, I must have a copy of Fulton & Harris.
Take my recommendation as a grain of salt as i didn't take my formal math education further than where you're currently at, but I felt the same way after similar classes learning the mechanics but not the motivations. Mathematics: Its Content, Methods and Meaning was recommended to me by a friend and I think it help fills the gaps in motivation and historical context/connecting different fields not covered in classes.
You need some grounding in foundational topics like Propositional Logic, Proofs, Sets and Functions for higher math. If you've seen some of that in your Discrete Math class, you can jump straight into Abstract Algebra, Rigorous Linear Algebra (if you know some LA) and even Real Analysis. If thats not the case, the most expository and clearly written book on the above topics I have ever seen is Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.
Some user friendly books on Real Analysis:
Some user friendly books on Linear/Abstract Algebra:
Topology(even high school students can manage the first two titles):
Some transitional books:
Plus many more- just scour your local library and the internet.
Good Luck, Dude/Dudette.
I actually bought http://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178
On chapter three now :D
Sorry, was a bad joke.
https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X
tl;dr: you need to learn proofs to read most math books, but if nothing else there's a book at the bottom of this post that you can probably dive into with nothing beyond basic calculus skills.
Are you proficient in reading and writing proofs?
If you aren't, this is the single biggest skill that you need to learn (and, strangely, a skill that gets almost no attention in school unless you seek it out as an undergraduate). There are books devoted to developing this skill—How to Prove It is one.
After you've learned about proof (or while you're still learning about it), you can cut your teeth on some basic real analysis. Basic Elements of Real Analysis by Protter is a book that I'm familiar with, but there are tons of others. Ask around.
You don't have to start with analysis; you could start with algebra (Algebra and Geometry by Beardon is a nice little book I stumbled upon) or discrete (sorry, don't know any books to recommend), or something else. Topology probably requires at least a little familiarity with analysis, though.
The other thing to realize is that math books at upper-level undergraduate and beyond are usually terse and leave a lot to the reader (Rudin is famous for this). You should expect to have to sit down with pencil and paper and fill in gaps in explanations and proofs in order to keep up. This is in contrast to high-school/freshman/sophomore-style books like Stewart's Calculus where everything is spelled out on glossy pages with color pictures (and where proofs are mostly absent).
And just because: Visual Complex Analysis is a really great book. Complex numbers, functions and calculus with complex numbers, connections to geometry, non-Euclidean geometry, and more. Lots of explanation, and you don't really need to know how to do proofs.
memory, just pick one book the basics are the same: A Sheep Falls Out of the Tree, Quantum Memory Power, not just memory techniques but with a section on Improve your intelligence
math: secrets of mental math
Among many others who can be given the title of the world's most intelligent person is Marilyn vos Savant: one of her books
I picked up the book by Simon Singh at a garage sale 10 or so years ago. Fascinating read. Looking forward to watching the doc now.
EDIT: evidently the book is now called Fermat's Enigma in the US...
I'm also planning on doing a Masters in Math or CS. What do you plan to write for your masters?
> Anybody else feels like this?
I think its natural to doubt yourself, sometimes. I dont know what else to say, but just try to be objective and emotionless about it (when you get stuck in a problem).
The following books that helped me improve my math problem solving skills when I was an undergrad:
I think this is the recommended replacement for Polya's "How to Solve It"
http://www.amazon.com/How-Solve-It-Mathematical-Princeton/dp/069111966X
Seriously what do you want to be "modernized?"
Try one (or a few) of these:
http://www.amazon.ca/Thinking-Mathematically-J-Mason/dp/0201102382
http://www.amazon.com/How-Think-Like-Mathematician-Undergraduate/dp/052171978X/
http://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X
www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/
www.amazon.com/Introduction-Mathematical-Thinking-Keith-Devlin/dp/0615653634/
https://www.coursera.org/course/maththink
I recommend you start studying proofs first. How to Prove It by Velleman is a great book for new math students. I went through the first three chapters myself before my first analysis course, and it made all the difference.
As you are taking a class than combines analysis and calculus, you might benefit from Spivak's book Calculus, which despite it's title, is precisely a combination of calculus and real analysis.
Wellll I'm going to speak in some obscene generalities here.
There are some philosophical reasons and some practical reasons that being a "pure" Bayesian isn't really a thing as much as it used to be. But to get there, you first have to understand what a "pure" Bayesian is: you develop reasonable prior information based on your current state of knowledge about a parameter / research question. You codify that in terms of probability, and then you proceed with your analysis based on the data. When you look at the posterior distributions (or posterior predictive distribution), it should then correctly correspond to the rational "new" state of information about a problem because you've coded your prior information and the data, right?
WELL let's touch on the theoretical problems first: prior information. First off, it can be very tricky to code actual prior information into a true probability distribution. This is one of the big turn-offs for frequentists when it comes to Bayesian analysis. "Pure" Bayesian analysis sees prior information as necessarily coming before the data is ever seen. However, suppose you define a "prior" whereby a parameter must be greater than zero, but it turns out that your state of knowledge is wrong? What if you cannot codify your state of knowledge as a prior? What if your state of knowledge is correctly codified but makes up an "improper" prior distribution so that your posterior isn't defined?
Now'a'days, Bayesians tend to think of the prior as having several purposes, but they also view it as part of your modeling assumptions - something that must be tested to determine if your conclusions are robust. So you might use a weakly regularizing prior for the purposes of getting a model to converge, or you might look at the effects of a strong prior based on other studies, or the effects of a non-informative prior to see what the data is telling you absent other information. By taking stock of the differences, you can come to a better understanding of what a good prediction might be based on the information available to you. But to a "pure" Bayesian, this is a big no-no because you are selecting the prior to fit together with the data and seeing what happens. The "prior" is called that because it's supposed to come before, not after. It's supposed to codify the current state of knowledge, but now'a'days Bayesians see it as serving a more functional purpose.
Then there are some practical considerations. As I mentioned before, Bayesian analysis can be very computationally expensive when data sets are large. So in some instances, it's just not practical to go full Bayes. It may be preferable, but it's not practical. So you wind up with some shortcuts. I think that in this sense, modern Bayesians are still Bayesian - they review answers in reference to their theoretical understanding of what is going on with the distributions - but they can be somewhat restricted by the tools available to them.
As always with Bayesian statistics, Andrew Gelman has a lot to say about this. Example here and here and he has some papers that are worth looking into on the topic.
There are probably a lot of other answers. Like, you could get into how to even define a probability distribution and whether it has to be based on sigma algebras or what. Jaynes has some stuff to say about that.
If you want a good primer on Bayesian statistics that has a lot of talking and not that much math (although what math it does have is kind of challenging, I admit, though not unreachable), read this book. I promise it will only try to brainwash you a LITTLE.
Read this book: How To Prove it
You need a good foundation: a little logic, intro to proofs, a taste of sets, a bit on relations and functions, some counting(combinatorics/graph theory) etc. The best way to get started with all this is an introductory discrete math course. Check these books out:
Mathematics: A Discrete Introduction by Edward A. Scheinerman
Discrete Mathematics with Applications by Susanna S. Epp
How to Prove It: A Structured Approach Daniel J. Velleman
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Combinatorics: A Guided Tour by David R. Mazur
Since nobody else has recommended it, I always recommend the book How to Prove it by Daniel J. Velleman for learning proofs. I always found proofs to be kind of black magic until I read that, which totally demystified them for me by revealing the structure of proofs and techniques for proving different kinds of statements. One of the best things about it is that it starts from square one with basic logic and builds from there in way that no prior knowledge is required beyond basic algebra skills.
You cannot go wrong with How To Prove It: A Structured Approach by Velleman https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/ref=sr_1_3?keywords=how+to+prove+it&amp;qid=1558195901&amp;s=gateway&amp;sr=8-3
I saw that book highly recommended, and after going through it myself a while ago I highly recommend it as well. When I do proofs I still maintain the mental model and use some of the mechanics that I learned from this book. You don't even have to read the whole thing in my opinion. Pick it up, work through a few pages per day, and stop when you feel like moving onto another subject-specific book, like Understanding Analysis.
Oh, and you might already know this, but do as many practice problems as you can! Learning proofs is all about practice.
https://www.amazon.com/dp/0521675995/ref=cm_sw_r_other_apa_Sn9NBbDH6MYPX not sure it this is exactly what you're asking for(might be more than you're asking for?) but this helped me a lot.
Have you read A Book of Abstract Algebra by Charles Pinter? https://www.amazon.co.uk/d/cka/Abstract-Algebra-Dover-Books-Mathematics-Charles-Pinter/0486474178
An Introduction to Ordinary Differential Equations - $7.62
Ordinary Differential Equations - $14.74
Partial Differential Equations for Scientists and Engineers - $11.01
Dover books on mathematics have great books for very cheap. I personally own the second and third book on this list and I thought they were a great resource, especially for the price.
Imagine you have a dataset without labels, but you want to solve a supervised problem with it, so you're going to try to collect labels. Let's say they are pictures of dogs and cats and you want to create labels to classify them.
One thing you could do is the following process:
(I'm ignoring problems like pictures that are difficult to classify or lazy or adversarial humans giving you noisy labels)
That's one way to do it, but is it the most efficient way? Imagine all your pictures are from only 10 cats and 10 dogs. Suppose they are sorted by individual. When you label the first picture, you get some information about the problem of classifying cats and dogs. When you label another picture of the same cat, you gain less information. When you label the 1238th picture from the same cat you probably get almost no information at all. So, to optimize your time, you should probably label pictures from other individuals before you get to the 1238th picture.
How do you learn to do that in a principled way?
Active Learning is a task where instead of first labeling the data and then learning a model, you do both simultaneously, and at each step you have a way to ask the model which next example should you manually classify for it to learn the most. You can than stop when you're already satisfied with the results.
You could think of it as a reinforcement learning task where the reward is how much you'll learn for each label you acquire.
The reason why, as a Bayesian, I like active learning, is the fact that there's a very old literature in Bayesian inference about what they call Experiment Design.
Experiment Design is the following problem: suppose I have a physical model about some physical system, and I want to do some measurements to obtain information about the models parameters. Those measurements typically have control variables that I must set, right? What are the settings for those controls that, if I take measurements on that settings, will give the most information about the parameters?
As an example: suppose I have an electric motor, and I know that its angular speed depends only on the electric tension applied on the terminals. And I happen to have a good model for it: it grows linearly up to a given value, and then it becomes constant. This model has two parameters: the slope of the linear growth and the point where it becomes constant. The first looks easy to determine, the second is a lot more difficult. I'm going to measure the angular speed at a bunch of different voltages to determine those two parameters. The set of voltages I'm going to measure at is my control variable. So, Experiment Design is a set of techniques to tell me what voltages I should measure at to learn the most about the value of the parameters.
I could do Bayesian Iterated Experiment Design. I have an initial prior distribution over the parameters, and use it to find the best voltage to measure at. I then use the measured angular velocity to update my distribution over the parameters, and use this new distribution to determine the next voltage to measure at, and so on.
How do I determine the next voltage to measure at? I have to have a loss function somehow. One possible loss function is the expected value of how much the accuracy of my physical model will increase if I measure the angular velocity at a voltage V, and use it as a new point to adjust the model. Another possible loss function is how much I expect the entropy of my distribution over parameters to decrease after measuring at V (the conditional mutual information between the parameters and the measurement at V).
Active Learning is just iterated experiment design for building datasets. The control variable is which example to label next and the loss function is the negative expected increase in the performance of the model.
So, now your procedure could be:
Or you could be a lot more clever than that and use proper reinforcement learning algorithms. Or you could be even more clever and use "model-independent" (not really...) rewards like the mutual information, so that you don't over-optimize the resulting data set for a single choice of model.
I bet you have a lot of concerns about how to do this properly, how to avoid overfitting, how to have a proper train-validation-holdout sets for cross validation, etc, etc, and those are all valid concerns for which there are answers. But this is the gist of the procedure.
You could do Active Learning and iterated experiment design without ever hearing about bayesian inference. It's just that those problems are natural to frame if you use bayesian inference and information theory.
About the jargon, there's no way to understand it without studying bayesian inference and machine learning in this bayesian perspective. I suggest a few books:
Is a pretty good introduction to Information Theory and bayesian inference, and how it relates to machine learning. The Machine Learning part might be too introductory if already know and use ML.
Some people don't like this book, and I can see why, but if you want to learn how bayesians think about ML, it is the most comprehensive book I think.
More of a philosophical book. This is a good book to understand what bayesians find so awesome about bayesian inference, and how they think about problems. It's not a book to take too seriously though. Jaynes was a very idiosyncratic thinker and the tone of some of the later chapters is very argumentative and defensive. Some would even say borderline crackpot. Read the chapter about plausible reasoning, and if that doesn't make you say "Oh, that's kind of interesting...", than nevermind. You'll never be convinced of this bayesian crap.
I'm going to shamelessly plug this book which I consider to be one of my favorite books ever. For the price it is definitely worth keeping a copy and reading it on the side if you're learning abstract algebra for the first time and it reads like a novel. It's definitely a small treasure I feel I discovered.
there's a lot going on here, so i'll try to take it a few steps at a time.
> how many REAL operators do we have?
you might be careful about your language here, as the word "real" has implications in the world of mathematics to mean "takes values in the real numbers", i.e., is non-complex. also, "real" in the normal sense of real or fake doesn't have a lot of meaning in mathematics. a better question might be "how many unique operators do we have?", but even that isn't quite good enough. you need to define context. a blanket answer to your question is that there are uncountably infinite amount of operators in mathematics that take all kinds of forms: linear operators, functional operators, binary operators, etc.
> taking a number to the power of another is just defined in terms of multiplication
similar to /u/theowoll's response, how would you define 2^(4.18492) in terms of multiplication? i know you're basing this question off of the interesting fact that 2^1 = 2, 2^2 = 2 2, 2^3 = 2 2 * 2, etc. and similarly for other certain classes of numbers, but how do you multiply 2 by itself 4.18492 times? it gets even more tricky to think of exponents like this if the base and power are non-rational (4.18492=418492/100000 is rational). what about the power of e^X, where e is the normal exponential and X is a matrix? take a look at wikipedia's article on exponentiation to see what a can of worms this discussion opens.
> So am I just plain wrong about all this, or there is some truth to it?
although there is a lot of incorrect things in your description when you consider general classes of "things you can multiply and add", what you are sort of getting at is what the theory of abstract algebra covers. in such a theory, it explores what it means to add, multiply, have inverses, etc. for varying collections of things called groups, rings, fields, vector spaces, modules, etc. and the relationships and properties of such things. you might take a look at a book of abstract algebra by charles pinter. you should be able to follow it, as it is an excellent book.
A book of abstract algebra by Charles Pinter is the best math book I've ever read in terms of readability, I think. The first chapter is an essay on the history of algebra and the book is worth it just for this chapter.
Try Pinter. If you think it is too simple for you go for Aluffi.
Learn math first. Physics is essentially applied math with experiments. Start with Calculus then Linear Algebra then Real Analysis then Complex Analysis then Ordinary Differential Equations then Partial Differential Equations then Functional Analysis. Also, if you want to pursue high energy physics and/or cosmology, Differential Geometry is also essential. Make sure you do (almost) all the exercises in every chapter. Don't just skim and memorize.
This is a lot of math to learn, but if you are determined enough you can probably master Calculus to Real Analysis, and that will give you a big head start and a deeper understanding of university-level physics.
If you are serious about this, then the best way to self learn Math is to learn to read Math books. This is a valuable skill. It stops you from having to rely on websites/tutorials and frees you to really read the stuff you're interested in.
Generally you probably want a more "back to basics" approach that will cover basic stuff and act as an introduction (again) to the topic (without handling you as if you're a child). I recommend Discrete Mathematics with Applications. Epp does a good job of starting at the beginning (with logic) and building a decent foundation through connectives, conditionals, existentials, universals, etc. eventually leading into proofs.
Her writing style is very readable IMO but still dense enough to help you learn how to read Math books.
If you're self motivated enough then start there. Read a chapter. Do the problems. Be confused. Do more problems. Still confused? Read the chapter again. Do more problems. Repeat. Eventually finish the book.
The next one will be faster and easier because of the work you put in. Eventually you'll be 3-5 books down, and you'll feel you know quite a bit. Then read more.. realize the field is huge and you know nothing. Read more to solve this.
Repeat
????
Success(?)
How to prove it is a great start. I think after that, you should focus on learning to think mathematically through practice instead of reading (at least, that's how I and most people learn best). Take classes or read and work through the textbooks of subjects that interest you. Discrete math would be a good place to start since it teaches proof techniques and basic probability and combinatorics; my class used this book which I thought was nice.
If you don't actually do the work, your thinking process isn't going to change.
Check out /r/compsci, /r/algorithms, and the subreddits in their sidebars.
Sure! There is a lot of math involved in the WHY component of Computer Science, for the basics, its Discrete Mathematics, so any introduction to that will help as well.
http://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328/ref=sr_sp-atf_title_1_1?s=books&amp;ie=UTF8&amp;qid=1368125024&amp;sr=1-1&amp;keywords=discrete+mathematics
This next book is a great theoretical overview of CS as well.
http://mitpress.mit.edu/sicp/full-text/book/book.html
That's a great book on computer programming, complexity, data types etc... If you want to get into more detail, check out: http://www.amazon.com/Introduction-Theory-Computation-Michael-Sipser/dp/0534950973
I would also look at Coursera.org's Algorithm lectures by Robert Sedgewick, thats essential learning for any computer science student.
His textbook: http://www.amazon.com/Algorithms-4th-Robert-Sedgewick/dp/032157351X/ref=sr_sp-atf_title_1_1?s=books&amp;ie=UTF8&amp;qid=1368124871&amp;sr=1-1&amp;keywords=Algorithms
another Algorithms textbook bible: http://www.amazon.com/Introduction-Algorithms-Thomas-H-Cormen/dp/0262033844/ref=sr_sp-atf_title_1_2?s=books&amp;ie=UTF8&amp;qid=1368124871&amp;sr=1-2&amp;keywords=Algorithms
I'm just like you as well, I'm pivoting, I graduated law school specializing in technology law and patents in 2012, but I love comp sci too much, so i went back into school for Comp Sci + jumped into the tech field and got a job at a tech company.
These books are theoretical, and they help you understand why you should use x versus y, those kind of things are essential, especially on larger applications (like Google's PageRank algorithm). Once you know the theoretical info, applying it is just a matter of picking the right tool, like Ruby on Rails, or .NET, Java etc...
Discrete Mathematics with Applications by Susanna Epp is pretty good, with a lot of exposition. In the introduction there is a guide on how to use the book, and the different sections to focus on if using it for a mainly mathematics-based class or for a computer science-based class.
http://www.amazon.com/gp/product/0495391328/ref=pd_lpo_sbs_dp_ss_2?pf_rd_p=1944687702&amp;pf_rd_s=lpo-top-stripe-1&amp;pf_rd_t=201&amp;pf_rd_i=0132122715&amp;pf_rd_m=ATVPDKIKX0DER&amp;pf_rd_r=0P155N9Y802PPMETYGVC
Ok, it's kinda expensive (~$60), but it's on amazon:
http://www.amazon.com/Probability-Theory-Logic-Science-Vol/dp/0521592712/ref=sr_1_1?ie=UTF8&s=books&qid=1252600006&sr=8-1
*edit for link pwnage.
https://www.amazon.com/Probability-Theory-Science-T-Jaynes/dp/0521592712
This book is full of proofs you can work through. It could keep you busy for quite a while and it's considered a standard for analysis.
https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X
Please take a look at What is mathematics by Courant, Robbins, Stewart. It is much leaner, yet it is accessible and was endorsed by Einstein:
"A lucid representation of the fundamental concepts and methods of the whole field of mathematics. It is an easily understandable introduction for the layman and helps to give the mathematical student a general view of the basic principles and methods."
I really like What is Mathematics by Richard Courant. It's aimed at the lay person and I think a 13 year old would enjoy it. It's a book you can jump around in too.
To me its about what you can do in your head. Get a book for example, BOOK is good.
Also, subscribe to /r/math. Finally, ANYTIME you see a number do something with it. Factor it, think of a historical significance, etc.
First off, I'd recommend looking into a book like this.
Second, when doing something like multiplication, it always helps to break a problem down into easier steps. Typically, you want to be working with multiples of 10/100/1000s etc.
For multiplying 32 by 32, I would break it into two problems: (32 x 30) + (32 x 2). With a moderate amount of practice, you should quickly be able to see that the first term is 960, and the second is 64. Adding them together gives the answer: 1024. It can be tricky to keep all these numbers in your head at once, but that honestly just comes down to practice.
Also, that same question can be expressed as 32^2 . These types of problems have a whole bunch of neat tricks. One that I recall from the book I linked above has to do with squaring any number ending in a 5, like 15 or 145. First, the number will always end in 25. For the leading digits, take the last 5 off the number, and multiply the remaining digits by their value +1. So, for 15 we just have 1x2=2. For 145, we have 14x15=210. Finally, tack 25 on the end of that, so you have 15^2 = (1x2)25 = 225, and 145^2 = (14x15)25 = 21025. Boom! Now you can square any number ending in 5 really quick.
Edit: Wanted to add some additional comments that have helped me out through the years. First, realize that
(1) Addition is easier than subtraction,
(2) Addition and subtraction are easier than multiplication,
(3) Multiplication is easier than division.
Let's go through these one by one. For (1), try to rewrite a subtraction problem as addition. Say you're given 31 - 14; then rephrase the question as, what plus 14 equals 31? You can immediately see that the ones digit is 7, since 4+7 = 11. We have to remember that we are carrying the ten over to the next digit, and solve 1 + (1 carried over) + what = 3. Obviously the tens digit for our answer is 1, and the answer is 17. I hope I didn't explain that too poorly.
For (2), that's pretty much what I was originally explaining at the start. Try to break a multiplication problem down to a problem of simple multiplication plus addition or subtraction. One more example: 37 x 40. Here, 40 looks nice and simple to work with; 37 is also pretty close to it, so let's add 3 to it and just make sure to subtract it later. That way, you end up with 40 x 40 - (3 x 40) = 1600 - 120 = 1480.
I don't really have any hints with division, unfortunately. I don't really run into it too often, and when I do, I just resort to some mental long division.
Good question OP! I drafted a blog article on this topic a while back but haven't published it yet. An excerpt is below.
--------
With equations, I sometimes just visualize what I'd usually do on paper. For arithmetic, there are actually a lot of computational methods that are better suited to mental computation than the standard pencil-and-paper algorithms.
In fact, mathematician Arthur Benjamin has written a book about this called Secrets of Mental Math.
There are tons of different options, often for the same problem. The main thing is to understand some general principles, such as breaking a problem down into easier sub-problems, and exploiting special features of a particular problem.
Below are some basic methods to give you an idea. (These may not all be entirely different from the pencil-and-paper methods, but at the very least, the format is modified to make them easier to do mentally.)
ADDITION
(1) Separate into place values: 27+39= (20+30)+(7+9)=50+16=66
We've reduced the problem into two easier sub-problems, and combining the sub-problems in the last step is easy, because there is no need to carry as in the standard written algorithm.
(2) Exploit special features: 298+327 = 300 + 327 -2 = 625
We could have used the place value method, but since 298 is close to 300, which is easy to work with, we can take advantage of that by thinking of 298 as 300 - 2.
SUBTRACTION
(1) Number-line method: To find 71-24, you move forward 6 units on the number line to get to 30, then 41 more units to get to 71, for a total of 47 units along the number line.
(2) There are other methods, but I'll omit these, since the number-line method is a good starting point.
MULTIPLICATION
(1) Separate into place values: 18*22 = 18*(20+2)=360+36=396.
(2) Special features: 18*22=(20-2)*(20+2)=400-4=396
Here, instead of using place values, we use the feature that 18*22 can be written in the form (a-b)*(a+b) to obtain a difference of squares.
(3) Factoring method: 14*28=14*7*4=98*4=(100-2)*4=400-8=392
Here, we've turned a product of two 2-digit numbers into simpler sub-problems, each involving multiplication by a single-digit number (first we multiply by 7, then by 4).
(4) Multiplying by 11: 11*52= 572 (add the two digits of 52 to get 5+2=7, then stick 7 in between 5 and 2 to get 572).
This can be done almost instantaneously; try using the place-value method to see why this method works. Also, it can be modified slightly to work when the sum of the digits is a two digit number.
DIVISION
(1) Educated guess plus error correction: 129/7 = ? Note that 7*20=140, and we're over by 11. We need to take away two sevens to get back under, which takes us to 126, so the answer is 18 with a remainder of 3.
(2) Reduce first, using divisibility rules. Some neat rules include the rules for 3, 9, and 11.
The rules for 3 and 9 are probably more well known: a number is divisible by 3 if and only if the sum of its digits is divisible by 3 (replace 3 with 9 and the same rule holds).
For example, 5654 is not divisible by 9, since 5+6+5+4=20, which is not divisible by 9.
The rule for 11 is the same, but it's the alternating sum of the digits that we care about.
Using the same number as before, we get that 5654 is divisible by 11, since 5-6+5-4=0, and 0 is divisible by 11.
PRACTICE
I think it's kind of fun to get good at finding novel methods that are more efficient than the usual methods, and even if it's not that fun, it's at least useful to learn the basics.
If you want to practice these skills outside of the computations that you normally do, there's a nice online arithmetic game I found that's simple and flexible enough for you to practice any of the four operations above, and you can set the parameters to work on numbers of varying sizes.
Happy calculating!
Greg at Higher Math Help
Edit: formatting
If you’d like a physical textbook, I’d recommend Basic Mathematics by Serge Lang, a celebrated mathematician and teacher. It’s an oldie but a goodie. https://www.amazon.com/dp/0387967877/
If you progress past that and want to refresh your calculus, it’s hard to go wrong with James Stewart’s Calculus. https://www.amazon.com/dp/B00YHKU50E/
A commonly used book for this exact purpose is Div, Grad, Curl by Schey.
For multivariable calculus I cannot recommend Div, Grad, Curl and All That enough. It’s got wonderful physically motivated examples and great problems. If you work through all the problems you’ll have s nice grasp on some central topics of vector calculus. It’s also rather thin, making it feel approachable for self learning (and easy to travel with).
If you're doing both applied and pure abstract algebra rather than elementary algebra, then you'll probably need to learn to write proofs for the pure side. I recommend Numbers, Groups, and Codes by J. F. Humphreys for an introduction to the basics and to some applied abstract algebra. If you need more work on proofs, the free Book of Proofs can help, and Fraleigh's A First Course in Abstract Algebra is a good book for pure abstract algebra. If you want something more advanced, I recommend the massive Abstract Algebra by Dummit and Foote.
Hi there,
For all intents and purposes, for someone your level the following will be enough material to stick your teeth into for a while.
Mathematics: Its Content, Methods and Meaning https://www.amazon.com/Mathematics-Content-Methods-Meaning-Volumes/dp/0486409163
This is a monster book written by Kolmogorov, a famous probabilist and educator in maths. It will take you from very basic maths all the way to Topology, Analysis and Group Theory. It is however intended as an overview rather than an exhaustive textbook on all of the theorems, proofs and definitions you need to get to higher math.
For relearning foundations so that they're super strong I can only recommend:
Engineering Mathematics
https://www.amazon.co.uk/Engineering-Mathematics-K-Stroud/dp/1403942463
Engineering Mathematics is full of problems and each one is explained in detail. For getting your foundational, mechanical tools perfect, I'd recommend doing every problem in this book.
For low level problem solving I'd recommend going through the ENTIRE Art of Problem Solving curriculum (starting from Prealgebra).
https://www.artofproblemsolving.com/store/list/aops-curriculum
You might learn a thing or two about thinking about mathematical objects in new ways (as an example. When Prealgebra teaches you to think about inverses it forces you to consider 1/x as an object in its own right rather than 1 divided by x and to prove things. Same thing with -x. This was eye opening for me when I was making the transition from mechanical to more proof based maths.)
If you just want to know about what's going on in higher math then you can make do with:
The Princeton Companion to Mathematics
https://www.amazon.co.uk/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809
I've never read it but as far as I understand it's a wonderful book that cherry picks the coolest ideas from higher maths and presents them in a readable form. May require some base level of math to understand
EDIT: Further down the Napkin Project by Evan Chen was recommended by /u/banksyb00mb00m (http://www.mit.edu/~evanchen/napkin.html) which I think is awesome (it is an introduction to lots of areas of advanced maths for International Mathematics Olympiad competitors or just High School kids that are really interested in maths) but should really be approached post getting a strong foundation.
http://www.amazon.com/Mathematics-Its-Content-Methods-Meaning/dp/0486409163
Bayes is the way to go: Ed Jayne's text Probability Theory is fundamental and a great read. Free chapter samples are here. Slightly off topic, David Mackay's free text is also wonderfully engaging.
How To Prove It
Have I got the book for you, op
https://www.amazon.com/dp/0521675995/ref=cm_sw_r_cp_apa_i_2EmtDbDDEMPG9
Learning proofs can mean different things in different contexts. First, a few questions:
The sort of recommendations for a pre-university student are likely to be very different from those for a university student. For example, high school students have a number of mathematics competitions that you could consider (at least in The United States; the structure of opportunities is likely different in other countries). At the university level, you might want to look for something like a weekly problem solving seminar. These often have as their nominal goal preparing for the Putnam, which can often feel like a VERY ambitious way to learn proofs, akin to learning to swim by being thrown into a lake.
As a general rule, I'd say that working on proof-based contest questions that are just beyond the scope of what you think you can solve is probably a good initial source of problems. You don't want something so difficult that it's simply discouraging. Further, contest questions typically have solutions available, either in printed books or available somewhere online.
This may be especially true for things like logic and very elementary set theory.
Some recommendations will make a lot more sense if, for example, you have access to a quality university-level library, since you won't have to spend lots of money out-of-pocket to get copies of certain textbooks. (I'm limiting my recommendations to legally-obtained copies of textbooks and such.)
Imagine trying to learn a foreign language without being able to practice it with a fluent speaker, and without being able to get any feedback on how to improve things. You may well be able to learn how to do proofs on your own, but it's orders of magnitude more effective when you have someone who can guide you.
Put differently, is your current goal to be able to produce a proof that will satisfy yourself, or to produce a proof that will satisfy someone else?
Have you had at least, for example, a geometry class that's proof-based?
Proofs are all about communicating ideas. If you struggle with writing in complete, grammatically-correct sentences, then that will definitely be a bottleneck to your ability to make progress.
---
With those caveats out of the way, let me make a few suggestions given what I think I can infer about where you in particular are right now.
Learning proofs is in many ways a skill that requires cultivation. Accordingly, you'll need to be patient and persistent, because proof-writing isn't a skill one typically can acquire passively.
---
How to improve at proofs is a big question beyond the scope of what I can answer in a single reddit comment. Nonetheless, I hope this helps point you in some useful directions. Good luck!
Read and work through this book: http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521675995/ref=sr_1_1?ie=UTF8&amp;qid=1301319337&amp;sr=8-1
How to Prove It: A Structured Approach by Velleman is good for developing general proof writing skills.
How to Think About Analysis by Lara Alcock beautifully deconstructs all the major points of Analysis(proofs included).
I think everyone is on point for the most part, but I'd like to be the devil's advocate and suggest a different route.
Learn logic, proof techniques and set theory as early as possible. It will aid you in further study of all 'types' of math and broaden your mind in a general sense. This book is a perfect place to start.
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
The best part is, when you start doing proofs you realize you've been thinking about math all wrong (at least I did). It's an exercise in creativity, not calculation.
In my mind, set theory & calculus are necessary pre-requisites to probability anyway, and linear algebra means much more once you have been introduced to inductive proofs, as well.
Perhaps rather than concentrating on these particular proofs you should look at something like How To Prove It.
If your intent is to take a class like analysis, you really should look into something like logic.
Daniel Velleman wrote an excellent little book called How to Prove It: A Structured Approach. It's actually designed for High School level students, but it works through the subject incredibly well.
Here's an Amazon link to the book:
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1333383091&amp;sr=1-1
For mathematical statistics: Statistical Inference.
Bioinformatics and Statistics: Statistical Methods in Bioinformatics.
R: R in a Nutshell.
Edit: The Elements of Statistical Learning (free PDF!!)
ESL is a great book, but it can get very difficult very quickly. You'll need a solid background in linear algebra to understand it. I find it delightfully more statistical than most machine learning books. And the effort in terms of examples and graphics is unparalleled.
I hope I am not too late.
You can post this to /r/suicidewatch.
Here is my half-baked attempt at providing you with some answers.
First of all let's see, what is the problem? Money and women. This sounds rather stereotypical but it became a stereotype because a lot of people had this kind of problems. So if you are bad at money and at women, join the club, everybody sucks at this.
Now, there are a few strategies of coping with this. I can tell you what worked for me and perhaps that will help you too.
I guess if there is only one thing that I would change in your attitude that would improve anything is learning the fact that "there is more where that came from". This is really important in girl problems and in money problems.
When you are speaking with a girl, I noticed that early on, men tend to start being very submissive and immature in a way. They start to offer her all the decision power because they are afraid not to lose her. This is a somehow normal response but it affects the relationship negatively. She sees you as lacking power and confidence and she shall grow cold. So here lies the strange balance between good and bad: you have to be powerful but also warm and magnanimous. You can only do this by experimenting without fearing the results of your actions. Even if the worst comes to happen, and she breaks up with you .... you'll always get a better option. There are 3.5 billion ladies on the planet. The statistics are skewed in your favor.
Now for the money issue. Again, there is more where that came from. The money, are a relatively recent invention. Our society is built upon them but we survived for 3 million years without them. The thing you need to learn is that your survival isn't directly related to money. You can always get food, shelter and a lot of other stuff for free. You won't live the good life, but you won't die. So why the anxiety then?
Question: It seems to me you are talking out of your ass. How do I put into practice this in order to get a girlfriend?
Answer: Talk to people. Male and female. Make the following your goals:
Talk to 1 girl each day for one month.
Meet a few friends each 3 days.
Make a new friend each two weeks.
Post your romantic encounters in /r/seduction.
This activities will add up after some time and you will have enough social skill to attract a female. You will understand what your female friend is thinking. Don't feel too bad if it doesn't work out.
Question: The above doesn't give a lot of practical advice on getting money. I want more of that. How do I get it?
Answer: To give you money people need to care about you. People only care about you when you care about them. This is why you need to do the following:
Start solving hard problems.
Start helping people.
Problems aren't only school problems. They refer to anything: start learning a new difficult subject (for example start learning physics or start playing an instrument or start writing a novel). Take up a really difficult project that is just above the verge of what you think you are able to do. Helping people is something more difficult and personal. You can work for charity, help your family members around the house and other similar.
Question: I don't understand. I have problems and you are asking me to work for charity, donate money? How can giving money solve anything?
Answer: If you don't give, how can you receive? Helping others is instilling a sense of purpose in a very strange way. You become superior to others by helping them in a dispassionate way.
Question: I feel like I am going to cry, you are making fun of me!
Answer: Not entirely untrue. But this is not the problem. The problem is that you are taking yourself too serious. We all are, and I have similar problems. The true mark of a person of genius is to laugh at himself. Cultivate your sense of humor in any manner you can.
Question: What does it matter then if I choose to kill myself?
Answer: There is this really good anecdote about Thales of Miletus (search wiki). He was preaching that there is no difference between life and death. His friends asked him: If there is no difference, why don't you kill yourself. At this, he instantly answered: I don't kill myself because there is no difference.
Question: Even if I would like to change and do the things you want me to do, human nature is faulty. It is certain that I would have relapses. How do I snap out of it?
Answer: There are five habits that you should instill that will keep bad emotions away. Either of this habits has its own benefits and drawbacks:
Question: Your post seems somewhat interesting but more in an intriguing kind of way. I would like to know more.
Answer: There are a few good books on these subjects. I don't expect you to read all of them, but consider them at least.
For general mental change over I recommend this:
http://www.amazon.com/Learned-Optimism-Change-Your-Mind/dp/1400078393/ref=sr_1_1?ie=UTF8&amp;qid=1324795853&amp;sr=8-1
http://www.amazon.com/Generous-Man-Helping-Others-Sexiest/dp/1560257288
For girl issues I recommend the following book. This will open up a whole bag of worms and you will have an entire literature to pick from. This is not going to be easy. Remember though, difficult is good for you.
http://www.amazon.com/GAME-UNDERCOVER-SOCIETY-PICK-UP-ARTISTS/dp/1841957518/ref=sr_1_1?ie=UTF8&amp;qid=1324795664&amp;sr=8-1 (lately it is popular to dish this book for a number of reasons. Read it and decide for yourself. There is a lot of truth in it)
Regarding money problem, the first thing is to learn to solve problems. The following is the best in my opinion
http://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X
The second thing about money is to understand why our culture seems wrong and you don't seem to have enough. This will make you a bit more comfortable when you don't have money.
http://www.amazon.com/Story-B-Daniel-Quinn/dp/0553379011/ref=sr_1_3?ie=UTF8&amp;qid=1324795746&amp;sr=8-3 (this one has a prequel called Ishmael. which people usually like better. This one is more to my liking.)
For mental contemplation there are two recommendations:
http://www.urbandharma.org/udharma4/mpe.html . This one is for meditation purposes.
http://www.amazon.com/Way-Pilgrim-Continues-His/dp/0060630175 . This one is if you want to learn how to pray. I am an orthodox Christian and this is what worked for me. I cannot recommend things I didn't try.
For exercising I found bodyweight exercising to be one of the best for me. I will recommend only from this area. Of course, you can take up weights or whatever.
http://www.amazon.com/Convict-Conditioning-Weakness-Survival-Strength/dp/0938045768/ref=sr_1_1?ie=UTF8&amp;qid=1324795875&amp;sr=8-1 (this is what I use and I am rather happy with it. A lot of people recommend this one instead: http://www.rosstraining.com/nevergymless.html )
Regarding friends, the following is the best bang for your bucks:
http://www.amazon.com/How-Win-Friends-Influence-People/dp/1439167346/ref=sr_1_1?ie=UTF8&amp;qid=1324796461&amp;sr=8-1 (again, lots of criticism, but lots of praise too)
The rest of the points are addressed in the above books. I haven't given any book on financial advices. Once you know how to solve problems and use google and try to help people money will start coming, don't worry.
I hope this post helps you, even though it is a bit long and cynical.
Merry Christmas!
On a more serious note, this book by Polya is wonderful.
This should keep you busy, but I can suggest books in other areas if you want.
Math books:
Algebra: http://www.amazon.com/Algebra-I-M-Gelfand/dp/0817636773/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1251516690&amp;sr=8
Calc: http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1356152827&amp;sr=1-1&amp;keywords=spivak+calculus
Calc: http://www.amazon.com/Linear-Algebra-Dover-Books-Mathematics/dp/048663518X
Linear algebra: http://www.amazon.com/Linear-Algebra-Modern-Introduction-CD-ROM/dp/0534998453/ref=sr_1_4?ie=UTF8&amp;s=books&amp;qid=1255703167&amp;sr=8-4
Linear algebra: http://www.amazon.com/Linear-Algebra-Dover-Mathematics-ebook/dp/B00A73IXRC/ref=zg_bs_158739011_2
Beginning physics:
http://www.amazon.com/Feynman-Lectures-Physics-boxed-set/dp/0465023827
Advanced stuff, if you make it through the beginning books:
E&M: http://www.amazon.com/Introduction-Electrodynamics-Edition-David-Griffiths/dp/0321856562/ref=sr_1_1?ie=UTF8&amp;qid=1375653392&amp;sr=8-1&amp;keywords=griffiths+electrodynamics
Mechanics: http://www.amazon.com/Classical-Dynamics-Particles-Systems-Thornton/dp/0534408966/ref=sr_1_1?ie=UTF8&amp;qid=1375653415&amp;sr=8-1&amp;keywords=marion+thornton
Quantum: http://www.amazon.com/Principles-Quantum-Mechanics-2nd-Edition/dp/0306447908/ref=sr_1_1?ie=UTF8&amp;qid=1375653438&amp;sr=8-1&amp;keywords=shankar
Cosmology -- these are both low level and low math, and you can probably handle them now:
http://www.amazon.com/Spacetime-Physics-Edwin-F-Taylor/dp/0716723271
http://www.amazon.com/The-First-Three-Minutes-Universe/dp/0465024378/ref=sr_1_1?ie=UTF8&amp;qid=1356155850&amp;sr=8-1&amp;keywords=the+first+three+minutes
Textbooks (calculus): Fundamentals of Physics: http://www.amazon.com/Fundamentals-Physics-Extended-David-Halliday/dp/0470469080/ref=sr_1_4?ie=UTF8&amp;qid=1398087387&amp;sr=8-4&amp;keywords=fundamentals+of+physics ,
Textbooks (calculus): University Physics with Modern Physics; http://www.amazon.com/University-Physics-Modern-12th-Edition/dp/0321501217/ref=sr_1_2?ie=UTF8&amp;qid=1398087411&amp;sr=8-2&amp;keywords=university+physics+with+modern+physics
Textbook (algebra): [This is a great one if you don't know anything and want a book to self study from, after you finish this you can begin a calculus physics book like those listed above]: http://www.amazon.com/Physics-Principles-Applications-7th-Edition/dp/0321625927/ref=sr_1_1?ie=UTF8&amp;qid=1398087498&amp;sr=8-1&amp;keywords=physics+giancoli
If you want to be competitive at the international level, you definitely need calculus, at least the basics of it.
Here is a good book: http://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536/ref=sr_1_1?ie=UTF8&amp;qid=1398087834&amp;sr=8-1&amp;keywords=calculus+kline
It is quite cheap and easy to understand if you want to self teach yourself calculus.
Another option would be this book:http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?ie=UTF8&amp;qid=1398087878&amp;sr=8-1&amp;keywords=spivak
If you can finish self teaching that to yourself, you will be ready for anything that could face you in mathematics in university or the IPhO. (However it is a difficult book)
Problem books: Irodov; http://www.amazon.com/Problems-General-Physics-I-E-Irodov/dp/8183552153/ref=sr_1_1?ie=UTF8&amp;qid=1398087565&amp;sr=8-1&amp;keywords=irodov ,
Problem Books: Krotov; http://www.amazon.com/Science-Everyone-Aptitude-Problems-Physics/dp/8123904886/ref=sr_1_1?ie=UTF8&amp;qid=1398087579&amp;sr=8-1&amp;keywords=krotov
You should look for problem sets online after you have finished your textbook, those are the best recourses. You can get past contests from the physics olympiad websites.
In addition to linear regression, do you need a reference for future use/other topics? Casella/Berger is a good one.
For linear regression, I really enjoyed A Modern Approach to Regression with R.
Casella and Berger is a fairly standard text for first-year graduate Math Stats courses. It's not the most detailed or exhaustive text on the topic, but it covers the main points and is fairly accessible.
>So, my question is- Would you recommend me to skip right into the formal logic parts (and things related, such as computer programs) when reading the book?
I dunno, it depends on what you're trying to get out of the book, I guess. If you just want an exposition of Gödel's incompleteness theorems you can skip to the logic parts, but if that's your goal then there are better books that will get you there faster and more rigorously, like Gödel's Proof by Newman and Nagel, and, incidentally, edited by Hofstadter.
If you're looking for a concise introductory level reference, I don't know of any at only the high-school level; additionally most undergrad level textbooks are gonna assume a certain level of sophistication w.r.t. the student.
However, if you are interested, the book "Godel's Proof" by Nagel, offers many accessible insights into the workings of mathemical logic
https://www.amazon.com/Gödels-Proof-Ernest-Nagel/dp/0814758371
I know this is not exactly what you had in mind, but one of the most significant proofs of the 20th century has an entire book written about it:
http://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371
The proof they cover is long and complicated, but the book is nonetheless intended for the educated layperson. It is very, very well written and goes to great lengths to avoid unnecessary mathematical abstraction. Maybe check it out.
Well I don't know how interested you are in this, but if you want to understand the incompleteness theorem and its implications without learning all of number theory, I ran across this book which provides the history leading up to Gödel, the mathematical context he was working in (e.g. Hilbert's project), and a full explanation of the proof itself in just over 100 pages. I read it in a day, and while I have a background in the area, even if you didn't know anything going into it, you could probably understand the whole thing with two days' careful reading.
This is a compilation of what I gathered from reading on the internet about self-learning higher maths, I haven't come close to reading all this books or watching all this lectures, still I hope it helps you.
General Stuff:
The books here deal with large parts of mathematics and are good to guide you through it all, but I recommend supplementing them with other books.
Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.
Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.
Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.
Abstract Algebra: One of the most important, and in my opinion fun, subjects in mathematics. Deals with algebraic structures, which are roughly sets with operations and properties of this operations.
There are many other beautiful fields in math full of online resources, like Number Theory and Combinatorics, that I would like to put recommendations here, but it is quite late where I live and I learned those in weirder ways (through olympiad classes and problems), so I don't think I can help you with them, still you should do some research on this sub to get good recommendations on this topics and use the General books as guides.
http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?ie=UTF8&amp;qid=1344481564&amp;sr=8-1&amp;keywords=spivaks+calculus
I haven't read all of it, but even the bit I did read was very challenging and it is generally recommended around here for a rigorous introduction to calculus. Be warned, it is pretty challenging, especially if you aren't comfortable with proofs.
Best:
Principles of Mathematical Analysis by Walter Rudin
Baby Rudin
Check this book out!
It absolutely changed my mental math ability. Arthur Benjamin also has videos all over the Internet with some quick mental math tricks.
I have a book on mental math, and this is essentially the technique that the author uses to square numbers mentally really quickly.
In other words,
x^2 = (x+k)(x-k) + k^2
where you substitute x's into the equation you gave.
This is the book.
Thanks for the suggestions! Just so you are aware the Fermat's Enigma link is a duplicate of Journey through Genius.
Journey through Genius sounds really interesting. I'm curious if you've ever read Gödel, Escher, Bach? If so how would you compare the two?
Lang's Basic mathematics might cover what you need.
The Art of Problem Solving has algebra books that focus a bit more on learning through problem solving than your average textbook. Also, Serge Lang's Basic Mathematics is a book about high school math written at a fairly high level.
I agree that there's an unfortunate tendency toward "cookbook mathematics" out there. On the topic you brought up, note that there isn't a general method of factoring polynomials by hand, so there isn't necessarily anything they could teach you that would subsume all other knowledge. However, I'd say learning by solving problems rather than memorizing unmotivated algorithms is better when possible.
We used the Dover textbook by Pinter. It's my favorite math textbook ever, the writing was just so clear, and even entertaining and funny. We had a good professor too.
A First Course in Graph Theory by Chartrand and Zhang
Combinatorics: A Guided Tour by Mazur
Discrete Math by Epp
For Linear Algebra I like these below:
Lecture Notes by Tao
Linear Algebra: An Introduction to Abstract Mathematics by Robert Valenza
Linear Algebra Done Right by Axler
Linear Algebra by Friedberg, Insel and Spence
Data Structures & Algorithms is usually the second course after Programming 101. Here is a progression (with the books I'd use) I would recommend to get started:
Edit: If you're feeling adventurous then after those you should look at
I'm only part way through it myself, but here's one I've been recomended in the past that I've been enjoying so far:
Probability Theory: The Logic of Science by E.T. Jaynes
http://www.amazon.com/Probability-Theory-The-Logic-Science/dp/0521592712
http://omega.albany.edu:8008/JaynesBook.html
The second link only appears to have the first three chapters in pdf (though it has everything as postscript files), but I would be shocked if you couldn't easilly find a free pdf off the whole thing online with a quick search.
Jaynes: Probability Theory. Perhaps 'rigorous' is not the first word I'd choose to describe it, but it certainly gives you a thorough understanding of what Bayesian methods actually mean.
This image was used for the cover of a famous text on error analysis.
Sorry, the solution is to do lots of proofs.
There's more to it, but honestly it's more of a thing that you have to read a book about rather than a message on reddit. How are you learning about this right now? Is it part of a course or self-study? I personally found How to Prove It to be a very useful textbook. Doesn't require any particular knowledge, and it builds out a nice foundation in logic and set theory.
How To Prove It. Read through the reviews. It's the best book for learning propositional and predicate logic for the first time.
You will first want to learn fundamental logic and set theory before diving into topics like analysis, algebra, and discrete topics. You will need an understanding of a rigorous proof -- not the hand-wavey kind of proof we've seen in our introductory calculus courses. This book is very readable and will prepare you for advanced mathematics. I've seen it work for many students.
After you're finished with it, you may want to study analysis which will build up the Calculus for you. If you don't care for calculus anymore, consider reading an abstract algebra text. Algebra is pretty fun. You can also pick a discrete topic like graph theory or combinatorics whose applications are very easy to see.
There are many ways to go, but in all of them you will absolutely need a a basic understanding of the use of logic in a mathematical proof.
What this expressions says
First of all let's specify that the domain over which these statements operate is the set of all people say.
Let us give the two place predicate P(x,y) a concrete meaning. Let us say that P(x,y) signifies the relation x loves y.
This allows us to translate the statement:
∀x∀yP(x,y) -> ∀xP(x,x)
What does ∀x∀yP(x,y) mean?
This is saying that For all x, it is the case that For all y, x loves y.
So you can interpret it as saying something like everyone loves everyone.
What does ∀xP(x,x) mean?
This is saying that For all x it is the case that x loves x. So you can interpret this as saying something like everyone loves themselves.
So the statement is basically saying:
Given that it is the case that Everyone loves Everyone, this implies that everyone loves themselves.
This translation gives us the impression that the statement is true. But how to prove it?
Proof by contradiction
We can prove this statement with a technique called proof by contradiction. That is, let us assume that the conclusion is false, and show that this leads to a contradiction, which implies that the conclusion must be true.
So let's assume:
∀x∀yP(x,y) -> not ∀xP(x,x)
not ∀xP(x,x) is equivalent to ∃x not P(x,x).
In words this means It is not the case that For all x P(x,x) is true, is equivalent to saying there exists x such P(x,x) is false.
So let's instantiate this expression with something from the domain, let's call it a. Basically let's pick a person for whom we are saying a loves a is false.
not P(a,a)
Using the fact that ∀x∀yP(x,y) we can show a contradiction exists.
Let's instantiate the expression with the object a we have used previously (as a For all statement applies to all objects by definition) ∀x∀yP(x,y)
This happens in two stages:
First we instantiate y
∀xP(x,a)
Then we instantiate x
P(a,a)
The statements P(a,a) and not P(a,a) are contradictory, therefore we have shown that the statement:
∀x∀yP(x,y) -> not ∀xP(x,x) leads to a contradiction, which implies that
∀x∀yP(x,y) -> ∀xP(x,x) is true.
Hopefully that makes sense.
Recommended Resources
Wilfred Hodges - Logic
Peter Smith - An Introduction to Formal Logic
Chiswell and Hodges - Mathematical Logic
Velleman - How to Prove It
Solow - How to Read and Do Proofs
Chartand, Polimeni and Zhang - Mathematical Proofs: A Transition to Advanced Mathematics
Try the two:
https://www.amazon.com/Introduction-Mathematical-Statistics-Robert-Hogg/dp/0321795431
https://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126
introduction to mathematical statistics by craig and statistical inference by george casella.
Axler's Linear Algebra Done Right is something you might enjoy looking at; since his basic point of view is that linear algebra is generally done wrong.
http://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/0387982582
Hey! I am a math major at Harvey Mudd College (who went to high school in the Pacific NW!). I'll answer from what I've seen.
End: Also, if you wanna learn something cool, I'd check out Discrete math. It's usually required for both a math or CS major, and it's some of the coolest undergraduate math out there. Oh, and, unlike some other math, it's not terrible to self-teach. :)
Good luck! Math is awesome!
Not sure if they qualify as "beautifully written", but I've got two that are such good reads that I love to go back to them from time to time:
You're English is great.
I'd like to reemphasize /u/Plaetean's great suggestion of learning the math. That's so important and will make your later career much easier. Khan Academy seems to go all through differential equations. All of the more advanced topics they have differential and integral calculus of the single variable, multivariable calculus, ordinary differential equations, and linear algebra are very useful in physics.
As to textbooks that cover that material I've heard Div, Grad, Curl for multivariable/vector calculus is good, as is Strang for linear algebra. Purcell an introductory E&M text also has an excellent discussion of the curl.
As for introductory physics I love Purcell's E&M. I'd recommend the third edition to you as although it uses SI units, which personally I dislike, it has far more problems than the second, and crucially has many solutions to them included, which makes it much better for self study. As for Mechanics there are a million possible textbooks, and online sources. I'll let someone else recommend that.
/u/LengthContracted this is a good book, as is Daphne Kollers book on PGMs as well as the associated course http://pgm.stanford.edu
A sample of what is on my reference shelf includes:
Real and Complex Analysis by Rudin
Functional Analysis by Rudin
A Book of Abstract Algebra by Pinter
General Topology by Willard
Machine Learning: A Probabilistic Perspective by Murphy
Bayesian Data Analysis Gelman
Probabilistic Graphical Models by Koller
Convex Optimization by Boyd
Combinatorial Optimization by Papadimitriou
An Introduction to Statistical Learning by James, Hastie, et al.
The Elements of Statistical Learning by Hastie, et al.
Statistical Decision Theory by Liese, et al.
Statistical Decision Theory and Bayesian Analysis by Berger
I will avoid listing off the entirety of my shelf, much of it is applications and algorithms for fast computation rather than theory anyway. Most of those books, though, are fairly well known and should provide a good background and reference for a good deal of the mathematics you should come across. Having a solid understanding of the measure theoretic underpinnings of probability and statistics will do you a great deal--as will a solid facility with linear algebra and matrix / tensor calculus. Oh, right, a book on that isn't a bad idea either... This one is short and extends from your vector classes
Tensor Calculus by Synge
Anyway, hope that helps.
Yet another lonely data scientist,
Tim.
http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178/ref=sr_1_1?ie=UTF8&amp;qid=1394386195&amp;sr=8-1&amp;keywords=pinter+abstract+algebra)
I've read some good reviews of Basic Mathematics by Serge Lang. It should prepare the reader for calculus.
Otherwise, many online and free books are already available. Here you find a list of free books approved by the American Institute of Mathematics.
If you want to understand the WHY, then you need to read proofs and at least be familiar with basic concepts of logic. I've found this site really helpful. It's a source for definitions and proofs.
https://www.amazon.fr/gp/product/0387967877/ref=ppx_yo_dt_b_asin_title_o09_s03?ie=UTF8&amp;psc=1
Il part vraiment de 0 et présente la construction des mathématiques à partir d'éléments très simples. Il faut comprendre l'anglais par contre, il y a peut-être des traductions.
If you're dislike of linear algebra comes from using the determinant and matrix calculations, you would love Axler's Linear Algebra Done Right.
There are some really good books that you can use to give yourself a solid foundation for further self-study in mathematics. I've used them myself. The great thing about this type of book is that you can just do the exercises from one side of the book to the other and then be confident in the knowledge that you understand the material. It's nice! Here are my recommendations:
First off, three books on the basics of algebra, trigonometry, and functions and graphs. They're all by a guy called Israel Gelfand, and they're good: Algebra, Trigonometry, and Functions and Graphs.
Next, one of two books (they occupy the same niche, material-wise) on general proof and problem-solving methods. These get you in the headspace of constructing proofs, which is really good. As someone with a bachelors in math, it's disheartening to see that proofs are misunderstood and often disliked by students. The whole point of learning and understanding proofs (and reproducing them yourself) is so that you gain an understanding of the why of the problem under consideration, not just the how... Anyways, I'm rambling! Here they are: How To Prove It: A Structured Approach and How To Solve It.
And finally a book which is a little bit more terse than the others, but which serves to reinforce the key concepts: Basic Mathematics.
After that you have the basics needed to take on any math textbook you like really - beginning from the foundational subjects and working your way upwards, of course. For example, if you wanted to improve your linear algebra skills (e.g. suppose you wanted to learn a bit of machine learning) you could just study a textbook like Linear Algebra Done Right.
The hard part about this method is that it takes a lot of practice to get used to learning from a book. But that's also the upside of it because whenever you're studying it, you're really studying it. It's a pretty straightforward process (bar the moments of frustration, of course).
If you have any other questions about learning math, shoot me a PM. :)
I learned lin. alg. from Axler's Linear Algebra Done Right. I found it extremely readable, with exercises that were not too hard to get through quickly.
I'd suggest Probability, Linear Algebra, Convex Optimization and ML in that order.
As for study materials, I'd suggest
That should keep you busy for a while.
I looked at the free pages on Amazon and it does seem a bit wordier than the physics books I remember. It could just be the chapter. Maybe it reads like a book; maybe it's incredibly boring :/
If money isn't an issue (or if you're resourceful and internet savvy ;) you can try the book by Serway & Jewett. It's fairly common.
http://www.amazon.com/Physics-Scientists-Engineers-Raymond-Serway/dp/1133947271
As for DE, this book really resonated with me for whatever reason. Your results may vary.
http://www.amazon.com/Course-Differential-Equations-Modeling-Applications/dp/1111827052/ref=sr_1_2?s=books&amp;ie=UTF8&amp;qid=1372632638&amp;sr=1-2&amp;keywords=differential+equations+gill
If your issue is with the technical nature of textbooks in general, then you'll either have to deal with it or look for some books that simplify/summarize the material in some way. The only example I can come up with is:
http://www.amazon.com/Div-Grad-Curl-All-That/dp/0393925161/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1372632816&amp;sr=1-1&amp;keywords=div+grad+curl
Although Div, Grad, Curl, and all That is intended for students in an Electromagnetics course (not Physics 2), it might be helpful. It's an informal overview of Calculus 3 integrals and techniques. The book uses electromagnetism in its examples. I don't think it covers electric circuits, which are a mess of their own. However, there are tons of resources on the internet for circuits. I hope all this was helpful :)
Div Grad Curl and all that
The Art of Electronics
Are you familiar with Div, Grad, Curl, & All That. If not you'd probably enjoy it.
let me give you a shortcut.
You want to know how partial derivatives work? Consider a function with two variables: f(x,y) = x^2 y^3, for a simple example.
here's what you do. Let's take the partial derivative with respect to x. What you do, is you consider all the other variables to be constant, and just take the standard derivative with respect to x. In this case, the partial derivative with respect to x is: 2xy^3. That's it, it's really that easy.
What about taking with respect to y? Same thing, now x is constant, and your answer is 3x^2 y^2.
This is an incredibly deep topic, but getting enough of an understanding to tackle gradient descent is really pretty simple. If you want to full on jump in though and get some exposure to way more than you need, check out div curl and grad and all that. It covers a lot, including a fair amount that you won't need for any ML algorithm I've ever seen (curl, divergence theorem, etc) but the intro section on the gradient at the beginning might be helpful... maybe see if you can find a pdf or something. There's probably other good intros too, but seriously... the mechanics of actually performing a partial derivative really are that easy. If you can do a derivative in one dimension, you can handle partial derivatives.
edit: I misread, didn't see you were a junior in highschool. Disregard div curl grad and all that, I highly recommend it, but you should be up through calc 3 and linear algebra first.
To change my advice to be slightly more relevant, learn how normal derivatives work. Go through the Kahn Academy calc stuff if the format appeals to you. Doesn't matter what course you go through though, you just need to go through a few dozen exercises (or a few hundred, depending on your patience and interest) and you'll get there. Derivatives aren't too complicated really, if you understand the limit definition of the derivative (taking the slope over a vanishingly small interval) then the rest is just learning special cases. How do you take the derivative of f(x)g(x)? f(g(x))? There's really not too many rules, so just spend a while practicing and you'll be right where you need to be. Once you're there, going up to understanding partial derivatives is as simple as I described above... if you can take a standard derivative, you can take a partial derivative.
Also: props for wading into the deep end yourself! I know some of this stuff might seem intimidating, but if you do what you're doing (make sure you understand as much as you can instead of blowing ahead) you'll be able to follow this trail as far as you want to go. Good luck, and feel free to hit me up if you have any specific questions, I'd be happy to share.
I'm soon starting my trek through every problem in the algebra text that Harvard's PhD prelim recommends for study:
Abstract Algebra by Dummit and Foote
I've started the first section of the first chapter, but that was only in a few hours of spare time. I'll be posting solutions by chapter soon and post my stories/insights on Hacker News. Here's section 1.1 (except the last problem, 36):
http://therobert.org/alg/1.1.pdf
Comments are appreciated. Better now than when I start the real journey. :)
Well, Hardy & Wright is the classic book for elementary stuff. It has almost everything there is to know. There is also a nice book by Melvyn Nathanson called Elementary Methods in Number Theory which I really like and would probably be my first recommendation. Beyond that, you need to decide which flavour you like. Algebraic and analytic are the big branches.
For algebraic number theory you'll need a solid grounding in commutative algebra and Galois theory - say at the level of Dummit and Foote. Lang's book is pretty classic, but maybe a tough first read. I might try Number Fields by Marcus.
For analytic number theory, I think Davenport is the best option, although Montgomery and Vaughan is also popular.
Finally, Serre (who is often deemed the best math author ever) has the classic Course in Arithmetic which contains a bit of everything.
If you are asking for classics, in Algebra, for example, there are(different levels of difficulty):
Basic Algebra by Jacobson
Algebra by Lang
Algebra by MacLane/Birkhoff
Algebra by Herstein
Algebra by Artin
etc
But there are other books that are "essential" to modern readers:
Chapter 0 by Aluffi
Basic Algebra by Knapp
Algebra by Dummit/Foot
I learned it out of Dummit and Foote originally, and I thought that was a pretty good book.
I know that in the long run competition math won't be relevant to graduate school, but I don't think it would hurt to acquire a broader background.
That said, are there any particular texts you would recommend? For Algebra, I've heard that Dummit and Foote and Artin are standard texts. For analysis, I've heard that Baby Rudin and also apparently the Stein-Shakarchi Princeton Lectures in Analysis series are standard texts.
I recommend this:
https://www.amazon.com/Mathematics-Content-Methods-Meaning-Volumes/dp/0486409163
Unlike most professional mathematical literature, it is aimed at novices and attempts to communicate ideas, not details. Unlike most popular treatments of mathematics, and in particular unlike the YouTubers you mention, it is written by expert mathematicians and is about advanced mathematical topics. I got a hardcover set from a used bookstore when I was young and enjoyed it very much. It's well worth your time.
Not quite encyclopaedic, but this gives a good overview of most topics you might encounter in an undergraduate course. The first section also gives a very good defense of the need for basic research into mathematics.
There's a lot of ground to cover in math, but completely doable. I'm going to recommend a dense book, but I truly think it's worth the read.
Let me leave you with this. You understand how number work correct? 1 + 1 = 2. It's a matter of fact. It's not up for debate and to question it would see you insane.
This is all of math. You need to truly understand
1 + 1 = 2
a + a = b everything is a function. There are laws to everything, even if people wish to deny it. If we don't understand it, it's easier to state that there are no laws that govern it, but there are. You just don't know them yet. Math isn't overwhelming when you think of it that way, at least to me. It's whole.
Ask yourself, 'why does 1 + 1 = 2 ?' If you were given 1 + x = 2, how would you solve it? Why exactly would you solve it that way? What governing set of rules are you using to solve the equation? You don't need to memorize the names of the rules, but how to use them. Understand the terminology in math, or any language, and it's easier to grasp that language.
The book Mathematics
https://www.amazon.com/gp/product/0486409163/ref=ya_st_dp_summary
This does not specifically target game programmers. However, it's not just specific categories of math that is important to game programmers. It's EVERYTHING math related. And knowing the meaning of it and understanding is more important than just a formula.
The book I just linked is an amazing book. It is well written, and avoids academia where possible. It's balance between math and explination is just right where it can effectively get the point across, and even help you understand more complex explinations.
This book features three volumes, and each volume goes over a wide array of topics in depth.
A Book of Abstract Algebra by Charles C. Pinter
I really enjoyed reading the book, almost reads like a novel. There is a great first chapter laying out the history of the subject and it just builds from there.
Math is essential the art pf careful reasoning and abstraction.
Do yes, definitely.
But it may be difficult at first, like training anything that’s not been worked.
Note: there are many varieties of math. I definitely recommend trying different ones.
A couple good books:
An Illustrated Theory of Numbers
Foolproof (first chapter is math history, but you can skip it to get to math)
A Book of Abstract Algebra
Also, formal logic is really fun, imk. And excellent st teaching solid thinking. I don’t know a good intro book, but I’m sure others do.
> I'm not sure what to read into before the Galois class begins.
"A Book of Abstract Algebra" by Charles Pinter
It depends on your interests. I thought the machine learning course on coursera was great. Antirez sometimes blogs about the internals of Redis on his blog, and he is a great writer. If you like math, this is the best math book I've read. Finally, you can always start contributing code to an open source project -- learn by doing!
Try this book
Definitely split up the load and take classes over the summer. I often hear people say Calculus II is the hardest of the EPIC MATH TRILOGY. I certainly agree. If you've done well in Calc I and II and have a notion of what 3d vectors are (physics should of covered this well) then you'll have no problem with Calc III (though series' and summations can be tough).
Differential equations will be your first introduction to hard "pure"-style math concepts. The language will take some time to understand and digest. I highly recommend you purchase this book to supplement your textbook. If you take notes on each chapter and work through the derivations, problems, and solutions, you'll be golden.
In my experience, materials is not math heavy for ME's. All of my tests were multiple choice and more concept based. It's not too bad.
Thermodynamics and Engineering Dynamics will be in the top three as far as difficulty goes. Circuits or Fluids will also be in there somewhere. Make sure you allow plenty of time to study these topics.
Good luck!
I used Tenenbaum. One of my favorite undergrad books. Only downside that it doesn't use any Linear Algebra
The second book that gerschgorin listed is very good, though a little old fashioned.
Since you are finishing up your math major, I'd recommend Hirsch & Smale & Devaney, an excellent book if you have a little bit of mathematical background.
There is also a video series I'm making meant to be a quick overview of many of the key topics. Maybe useful, maybe not. Also, the MIT lectures are excellent.
I don't think it contains any group theory, but everything else is there:
Discrete Mathematics with Applications by Susanna S. Epp (
This one below contains some algebra(groups):
Mathematics: A Discrete Introduction by Edward A. Scheinerman
Both are pretty elementary.
I have not taken the class yet (I'm taking 161 and 225 in January), but I looked at the syllabi already and here's the textbook for the class:
http://www.amazon.com/Discrete-Mathematics-Applications-Kenneth-Rosen/dp/0073383090/ref=sr_1_1?ie=UTF8&amp;qid=1417826968&amp;sr=8-1&amp;keywords=Rosen+Discrete+Math
You may want to go ahead and pick this up and start looking through it prior to January. I already grabbed a copy; I finish Calculus II tomorrow at my community college and I am going to be starting Rosen very soon.
This book is also commonly recommended:
http://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328/ref=sr_1_1?ie=UTF8&amp;qid=1417827137&amp;sr=8-1&amp;keywords=Epp
I'm not sure what your math background is, but one of the most important success factors (in my experience) in math classes is a lot of practice. If you start working through either of those books now, you'll probably be in a good place once class starts in January.
We could also probably get a study group going on in here; I'm pretty comfortable with math, so I am happy to help out anyone else who needs help.
Casella & Berger is the go-to reference (as Smartless has already pointed out), but you may also enjoy Jaynes. I'm not sure I'd say it's quick but if gaps are your concern, it's pretty drum-tight.
it most certainly is! There's a whole approach to statistics based around this idea of updating priors. If you're feeling ambitious, the book Probability theory by Jaynes is pretty accessible.
You received A's in your math classes at a major public university, so I think you're in pretty good shape. That being said, have you done proof-based math? That may help tremendously in giving intuition because with proofs, you are giving rigor to all the logic/theorems/ formulas, etc that you've seen in your previous math classes.
Statistics will become very important in machine learning. So, a proof-based statistics book, that has been frequently recommended by /r/math and /r/statistics is Statistical Inference by Casella & Berger: https://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126
I've never read it myself, but skimming through some of the beginning chapters, it seems pretty solid. That being said, you should have an intro to proof-course if you haven't had that. A good book for starting proofs is How to Prove It: https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
See Casella and Berger chapter 2, theorem 2.1.5
Casella and Berger is one of the go-to references. It is at the advanced undergraduate/first year graduate student level. It's more classical statistics than data science, though.
Good statistical texts for data science are Introduction to Statistical Learning and the more advanced Elements of Statistical Learning. Both of these have free pdfs available.
>Can they be shown to be consistent?
Not by the metamathematics itself, no. It's a result from Gödel's Incompleteness Theorems that no consistent mathematical system that can be mapped into arithmetics can demonstrate it's own consistency.
This book does a good job of explaining Gödel's work, you should consider reading it.
For something more rigorous than "Godel, Escher, Bach" try "Godel's Proof" by Nagel and Newman.
There is a book entitled Godel's proof that was written that was written to explain the ideas of Godel's proof without requiring too much background.
http://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371/ref=sr_1_1?ie=UTF8&amp;qid=1342841590&amp;sr=8-1&amp;keywords=godel%27s+proof
It is hard for me to offer to much advice beyond that because I am in a different field of mathematics (number theory).
it's not a bad book but it's got a bad rap
hofstadter writes the foreword to my favorite book on godel's work, this guy
This is the one I read:
http://www.amazon.com/Gödels-Proof-Ernest-Nagel/dp/0814758371
I like Algebra and Trigonometry by I.M. Gelfand. They are cheap books too.
I also have scans of them, PM me if you want to check them out.
Edit:
Also, Khan Academy is great resource for explanations. But I would recommend aiding Khan Academy with a text just for the problem set and solutions.
https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918
From my experience, Calculus in America is taught in 2 different ways: rigorous/mathematical in nature like Calculus by Spivak and applied/simplified like the one by Larson.
Looking at the link, I dont think you need to know sets and math induction unless you are about to start learning Rigorous Calculus or Real Analysis. Also, real numbers are usually introduced in Real Analysis that comes after one's exposure to Applied/Non-Rigorous Calculus. Complex numbers are, I assume, needed in Complex Analysis that follows Real Analysis, so I wouldn't worry about sets, real/complex numbers beyond the very basics. Math induction is not needed in non-proof based/regular/non-rigorous Calculus.
From the link:
For Calc 1(applied)- again, in my experience, this is the bulk of what's usually tested in Calculus placement exams:
Solving inequalities and equations
Properties of functions
Composite functions
Polynomial functions
Rational functions
Trigonometry
Trigonometric functions and their inverses
Trigonometric identities
Conic sections
Exponential functions
Logarithmic functions
For Calc 2(applied) - I think some Calc placement exams dont even contain problems related to the concepts below, but to be sure, you, probably, should know something about them:
Sequences and series
Binomial theorem
In Calc 2(leading up to multivariate Calculus (Calc 3)). You can pick these topics up while studying pre-calc, but they are typically re-introduced in Calc 2 again:
Vectors
Parametric equations
Polar coordinates
Matrices and determinants
As for limits, I dont think they are terribly important in pre-calc. I think, some pre-calc books include them just for good measure.
If you are interested enough in machine learning that you are going to work through ESL, you may benefit from reading up on some math first. For example:
Without developing some mathematical maturity, some of ESL may be lost on you. Good luck!
My buddy (phd student) told me that if I were to do a reading course, or just want to do self study that I should use Munkres. I think you can find international editions for much cheaper than that. We were using Rudin for our analysis class and spent a lot of time on ch.2. These are my only suggestions because I haven't done much with topology or analysis.
I would try http://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1375896302&amp;sr=1-1 and if that does not truly challenge you, then we can chat some more.
Here are some suggestions :
https://www.coursera.org/course/maththink
https://www.coursera.org/course/intrologic
Also, this is a great book :
http://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X/ref=sr_1_5?ie=UTF8&amp;qid=1346855198&amp;sr=8-5&amp;keywords=history+of+mathematics
It covers everything from number theory to calculus in sort of brief sections, and not just the history. Its pretty accessible from what I've read of it so far.
EDIT : I read what you are taking and my recommendations are a bit lower level for you probably. The history of math book is still pretty good, as it gives you an idea what people were thinking when they discovered/invented certain things.
For you, I would suggest :
http://www.amazon.com/Principles-Mathematical-Analysis-Third-Edition/dp/007054235X/ref=sr_1_1?ie=UTF8&amp;qid=1346860077&amp;sr=8-1&amp;keywords=rudin
http://www.amazon.com/Invitation-Linear-Operators-Matrices-Bounded/dp/0415267994/ref=sr_1_4?ie=UTF8&amp;qid=1346860052&amp;sr=8-4&amp;keywords=from+matrix+to+bounded+linear+operators
http://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753/ref=sr_1_5?ie=UTF8&amp;qid=1346860077&amp;sr=8-5&amp;keywords=rudin
http://www.amazon.com/DIV-Grad-Curl-All-That/dp/0393969975
http://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0738204536/ref=sr_1_2?s=books&amp;ie=UTF8&amp;qid=1346860356&amp;sr=1-2&amp;keywords=chaos+and+dynamics
http://www.amazon.com/Numerical-Analysis-Richard-L-Burden/dp/0534392008/ref=sr_1_5?s=books&amp;ie=UTF8&amp;qid=1346860179&amp;sr=1-5&amp;keywords=numerical+analysis
This is from my background. I don't have a strong grasp of topology and haven't done much with abstract algebra (or algebraic _____) so I would probably recommend listening to someone else there. My background is mostly in graduate numerical analysis / functional analysis. The Furata book is expensive, but a worthy read to bridge the link between linear algebra and functional analysis. You may want to read a real analysis book first however.
One thing to note is that topology is used in some real analysis proofs. After going through a real analysis book you may also want to read some measure theory, but I don't have an excellent recommendation there as the books I've used were all hard to understand for me.
I'm not sure I understand your concern, but if you struggle with math, it may help to start with coding. It can make things a little more concrete. You might try code academy, a coding bootcamp, or MIT open courseware.
An Emory prof has a great intro stats course online: https://www.youtube.com/user/RenegadeThinking
Linear algebra is the foundation of the most widely used branch of stats. This book teaches it by coding example. It's full of interesting practical applications (there's a coursera course to go with it): https://www.amazon.com/Coding-Matrix-Algebra-Applications-Computer/dp/0615880991/ref=sr_1_1?ie=UTF8&amp;qid=1469533241&amp;sr=8-1&amp;keywords=coding+the+matrix
Once you start to feel comfortable, this book offers a great (albeit dense) introduction to mathematics. It used to be used in freshman gen ed math courses, but sadly, American unis decided that actually doing math/logic isn't a priority anymore: https://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192/ref=sr_1_1?ie=UTF8&amp;qid=1469533516&amp;sr=8-1&amp;keywords=what+is+mathematics
"What is Mathematics ? An elementary approach to Ideas and methods ( 2nd Edition) by Richard Courant (Author), Herbert Robbins (Author), Ian Stewart (Editor) "
Try this book for help with understanding Algebra. My uncle had left a copy at my grandparents house, and I picked it up when I was there when I was in the third grade (we were working on multiplication and division). I made a perfect score in the state tests for Algebra 1, Algebra 2, Geometry, and Trigonometry.
I read this book in high school, and it really helped me figure out how to think about breaking down more complex problems.
This book made math very clear for me as well.
I think these books may help you because you could do the math he read to you. These books helps give you an understanding of what is actually happening. Foe example, most people do not understand that multiplication is nothing more than extended addition, until you explain it to them. If you can think about the problems and understand what the problem is saying, it will be easier to figure out. I did a lot of math in my head that would have taken several pages to write it out the way I did it, but if you wrote it the way they expect would only take a few lines.
I am very happy for you for finally finding someone who knew what was going on with you. I had a similar problem in elementary school, but my parents did not trust the school and had me tested on their own. They decided that I had a "social communication disorder, kind of like a really weak autism" (This is what my parents ended up telling me anyway). The school thought I was "developmentally challenged" ("borderline retarded" was the phrase that was bandied about) but when my parents had my IQ tested, it was a 141, which is not quite what was expected, they decided that the problems were elsewhere.
One thing that is very important in math is that if you do not understand, you can go back and work on fundamentals and build up your foundation, and the more advanced stuff will be easier.
Good luck, and I believe you really are an adept writer. What you wrote grabbed my interest and was compelling.
Try What is Mathematics?, by Courant & Robbins. It's a good overview of mathematics beyond the elementary level you've completed. Another good book like that is Geometry and the Imagination, by Hilbert & Cohn-Vesson.
I have the same problem. Its a lot about efficiency. Ive been reading secrets to mental math and that's helpful.
https://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401
If you want to learn to calculate quickly in your head, probably the most fruitful thing is to pick up a bunch of tricks for mental math. One good video course for this is Secrets of Mental Math put out by The Great Courses. The same lecturer published out a very good book on the subject as well.
Of course, if you want to go old school, then it's hard to beat the utility of memorizing logarithm tables...
The book The Secrets of Mental Math has some great tricks in it to help you along.
I read this book a few years ago, and it is pretty much the way I do any basic arithmetic in my head now. http://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401/ref=sr_1_1?ie=UTF8&amp;qid=1333153637&amp;sr=8-1
These might interest you:
Poincare's Prize
Prime Obsession
Fermat's Enigma
The Code Book
I think Basic Mathematics is basically a precalculus text. I can't stand normal textbooks, everything is disconnected and done for you. This is written by one of the best mathematicians and will provoke thought and understanding. He knows his audience too, he's good with kids, check out his book Math! Encounters with High School Students. He's also written a 2-volume calculus text that I know has been used well in high school settings.
What class were you previously in? What class are you going to now? Honestly, if you just practice an hour a day going through a textbook like Lang's Basic Mathematics, then you'll be fine. The summer is a great time to not only review but to get ahead. Bored of your previous material? Go learn something new!
If you can find this at your library, I suggest you pour over it in the weekend. You will not regret it.
I used Susanna Epp's Discrete Mathematics text and rather enjoyed it. Velleman's How To Prove It is also quite good.
http://www.abebooks.com/servlet/SearchResults?bi=0&amp;bx=off&amp;ds=20&amp;kn=Epp+discrete+applications+3&amp;recentlyadded=all&amp;sortby=17&amp;sts=t
How to Prove It: A Structured Approach by Daniel J. Velleman http://www.amazon.com/dp/0521675995/ref=cm_sw_r_udp_awd_ff3Vtb08QR4FZ
For proof writing techniques I highly recommend Velleman's "How to Prove It" link
I'm going to recommend the book How to Prove It. Its all about learning the logic for proofs and strategies for writing proofs. Its one of those books that you work through slowly and complete all the exercises. Its recommended around here a-lot. I'd also suggest using the search feature if you ever want to look for other recommended books because those threads come up often around here.
Best wishes.
So here are some options I recommend:
You can find all the textbooks I mentioned online, if you know what I mean. All of these assume you haven't seen math in a while, and they all start from the very basics. Take your time with the material, play around with it a bit, and enjoy your summer :D
EditL this article describes one way you can go about your studies
How to Prove It is only 20 bucks.
Hey mathit.
I'm 32, and just finished a 3 year full-time adult education school here in Germany to get the Abitur (SAT-level education) which allows me to study. I'm collecting my graduation certificate tomorrow, woooo!
Now, I'm going to study math in october and wanted to know what kind of extra prep you might recommend.
I'm currently reading How to Prove It and The Haskell Road to Logic, Maths and Programming.
Both overlap quite a bit, I think, only that the latter is more focused on executing proofs on a computer.
Now, I've just been looking into books that might ease the switch to uni-level math besides the 2 already mentioned and the most promising I found are these two:
How to Study for a Mathematics Degree and Bridging the Gap to University Mathematics.
Do you agree with my choices? What else do you recommend?
I found online courses to be ineffective, I prefer books.
What's your opinion, mathit?
Cheers and many thanks in advance!
Спасибо за ссылку. Я обязательно это проверю. Думаю, надо было быть более конкретным. Я читал книгу, которая учит своих читателей, как строить математические доказательства. В книге дается очень общий обзор этих тем, которые я перечислил выше. Я проверю ссылку, но если вы знаете книгу на русском языке, которая учит строить математические доказательства студентам, которые начинают изучать продвинутую математику, напишите Мне пожалуйсте.
Вот книга Для справки. (в случае, если вы знаете английского языка).
How to Prove it - A structured approach
I would recommend the book "How To Prove It".
https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
It helped me in my transition into proof based mathematics. It will teach common techniques used in proofs and provides a bunch of practice problems as well.
How to Prove It
It's cheap, highly rated, starts with the basics, and as the title says, shows you how to prove it!
“How to Prove it”. D. Velleman: Amazon US Link
Probably the best resource on the topic!
The important part of this question is what do you know? By saying you're looking to learn "a little more about econometrics," does that imply you've already taken an undergraduate economics course? I'll take this as a given if you've found /r/econometrics. So this is a bit of a look into what a first year PhD section of econometrics looks like.
My 1st year PhD track has used
-Casella & Berger for probability theory, understanding data generating processes, basic MLE, etc.
-Greene and Hayashi for Cross Sectional analysis.
-Enders and Hamilton for Time Series analysis.
These offer a more mathematical treatment of topics taught in say, Stock & Watson, or Woodridge's Introductory Econometrics. C&B will focus more on probability theory without bogging you down in measure theory, which will give you a working knowledge of probability theory required for 99% of applied problems. Hayashi or Greene will mostly cover what you see in an undergraduate class (especially Greene, which is a go to reference). Hayashi focuses a bit more on general method of moments, but I find its exposition better than Greene. And I honestly haven't looked at Enders or Hamilton yet, but they will cover forecasting, auto-regressive moving average problems, and how to solve them with econometrics.
It might also be useful to download and practice with either R, a statistical programming language, or Python with the numpy library. Python is a very general programming language that's easy to work with, and numpy turns it into a powerful mathematical and statistical work horse similar to Matlab.
I was an Actuary (so I took the Financial Engineering exams) before I went back to get my PhD in Statistics. If you're familiar with:
You should be fine in a PhD stats program. It's easy enough to learn the statistics but harder to learn the math (specifically you're going to want strong analysis and calculus skills).
Check out Statistical Inference - Casella & Berger it's a pretty standard 1st year theory text in Statistics, flip through the book and see how challenging the material looks to you. If it seems reasonable (don't expect to know it -- this is stuff you're going to learn!) then you ought to be fine.
This is a classic. I took a grad level course with this textbook and every problem is nasty. But yes, it is really a classic.
Also, I just begun Data Analysis Using Regression and Multilevel/Hierarchical Models by Andrew Gelman and Jennifer Hill. Love his interpretation of linear regression. Linear regression might sound like basics, but it lays the foundation work for everything else and from time to time I feel compelled to review it. This book gave me a new way to look at a familiar topic.
If you are familiar with any statistical programming language/packages, I would highly suggest you implement the learnings from any books you have.
Beginner Resources: These are fantastic places to start for true beginners.
Introduction to Probability is an oldie but a goodie. This is a basic book about probability that is suited for the absolute beginner. Its written in an older style of english, but other than that it is a great place to start.
Bayes Rule is a really simple, really basic book that shows only the most basic ideas of bayesian stats. If you are completely unfamiliar with stats but have a basic understanding of probability, this book is pretty good.
A Modern Approach to Regression with R is a great first resource for someone who understands a little about probability but wants to learn more about the details of data analysis.
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Advanced resources: These are comprehensive, quality, and what I used for a stats MS.
Statistical Inference by Casella and Berger (2nd ed) is a classic text on maximum likelihood, probability, sufficiency, large sample properties, etc. Its what I used for all of my graduate probability and inference classes. Its not really beginner friendly and sometimes goes into too much detail, but its a really high quality resource.
Bayesian Data Analysis (3rd ed) is a really nice resource/reference for bayesian analysis. It isn't a "cuddle up by a fire" type of book since it is really detailed, but almost any topic in bayesian analysis will be there. Although its not needed, a good grasp on topics in the first book will greatly enhance the reading experience.
An Introduction to Error Analysis by John R. Taylor is the text that undergrads at UC Berkeley use. It's pretty decent.
As an aside, I think that the undergraduate sequence at most schools does a terrible job of teaching about uncertainty and error analysis. I'm a PhD candidate at Berkeley (graduating in December!), and my dissertation involves high precision measurements that test the Standard Model. Thus, analyzing sources uncertainty is my bread and butter. I really appreciate how approximations, models, and measurement precision are interrelated.
I'm really curious to see what resources other people put here.
It’s the cover of one of my favorite books I used in college. I still keep it on my desk. Error Analysis by John Taylor
In fact, it is possible to give error bars from one exact measurement. For example, let's say I count how many rain drops hit my hand in 5 seconds and the result is 25. The number of rain drops striking my hand in a given length of time will form a Poisson distribution. One can argue that based on my one measurement, the best estimate I can make of the true rate of rain striking my hand each 5 seconds is 25 +/- sqrt(25) = 25 +/- 5.
As you might intuit, the uncertainty of the mean number of drops striking my hand will decrease as more measurements are taken. This tends to drop like 1/Sqrt(N), where N is the number of 5-second raindrop measurements I make.
This style of problem is very standard in any introductory statistics textbook, but I can give you a particular book if you'd like to look into it further:
An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements by John R. Taylor
These plots are "distributions" in the sense that I meant the word distribution. Distributions are simply a collection of values placed side by side. When you arrange each month's datapoint side by side, that's a distribution.
The idea of significant figures is a simplification of error analysis. It doesn't produce perfect results, as you've found in your example. It's useful as a simple rule of thumb, especially for students, but any proper analysis would use real error analysis. Your approach of looking at the range of possible values is good, and is basically the next level of complexity after sig figs.
The problem with error analysis is that it's a bit of a bottomless rabbit-hole in terms of complexity: you can make things very complicated very quickly if you try to do things as accurately as possible (for example: the extreme values in your range of possible times are less likely than the central values, and since your using the inverse of the time, that produces a non-uniform distribution in the velocities. Computing the actual probability distribution is a proper pain in the ass).
My advice is this: if you're a highschool student or non-physics university student, stick to sig-figs: it's not perfect, but it's good enough for the sorts of problems you'll be working with. If you're a physics major, you should learn some basic error analysis from your lab courses. If you're really interested in learning to do it properly, I think the most common textbook is the 'Introduction to Error Analysis' by Taylor.
Are you interested in systematic errors, random errors, or both? Ignoring systematic errors, with the information that you've given, here are the obvious things to consider:
For each of the items above, you can determine the uncertainties with a simple design-of-experiments. For validated instrumentation, the uncertainties will be specified as part of the IQ/OQ/PQ process, but even so, you should still verify them yourself.
Once you have these values, calculating how each of them contribute to the final error is relatively straightforward using principles of error propagation. There are many books and websites devoted to the topic of error propagation. I have a copy of John Taylor's book, which I like. It does have a significant number of errors in the book because it contains so many equations and works them out in detail. However, the principles of error propagation are taught very well in the book, and the minor math errors (I know it's ironic) are easy to spot.
We used this in my undergrad Experimental Physics course: https://www.amazon.com/Introduction-Error-Analysis-Uncertainties-Measurements/dp/093570275X
Here's my radical idea that might feel over-the-top and some here might disagree but I feel strongly about it:
In order to be a grad student in any 'mathematical science', it's highly recommended (by me) that you have the mathematical maturity of a graduated math major. That also means you have to think of yourself as two people, a mathematician, and a mathematical-scientist (machine-learner in your case).
AFAICT, your weekends, winter break and next summer are jam-packed if you prefer self-study. Or if you prefer classes then you get things done in fall, and spring.
Step 0 (prereqs): You should be comfortable with high-school math, plus calculus. Keep a calculus text handy (Stewart, old edition okay, or Thomas-Finney 9th edition) and read it, and solve some problem sets, if you need to review.
Step 0b: when you're doing this, forget about machine learning, and don't rush through this stuff. If you get stuck, seek help/discussion instead of moving on (I mean move on, attempt other problems, but don't forget to get unstuck). As a reminder, math is learnt by doing, not just reading. Resources:
math on irc.freenode.net
Here are two possible routes, one minimal, one less-minimal:
Minimal
Less-minimal:
NOTE: this is pure math. I'm not aware of what additional material you'd need for machine-learning/statistical math. Therefore I'd suggest to skip the less-minimal route.
This book about error analysis is really good
I think the rule about sig figs is that you want the sig figs on your error to be of the same place as your last sig fig in your calculation. So your numbers would be 5.77 ± 0.31.
>When's the last time someone flew a train into a building?
It's happened before.
This was on the cover of a college text book I had: An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements
I'm potentially interested in picking up a textbook on error analysis. How do we feel about John R. Taylor's book?
I took a Stats for Sci & Eng class (it had this book). All I learned was that stats is really hard and you have to use way more calculus than I initially thought.
Maybe Casella and Berger? If you can get hold of a copy, take a look. If that's more theoretical than you were looking for to start we might be able to suggest something else.
What is your background?
http://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126
Is a fairly standard first year grad textbook with I quite enjoy. Gives you a mathematical statistics foundation.
http://www.amazon.com/All-Statistics-Concise-Statistical-Inference/dp/1441923225/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1278495200&amp;sr=1-1
I've heard recommended as an approachable overview.
http://www.amazon.com/Modern-Applied-Statistics-W-Venables/dp/1441930086/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1278495315&amp;sr=1-1
Is a standard 'advanced' applied statistics textbook.
http://www.amazon.com/Weighing-Odds-Course-Probability-Statistics/dp/052100618X
Is non-standard but as a mathematician turned probabilist turned statistician I really enjoyed it.
http://www.amazon.com/Statistical-Models-Practice-David-Freedman/dp/0521743850/ref=pd_sim_b_1
Is a book which covers classical statistical models. There's an emphasis on checking model assumptions and seeing what happens when they fail.
This book has a fairly good introduction to probability theory if you don't need it to be measure theoretic. Statistical Inference
Berger and Casella's Statistical Inference is what you need if you want a mathematical approach.
How to Solve It by Polya is a great book about the use of critical thinking in the process of solving mathematics problems.
http://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X
I highly recommend Polya's How to Solve it too.
The best advice to get better at solving these problems is to persist. You should have to try, to think, to fail slowly building a picture until you find the solution. Have patience with not knowing exactly what to do.
For more technical general advice Polya's lovely book How to Solve it is excellent.
how to solve it by g. polya
book wiki
book on amazon
George Pólya wiki
Polya's How to Solve It:
http://www.amazon.com/How-Solve-It-Mathematical-Princeton/dp/069111966X
http://math.hawaii.edu/home/pdf/putnam/PolyaHowToSolveIt.pdf
My favorite book on problem solving is Problem Solving Through Problems. There's an online copy, too. (I recommend you print it and get it bound at Kinkos if you intend to seriously work through it, though. This type of thing sucks on a screen.)
How To Solve It is another popular recommendation for that topic. Personally, I only read part of it. It's alright.
I can recommend other stuff if you tell me what level of math you're at, what you're interested in learning, etc.
What's Math Got to Do With It?
How to Solve It
Check out "How to Solve It" It's a small book but well worth the price. It talks about how to think critically and creatively go about solving problems.
I'm a former physicists The way I felt I got smarter over the years as an undergraduate and graduate student was by continuing to solve hard and harder math and physics problems. Throwing yourself at increasingly difficult problems forces you to think systematically (so that you aren't considering the same solution again and again) and creatively (bring in other concepts and apply them to new situations) and perhaps, most importantly, to not give up. I found myself just being able to solve technical problems in other areas faster. My brain naturally got faster just like how someone who continually runs a slightly greater distances or just a little bit faster everyday is going to just naturally develop the muscles to make that possible. Also having a repository of solved problems as reference helps you solve future problems.
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I found this book useful for problem solving:
https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X
this book is quite short but perfect for an aspiring mathematician that is going to start hearing about Gödel's proof in casual conversation. This provides a concise easy treatment of it's importance and how the proof works. Also, see it's reviews on goodreads
I was totally enthralled with the philosophy of Mathematics when I was in college. One of the books I found interesting -- before I had progressed in mathematical logic -- was this one on Godel's Proof.
Gödel's Proof is a good starting point for the incompleteness theorem. Covers the basics of the theorem and its impacts. Unless you are prepping for coursework in logic than this book likely has the right amount of depth for you.
I don't have a recommendation for Tarski. Hopefully someone else has something for you.
160 pages:
https://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371/ref=sr_1_1?ie=UTF8&amp;qid=1486067268&amp;sr=8-1&amp;keywords=godels+proof
https://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371
Book recommendation for an intro to Godel's Theorem: Gödel's Proof - Ernest Nagel and James Newman. Well written, concise and requires no prior mathematical knowledge.
Edit: Never mind. misread "I do have an introductory understanding..." as "I don't have an introductory understanding...". Still a good recommendation for anyone else who is interested!
Agree. I picked up on that from the intro to GEB, stopped reading GEB, and decided to get a better understanding of Gödel's proof by reading the book Hofstadter says introduced him to Gödel - Gödel's Proof, by Ernest Nagel and James R. Newman. I recommend it as a very approachable introduction to Gödel's incompleteness theorems. Even now I can recall moments reading that little book where I'd get a big smile on my face as the force of his argument and conclusion would bear down on me. What Gödel did is nothing short of mind blowing.
After that, if you want more, then go to Gödel's Incompleteness Theorems by Raymond M. Smullyan (You'll want to buy this one used). This one is a much more technical, though still approachable if you're prepared at an undergrad level, guide through to Gödel's conclusions. You should go into it with an undergrad level of fluency in propositional and predicate logic.
You can read GEB without all that, certainly without the second book, but I've found it a better experience having more familiarity with Gödel as I work through it.
Highly recommend Godel's Proof for anybody looking to jump into the question of how well founded modern mathematics is.
Try to find entry points that interest you personally, and from there the next steps will be natural. Most books that get into the nitty-gritty assume you're in school for it and not directly motivated, at least up to early university level, so this is harder than it should be. But a few suggestions aimed at the self-motivated: Lockhart Measurement, Gelfand Algebra, 3blue1brown's videos, Calculus Made Easy, Courant & Robbins What Is Mathematics?. (I guess the last one's a bit tougher to get into.)
For physics, Thinking Physics seems great, based on the first quarter or so (as far as I've read).
If you need to brush up on some of the more basic topics, there's a series of books by IM Gelfand:
Algebra
Trigonometry
Functions and Graphs
The Method of Coordinates
The following are all really good and all very different. Check out the reviews and decide what fits you best. If I had to pick only one, I would pick one of the first three listed.
http://www.amazon.com/gp/product/0471530123
http://www.amazon.com/gp/product/1592577229
http://www.amazon.com/The-Humongous-Book-Calculus-Problems/dp/1592575129
http://www.amazon.com/gp/product/0817636773
http://www.amazon.com/gp/product/0760706603
http://www.amazon.com/gp/product/B000ZEC19Y
http://www.amazon.com/gp/product/0070293317
I've heard good things about: http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773/ref=cm_cr_pr_product_top?ie=UTF8
but I admit I haven't read it.
Read this: https://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773 and you're more than set for algebraic manipulation.
And if you're looking to get super fancy, then some of that: https://www.amazon.com/Method-Coordinates-Dover-Books-Mathematics/dp/0486425657/
And some of this for graphing practice: https://www.amazon.com/Functions-Graphs-Dover-Books-Mathematics/dp/0486425649/
And if you're looking to be a sage, these: https://www.amazon.com/Kiselevs-Geometry-Book-I-Planimetry/dp/0977985202/ + https://www.amazon.com/Kiselevs-Geometry-Book-II-Stereometry/dp/0977985210/
If you're uncomfortable with mental manipulation of geometric objects, then, before anything else, have a crack at this: https://www.amazon.com/Introduction-Graph-Theory-Dover-Mathematics/dp/0486678709/
These are, in my opinion, some of the best books for learning high school level math:
These are all 1900's Russian math text books (probably the type that /u/oneorangehat was thinking of) edited by I.M. Galfand, who was something like the head of the Russian School for Correspondence. I basically lived off them during my first years of high school. They are pretty much exactly what you said you wanted; they have no pictures (except for graphs and diagrams), no useless information, and lots of great problems and explanations :) There is also I.M Gelfand Trigonometry {[.pdf] (http://users.auth.gr/~siskakis/GelfandSaul-Trigonometry.pdf) | Amazon} (which may be what you mean when you say precal, I'm not sure), but I do not own this myself and thus cannot say if it is as good as the others :)
I should mention that these books start off with problems and ideas that are pretty easy, but quickly become increasingly complicated as you progress. There are also a lot of problems that require very little actual math knowledge, but a lot of ingenuity.
Sorry for bad Englando, It is my native language but I haven't had time to learn it yet.
I'll be working through Spivak's calculus for fun. Wish me luck!
For getting more intuition on proofs I would suggest the following book
http://www.amazon.com/Nuts-Bolts-Proofs-Third-Introduction/dp/0120885093/ref=sr_1_1?ie=UTF8&amp;qid=1311007015&amp;sr=8-1
I think Rudin might be really tricky at your level, you can keep with it if you want, but I think Calculus by Michael Spivak would be much more approachable for you.
http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1311007057&amp;sr=1-1
i personally prefer Yurope! Hillary's Invasion. Very insightful reading.
I didn't mean to make it sound so serious :) However, stress, drinking, and insomnia can all have some unexpectedly large effects, so it may be worth dropping into a counseling session if your university has one.
In regards to math education and intuition, something I found very useful was to read some books that start from scratch, like Burn Math Class, or Spivak's calculus for a real challenge. You're at a point in your education where you have the sophistication to understand the foundations of math, so you can start to rebuild intuition about a lot of things that will make university-level math much more sensible.
Start with 3 Blue 1 Brown's Essence of Calculus Series - https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr
and follow the following books -
Calculus by Spivak - https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918
Calculus Made Easy - http://calculusmadeeasy.org/
Follow all the concepts and solve the examples and exercises.
Feel free to ask the questions here or in mathsoverflow.
Last but not the least, PRACTICE, PRACTICE, PRACTICE........!
I am surprised no one has mentioned M. Spivak's very well known text Calculus. I thought this book was a pleasure to read. His writing was very fun and lighthearted and the book certainly teaches the material very well. In my opinion this is the best introductory calculus text there is.
When I first started learning math on my own, I started learning calculus from something like this. Though I enjoyed it, it didn't really show me what 'real math' was like. For learning something closer to higher math, a more rigorous version would be something like this. It's all preference, though.
If you don't know much about calculus at all, start with the first one, and then work your way up to Spivak.
If you want to learn serious mathematics, start with a theoretical approach to calculus, then go into some analysis. Introductory Real Analysis by Kolmogorov is pretty good.
As far as how to think about these things, group theory is a strong start. "The real numbers are the unique linearly-ordered field with least upper bound property." Once you understand that sentence and can explain it in the context of group theory and the order topology, then you are in a good place to think about infinity, limits, etc.
Edit: For calc, Spivak is one of the textbooks I have heard is more common, but I have never used it so I can't comment on it. I've heard good things, though.
A harder analysis book for self-study would be Principles of Mathematical Analysis by Rudin. He is very terse in his proofs, so they can be hard to get through.
If you have a chance, I recommend checking out some textbooks on real analysis, which will guide you through the derivations and proofs of many theorems in calculus that you've thus far been expected to take for granted.
Some would recommend starting with Rudin's Principles of Mathematical Analysis, and it's certainly a text that I plan to read at some point. For your purposes, I might recommend Spivak's Calculus since it expects you to rigorously derive some of the most important results in calculus through proof-writing exercises. This was my first introduction to calculus during high-school. While it was overwhelming at first, it prepared me for some of my more advanced undergraduate courses (including real analysis and topology), and it seems to be best described as an advanced calculus textbook.
Linear algebra is about is about linear functions and is typically taken in the first or second year of college. College algebra normally refers to a remedial class that covers what most people do in high school. I highly recommend watching this series of videos for getting an intuitive idea of linear algebra no matter what book you go with. The book you should use depends on how comfortable you are with proofs and what your goal is. If you just want to know how to calculate and apply it, I've heard Strang's book with the accompanying MIT opencourseware course is good. This book also looks good if you're mostly interested in programming applications. A more abstract book like Linear Algebra Done Right or Linear Algebra Done Wrong would probably be more useful if you were familiar with mathematical proofs beforehand. How to Prove it is a good choice for learning this.
I haven't seen boolean algebra used to refer to an entire course, but if you want to learn logic and some proof techniques you could look at How to Prove it.
Most calculus books cover both differential and integral calculus. Differential calculus refers to taking derivatives. A derivative essentially tells you how rapidly a function changes at a certain point. Integral calculus covers finding areas under curves(aka definite integrals) and their relationship with derivatives. This series gives some excellent explanations for most of the ideas in calculus.
Analysis is more advanced, and is typically only done by math majors. You can think of it as calculus with complete proofs for everything and more abstraction. I would not recommend trying to learn this without having a strong understanding of calculus first. Spivak's Calculus is a good compromise between full on analysis and a standard calculus class. It's possible to use this as a first exposure to calculus, but it would be difficult.
The popular opinion by some mathematical elite is that Stewart dumbs down calculus, focuses too much on applications, and not enough on theory, which is important for those moving beyond to real analysis and other upper division courses. You should read the reviews of Spivak's or Apostol's calculus text books to see what I mean.
You do realize that there is guesswork but the extremes of the confidence interval are strictly positive right? In other words, no one is certain but what we are certain about is that optimum homework amount is positive. Maybe it's 4 hours, maybe it's 50 hours. But it's definitely not 0.
I don't like homework either when I was young. I dreaded it, and I skipped so many assignments, and I regularly skipped school. I hated school. In my senior year I had such severe senioritis that after I got accepted my grades basically crashed to D-ish levels. (By the way this isn't a good thing. It makes you lazy and trying to jumpstart again in your undergrad freshman year will feel like a huge, huge chore)
Now that I'm older I clearly see the benefits of homework. My advice to you is not to agree with me that homework is useful. My advice is to pursue your dreams, but when doing so be keenly aware of the pragmatical considerations. Theoretical physics demands a high level of understanding of theoretical mathematics: Lie groups, manifolds and differential algebraic topology, grad-level analysis, and so on. So get your arse and start studying math; you don't have to like your math homework, but you'd better be reading Spivak if you're truly serious about becoming a theoretical physicist. It's not easy. Life isn't easy. You want to be a theoretical physicist? Guess what, top PhD graduate programs often have acceptance rates lower than Harvard, Yale, Stanford etc. You want to stand out? Well everyone wants to stand out. But for every 100 wannabe 15-year-old theoretical physicists out there, only 1 has actually started on that route, started studying first year theoretical mathematics (analysis, vector space), started reading research papers, started really knowing what it takes. Do you want to be that 1? If you don't want to do homework, fine; but you need to be doing work that allows you to reach your dreams.
If you're looking at it from a mathematical "I want to prove things" standpoint, I'd recommend Apostol. I've also heard good things about Spivak, although I've never read that book.
If you're looking at it from an engineering "Just tell me how to do the damn problem" perspective, I'm no help to you.
I have yet to read it myself, but the classic text for Calculus is Spivak's Calculus. It is very highly recommended.
Hey y'all! I'm 16, and am about to finish Spivak's Calculus. Assuming that I know everything up to Algebra II, AP Statistics, Trigonometry, a bit of linear algebra (please specify if the subject requires extensive knowledge here), and have thoroughly gone through Spivak's Calculus, what should be the next thing I study? And what textbook(s) would you recommend for learning that subject?
Right now I'm leaning towards Real Analysis, or Multi Variable Calculus, or maybe Topology, or... case in point, I am very undecided and am in need of recommendations.
Honestly, I think you should be more realistic: doing everything in that imgur link would be insane.
You should try to get a survey of the first 3 semesters of calculus, learn a bit of linear algebra perhaps from this book, and learn about reading and writing proofs with a book like this. If you still have time, Munkres' Topology, Dummit and Foote's Abstract Algebra, and/or Rudin's Principles of Mathematical Analysis would be good places to go.
Roughly speaking, you can theoretically do intro to proofs and linear algebra independently of calculus, and you only need intro to proofs to go into topology (though calculus and analysis would be desirable), and you only need linear algebra and intro to proofs to go into abstract algebra. For analysis, you need both calculus and intro to proofs.
Linear Algebra can be of different levels of difficulty:
I've never taught the course, but a couple of my colleagues are very fond of Linear Algebra Done Wrong and would willingly teach from it if (1) the title wouldn't immediately turn students off of it and (2) the school would be okay with sacrificing some income from students having to purchase a book.
If you're curious, the book title is a play on the title of another well-known linear algebra book.
It's aight. Just read linear algebra, and mv calculus. Maybe some statistical mechanics, read some thermo and kinetics. Atkins for kinetics and thermo, McQuarrie for stat mech. For linear algebra read get this. You'll still have to take classes on it, so it's cool. The worst you may have to do is take some UG classes to get up to speed.
Math 53 isn't heavy on proofs at all except possibly near the tail end of the course. Actually, the whole purpose of Math 53 really is the last 2 weeks when it gets into the Stoke's and Divergence Theorem. If you want to get started early on that I recommend the excellent Div, Grad, Curl, and All That which is a short text you can get online or the library that really makes the topic more manageable. Be prepared for it because it will hit you right at the end of the semester although the curve is generally nicer than Math 1B.
Math 54, or linear algebra in general, is for a lot of people the "intro to proofs" course. Right around the time Math 53 goes at breakneck speed, Math 54 finishes up with fourier analysis. It's doable but you have to stay on top of things the whole semester or have a miserable few weeks near the end.
As others have mentioned, there are a lot of good books on Math Methods of Physics out there (I used Hassani's Mathematical Methods: For Students of Physics and Related Fields).
That said, if you're having trouble with calculus, I'd recommend going back and really understanding that well. It underlies more or less all the mathematics found in physics, and trying to learn vector calculus (essential for E&M) without having a solid understanding of single-variable calculus is just asking for trouble.
There are a number of good books out there. Additionally, Khan Academy covers calculus very well. The videos on this page cover everything you'd encounter in your first year, and maybe a smidge more.
Once you move on to vector calculus, Div, Grad, Curl and All That is without equal.
I found the book Div Grad Curl and All That to explain it pretty well. The book is short enough to read through in a couple hours.
I thought of some books suggestions. If you're going all in, go to the library and find a book on vector calculus. You're going to need it if you don't already know spherical coordinates, divergence, gradient, and curl. Try this one if your library has it. Lots of good books on this though. Just look for vector calculus.
Griffiths has a good intro to E&M. I'm sure you can find an old copy on a bookshelf. Doesn't need to be the new one.
Shankar has a quantum book written for an upper level undergrad. The first chapter does an excellent job explaining the basic math behind quantum mechanics .
Is it this one?
For vector calculus: Div, Grad, Curl, and All That: An Informal Text on Vector Calculus
For complex variables/Laplace: Complex Variables and the Laplace Transform for Engineers - Caution! Dover book! Slightly obtuse at times!
For the finite difference stuff I would wait until you have a damn good reason to learn it, because there are a hundred books on it and none of them are that good. You're better off waiting for a problem to come along that really requires it and then getting half a dozen books on the subject from the library.
I can't help with the measurement text as I'm a physicist, not an engineer. Sorry. Hope the rest helps.
For vector calculus, you might enjoy the less formal British text Div, Grad, Curl, and All That by H. M. Schey; for group theory in brief, consider the free textbook Elements of Abstract and Linear Algebra by Edwin H. Connell.
Alternatives to Schey's book include the much more formal Calculus on Manifolds by Michael Spivak, which does have more exercises than Schey but uses most of them to develop the theory, rather than as the mindless drills that fill an ordinary textbook; Michael E. Corral's free textbook Vector Calculus isn't huge but is written closer to an ordinary textbook.
Haven't used it myself, but you might want to check out Div,Grad,Curl by Schey.
Friendly info:
"College Algebra" = Elementary Algebra.
College Level Algebra = Abstract Algebra.
Example: Undergrad Algebra book.
Example: Graduate Algebra book.
If you can read through gallian's book, I consider dummit and foote's book (http://www.amazon.com/Abstract-Algebra-3rd-David-Dummit/dp/0471433349) as the best math textbook i've ever read. tons of examples, thorough treatment of material, and tons of exercises.
I am a master's student with interests in algebraic geometry and number theory. And I have a good collection of textbooks on various topics in these two fields. Also, as part of my undergraduate curriculum, I learnt abstract algebra from the books by Dummit-Foote, Hoffman-Kunze, Atiyah-MacDonald and James-Liebeck; analysis from the books by Bartle-Sherbert, Simmons, Conway, Bollobás and Stein-Shakarchi; topology from the books by Munkres and Hatcher; and discrete mathematics from the books by Brualdi and Clark-Holton. I also had basic courses in differential geometry and multivariable calculus but no particular textbook was followed. (Please note that none of the above-mentioned textbooks was read from cover to cover).
As you can see, I didn't learn much geometry during my past 4 years of undergraduate mathematics. In high school, I learnt a good amount of Euclidean geometry but after coming to university geometry appears very mystical to me. I keep hearing terms like hyperbolic/spherical geometry, projective geometry, differential geometry, Riemannian manifold etc. and have read general maths books on them, like the books by Hartshorne, Ueno-Shiga-Morita-Sunada and Thorpe.
I will be grateful if you could suggest a series of books on geometry (like Stein-Shakarchi's Princeton Lectures in Analysis) or a book discussing various flavours of geometry (like Dummit-Foote for algbera). I am aware that Coxeter has written a series of textbooks in geometry, and I have read Geometry Revisited in high school (which I enjoyed). If these are the ideal textbooks, then where to start? Also, what about the geometry books by Hilbert?
If you are enjoying your Calc 3 book, I highly recommend reading Topology, which provides the foundations of analysis and calculus. Two other books I would highly recommend to you would be Abstract Algebra and Introduction to Algorithms, though I suspect you're well aware of the latter.
Dummit and Foote's Abstract Algebra is an excellent book for the algebra side of things. It can be a little dense, but it's chock full of examples and is very thorough.
To help get through the first ten or so chapters, Charles Pinter's A Book of Abstract Algebra is an incredible resource. It does wonders for building up an intuition behind algebra.
Before the Princeton Companion to Mathematics, there were:
What Is Mathematics? by Courant and Robbins
Mathematics: Its Content, Methods and Meaning by Aleksandrov, Kolmogorov, and Lavrent'ev
Concepts of Modern Mathematics by Ian Stewart
Mathematics - Its Content, Methods and Meaning gives you an overview of the major topics covered in university Math curriculum.
Recentemente estive procurando algo interessante pra ler e me deparei com várias recomendações do livro How to solve it: A New Aspect of Mathematical Method.
Um livro extremamente denso mas com muito conteúdo é o Mathematics: Its Content, Methods and Meaning. Comecei a ler esse livro, mas outras atividades me fizeram dar uma pausa. Vou tentar voltar a ele e colocar como meta terminar antes de 2020 rs.
Já li alguns livros explicando a origem dos números. Mas, de todos que li, Os números é imbatível.
Check out:
If you only want one math book, SO said this is it: Mathematics
For those interested in the "abstractness" of non-natural numbers, there's a phenomenal brief introduction in one of my favorite math texts, Mathematics: Its content, methods and meaning. A cold war Russian standard that covers a helluva lot of ground in applied math.
They make the point that the number "1" seems pretty intuitive to humans... you can have "1" of something, or "2" of something. But having "0" of something doesn't really make any sense, and for a long time it was argued whether or not "0" was even a "number". You certainly can't have "1/2" of a thing. If you cut an object in half, you just have 2 things now. And to have negative something is just absurd. There's a blurb about some primitive isolated tribes that have words for the number "1", "2", and "many". The number 1,237,298 is still pretty abstract to a human, because it's not like you can count that or really visualize that many things, but we acknowledge such a quantity can be useful.
Κατ' αρχάς συγχαρητήρια και καλή αρχή. Έχεις επιλέξει φοβερά ενδιαφέρον πεδίο κατά τη γνώμη μου και ζηλεύω λίγο :P
Ήθελα κι εγώ να μάθω σωστά μαθηματικά κάποια περίοδο και είχα ψάξει σε φόρουμ για κάποιο προτεινόμενο βιβλίο που να είναι ολοκληρωμένο και εύκολα κατανοητό. Ήταν πολλοί που πρότειναν αυτό το βιβλίο: https://www.amazon.com/dp/0486409163/?coliid=IFD6IMMG22STW&colid=3J1YAUNLTYQCX&psc=0&ref_=lv_ov_lig_dp_it
Δυστυχώς τελικά δε το αγόρασα επειδή δεν είχα χρόνο να αφιερώσω αλλά τα σχόλια που διάβασα με είχαν πείσει. Ίσως σε βοηθήσει.
Are you thinking of this one?
Mathematics: its Content, Methods, and Meaning by Alexandrov, Kolomogrov, and Lavrent'ev.
For your situation I would highly recommend Mathematics: Its Content, Methods and Meaning, which is ~1000 page survey of mathematics topics.
I would also highly suggest the 3 volume set, Mathematical Thought from Ancient to Modern Times by Morris Kline. I'm not finding the words for why I think anyone, but particularly teachers, to have a historical context for mathematics, but I strongly believe it.
It also helps to read about what sort of problems people were interested in when they came up with things such as groups, or sqrt(-1), etc.
Another book that you wouldn't use in a class: "Mathematics: Its Content, Meaning, and Methods"
http://www.amazon.com/Mathematics-Content-Methods-Meaning-Dover/dp/0486409163
I'm working through the exercises in Pinter's Abstract Algebra.
If you're not afraid of math there are some cheap introductory textbooks on topics that might be accessible:
For abstract algebra: http://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178/ref=sr_1_1?ie=UTF8&amp;qid=1459224709&amp;sr=8-1&amp;keywords=book+of+abstract+algebra+edition+2nd
For Number Theory: http://www.amazon.com/Number-Theory-Dover-Books-Mathematics/dp/0486682528/ref=sr_1_1?ie=UTF8&amp;qid=1459224741&amp;sr=8-1&amp;keywords=number+theory
These books have complimentary material and are accessible introductions to abstract proof based mathematics. The algebra book has all the material you need to understand why quintic equations can't be solved in general with a "quintic" formula the way quadratic equations are all solved with the quadratic formula.
The number theory book proves many classic results without hard algebra, like which numbers are the sum of two squares, etc, and has some of the identities ramanujan discovered.
For an introduction to analytic number theory, a hybrid pop/historical/textbook is : http://www.amazon.com/Gamma-Exploring-Constant-Princeton-Science/dp/0691141339/ref=sr_1_1?ie=UTF8&amp;qid=1459225065&amp;sr=8-1&amp;keywords=havil+gamma
This book guides you through some deep territory in number theory and has many proofs accessible to people who remember calculus 2.
I like this abstract algebra book: A Book of Abstract Algebra
This is one of the best books of abstract algebra I've seen, very well explained, favoring clear explanations over rigor, highly recommended (take your time to read the reviews, the awesomeness of this book is real :P): http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178/ref=sr_1_6?ie=UTF8&amp;qid=1345229432&amp;sr=8-6&amp;keywords=introduction+to+abstract+algebra
On a side note, trust me, Dummit or Fraileigh are not what you want.
"A Book of Abstract Algebra" by Charles C. Pinter is nice, from what I've seen of it--which is about the first third. I'm going through it in an attempt to relearn the abstract algebra I've forgotten.
I was using Herstein (which was what I learned from the first time), and was doing fine, but saw the Pinter book at Barnes & Noble. I've found it is often helpful when relearning a subject to use a different book from the original, just to get a different approach, so gave it a try (it's a Dover, so was only ten bucks).
What is nice about the Pinter book is that it goes at a pretty relaxed pace, with a good variety of examples. A lot of the exercises apply abstract algebra to interesting things, like error correcting codes, and some of these things are developed over the exercises in several chapters.
You don't have to be a prodigy to be able to understand some real mathematics in middle school or early high school. By 9th grade, after a summer of reading calculus books from the local public library, I was able to follow things like Niven's proof that pi is irrational, for instance, and I was nowhere near a prodigy.
From the ground up, I dunno. But I looked through my amazon order history for the past 10 years and I can say that I personally enjoyed reading the following math books:
An Introduction to Graph Theory
Introduction to Topology
Coding the Matrix: Linear Algebra through Applications to Computer Science
A Book of Abstract Algebra
An Introduction to Information Theory
For ODEs, I'd seriously suggest buying this. Lots and lots of exercises, and full solutions. Plus, at $15, it hopefully won't break the bank too badly.
this one
i used this book, the one that was required for the class sucked, this one is much better and it's super cheap. Also, answers and steps are included in the sections, so you can actually check if you're doing it correctly or not.
I've never really used MIT OCW however I've used Paul's OMN a lot back when I was studying multivar calc. I do recommend books, though. I have books both on multivar calc and differential equations and they're both well, however, I've moved on from calculus (that is, I don't actively study it anymore) so I can't really say much more.
The books I have:
> https://www.amazon.com/Multivariable-Calculus-Clark-Bray/dp/1482550741/ref=sr_1_3?s=books&amp;ie=UTF8&amp;qid=1500976188&amp;sr=1-3&amp;keywords=multivariable+calculus
> https://www.amazon.com/Ordinary-Differential-Equations-Dover-Mathematics/dp/0486649407/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1500976233&amp;sr=1-1&amp;keywords=differential+equations
There seems to often be this sort of tragedy of the commons with the elementary courses in mathematics. Basically the issue is that the subject has too much utility. Be assured that it is very rich in mathematical aesthetic, but courses, specifically those aimed at teaching tools to people who are not in the field, tend to lose that charm. It is quite a shame that it's not taught with all the beautiful geometric interpretations that underlie the theory.
As far as texts, if you like physics, I can not recommend highly enough this book by Lanczos. On the surface it's about classical mechanics(some physics background will be needed), but at its heart it's a course on dynamical systems, Diff EQs, and variational principles. The nice thing about the physics perspective is that you're almost always working with a physically interpretable picture in mind. That is, when you are trying to describe the motion of a physical system, you can always visualize that system in your mind's eye (at least in classical mechanics).
I've also read through some of this book and found it to be very well written. It's highly regarded, and from what I read it did a very good job touching on the stuff that's normally brushed over. But it is a long read for sure.
Ordinary Differential Equations by Tenenbaum and Pollard is a classic. I thought it explained things well and was more rigorous than some other treatments of subject that I've come across.
15! Well then, you have plenty of time to figure this out. Well, a few years, in any case.
I think what you should do is learn some programming as soon as possible (assuming you don't already). It's easy, trust me. Start with C, C++, Python or Java. Personally, I started with C, so I'll give you the tutorials I learned from: http://www.cprogramming.com/tutorial/c/lesson1.html
You should also try out some electronics. There's too much theory for me to really explain here, but try and maybe get a starter's kit with a book of tutorials on basic electronics. Then, move onto some more complicated projects. It wouldn't hurt to look into some circuit theory.
For mechanical, well... that one is kind of hard to get practical experience for on a budget, but you can still try and learn some of the theory behind it. Start with learning some dynamics and then move onto statics. Once you've got that down, try learning about the structure and property of materials and then go to solid mechanics and machine design. There's a lot more to mechanical engineering than that, but that's a good starting point.
There's also, of course, chemical engineering, civil engineering, industrial engineering, aerospace engineering, etc, etc... but the main ones I know about are mechanical (what I'm currently studying), electrical and computer.
Hope this helped. I wasn't trying to dissuade you from pursuing engineering, but instead I'm just forewarning you that a lot of people go into it with almost no actual engineering skills and well, they tend to do poorly. If you start picking up some skills now, years before college, you'll do great.
EDIT: Also, try learning some math! It would help a lot to have some experience with linear algebra, calculus and differential equations. This book should help.
http://www.amazon.com/Ordinary-Differential-Equations-Dover-Mathematics/dp/0486649407/ref=mt_paperback?_encoding=UTF8&amp;me=
Stay away from Youtube and Khan Academy unless you need reinforcement on a specific topic. Go through this book, page by page, learn the material, and do every problem.
I've used Tenenbaum to teach myself ODEs. Got an A in my class. Arnold is cannon, but you need mathematical maturity so YMMV.
With regards to your edit, if your friend is still incarcerated after reading his calculus text, send him Ordinary Differential Equations by Tenenbaum and Pollard. It contains zillions of worked problems showing how ODEs can be applied to physical problems.
The one and only , if you're willing to dedicate the time
Ordinary Differential Equations (Dover Books on Mathematics)
https://www.amazon.com/dp/0486649407/
For discrete math I like Discrete Mathematics with Applications by Suzanna Epp.
It's my opinion, but Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers is much better structured and more in depth than How To Prove It by Velleman. If you follow everything she says, proofs will jump out at you. It's all around great intro to proofs, sets, relations.
Also, knowing some Linear Algebra is great for Multivariate Calculus.
This is how I learned logic, for computer science.
First chapter of this Discrete mathematics book in my discrete math class
https://www.amazon.ca/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328
Then, using The Logic Book for a formal philosophy logic 1 course.
https://www.amazon.ca/Logic-Book-Merrie-Bergmann/product-reviews/0078038413/ref=dpx_acr_txt?showViewpoints=1
The second book was horrid on itself, luckily my professor's academic lineage goes back to Tarski. He's an amazing Professor and knows how to teach...that was a god send. Ironically, he dropped the text and I see that someone has posted his openbook project.
The first book (first chapter), is too applied I imagine for your needs. It would also only be economically feasible if well, you disregarded copyright law and got a "free" PDF of it.
Try these books(the authors will hold your hand tight while walking you through interesting math landscapes):
Discrete Mathematics with Applications by Susanna Epp
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
A Friendly Introduction to Number Theory Joseph Silverman
A First Course in Mathematical Analysis by David Brannan
The Foundations of Analysis: A Straightforward Introduction: Book 1 Logic, Sets and Numbers by K. G. Binmore
The Foundations of Topological Analysis: A Straightforward Introduction: Book 2 Topological Ideas by K. G. Binmore
Introductory Modern Algebra: A Historical Approach by Saul Stahl
An Introduction to Abstract Algebra VOLUME 1(very elementary)
by F. M. Hall
There is a wealth of phenomenally well-written books and as many books written by people who have no business writing math books. Also, Dover books are, as cheap as they are, usually hit or miss.
One more thing:
Suppose your chosen author sets the goal of learning a, b, c, d. Expect to be told about a and possibly c explicitly. You're expected to figure out b and d on your own. The books listed above are an exception, but still be prepared to work your ass off.
>My first goal is to understand the beauty that is calculus.
There are two "types" of Calculus. The one for engineers - the plug-and-chug type and the theory of Calculus called Real Analysis. If you want to see the actual beauty of the subject you might want to settle for the latter. It's rigorous and proof-based.
There are some great intros for RA:
Numbers and Functions: Steps to Analysis by Burn
A First Course in Mathematical Analysis by Brannan
Inside Calculus by Exner
Mathematical Analysis and Proof by Stirling
Yet Another Introduction to Analysis by Bryant
Mathematical Analysis: A Straightforward Approach by Binmore
Introduction to Calculus and Classical Analysis by Hijab
Analysis I by Tao
Real Analysis: A Constructive Approach by Bridger
Understanding Analysis by Abbot.
Seriously, there are just too many more of these great intros
But you need a good foundation. You need to learn the basics of math like logic, sets, relations, proofs etc.:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers
Discrete Mathematics with Applications by Epp
Mathematics: A Discrete Introduction by Scheinerman
You should be able to see the textbook that is required by looking on your MyOSU page. This was the book used when I took it last year: https://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328/ref=sr_1_2?crid=3TYI2C38O2DKB&keywords=susanna+epp+discrete+mathematics+with+applications+4th+edition&qid=1568344545&sprefix=susannah+epp%2Caps%2C175&sr=8-2
I should note that topics like graph theory, combinatorics, areas otherwise under the "discrete math" category, don't really require calculus, analysis, and other "continuous math" subjects to learn them. Instead, you can get up to college level algebra, then get a book like
Discrete Mathematics and Its Applications Seventh Edition (Higher Math) https://www.amazon.com/dp/0073383090/ref=cm_sw_r_cp_api_U6Zdzb793HMA7
Or the more highly regarded but less problem set answers,
Discrete Mathematics with Applications https://www.amazon.com/dp/0495391328/ref=cm_sw_r_cp_api_d7ZdzbQ77B65P
This will be enough to tackle ideas from discrete math. I'd recommend reading a book on logic to help with proof techniques and the general idea for rigorously proving statements.
Gensler is a great one but can require a computer if you want more extensive feedback and problem sets.
I personally didn't find discrete maths difficult, but I know it's the big hurdle for most people. Also since my degrees were primarily in mathematics, discrete maths wasn't my first proof-based subject so my experience is very different from most people.
I certainly don't have a comprehensive background in computer science, I've only taken 4 computer science subjects in my 5 years of university education and find my training in mathematics to be highly useful for programming at work and self-study of computer science.
If you can understand and absorb the first four chapters of Discrete Mathematics with Applications (only ~200 pages and should be available in any Uni library) then you'll be well set up for most of the maths that seems to trip people up.
This is the textbook we used and it was incredibly dry (and expensive). I highly recommend this book to ease you into it.
This is the book we are using for our class.
Is the book for 225 necesary? The book store said that there were no required materials for the class but I came across this one several times Discrete Mathematics with Applications 4th Edition
You can read the "Highlights of the Fourth Edition" on Page xvii through Amazon's preview.
Edit: This Amazon review is also relevant to you:
> I've taught discrete math from the 3rd Edition of this book at least 6 times, and struggled with several issues. (The textbook for our Discrete Math course is chosen by a committee in our department.) Much of a discrete math course involves looking closely at some very simple mathematics. Most of the mathematics is already known to a typical university freshman; what a set is, what a prime is, what an ordered pair is, etc. Of course they have had little rigor in these elementary topics, but still, they have the notions and vocabulary. The 3rd Edition pretended that sets, e.g., did not exist until one finally arrived at the chapter on sets. It's unnatural to lecture one's way through two chapters on logic and a chapter on techniques of proof, without being able to draw on simple examples from set theory. One gets tired quickly of examples of dogs and cats in highly artificial situations, and would like to say something about primes or the set of even integers.
>The 4th Edition corrects this problem by the addition of an introductory chapter which fixes the vocabulary and notation. This was a needed change. The 3rd Edition required considerable acrobatics in avoiding words like "is an element of" until Chapter 5 (Set Theory.) Really? I'm supposed to cover the proof technique of "division into cases" and I can't say "the set of integers of the form 4k+1?" So good change.
>Every semester, I get e-mails from my students asking if the previous edition of the text will suffice for my course. Usually, I say yes. In the case of my discrete math course, I'll have to say no. The modifications of this text are substantial. Besides the above, the old Chapter 8 (Recursion) is now incorporated into the new (much expanded) Chapter 5 (Sequences and Induction.) That is also a sensible change.
>My remaining complain about the text is that it's a bit condescending. I think it's bad form to always present mathematics apologetically. "There, there now, I know it's difficult, but we'll go extremely slowly and take tiny, tiny bites covered in catsup so you can scarcely taste them." There's no need for us at the university level to re-enforce the bad attitudes the students learned in high school. It's math. It's hard. You can do it, not because math is made easy, but because you are, in fact, clever enough.
>I would not have recommended the 3rd Edition to anyone, but I would recommend the 4th. I'm very happy with the changes.
Yes, the arrow is obvious when you stop to think about it, but everything becomes obvious after given enough thought. If they had used "=" it would have required slightly less thought and made it easier on the reader. What is the downside? I had no trouble reading the psuedo code, I just take issue with the useless proof ridden overly dense textbooks. You're saying that because it's possible to understand something, we should make no effort to make it easier to understand? This is where a lot of textbooks fail, in my opinion.
And just because a text is easier to understand doesn't make it dumbed down. I had a discrete math class that used this textbook:
http://www.amazon.com/Discrete-Mathematics-Computer-Scientists-Cliff/dp/0132122715/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1345561997&amp;sr=1-1&amp;keywords=discrete+math+for+computer+scientists
It was horribly dense and consisted of proof after proof with no real world examples. I could have learned from it if I so chose to, but I'd rather not spend 30 minutes parsing each page for what they are actually trying to teach me. I instead downloaded:
http://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328/ref=sr_1_2?ie=UTF8&amp;qid=1345561973&amp;sr=8-2&amp;keywords=susanna+epp
And got a 95 in the class. Did I learn less because I used an easier textbook? I would say that I actually learned more, because it took significantly less time to learn the concepts presented, leaving me with more time to learn overall.
Nowhere is this disparity more clear than in data structures and algorithms books.
The book that really helped me prepare for CS 6505 this fall was Discrete Mathematics with Applications by Susanna Epp. I found it easy to digest and it seemed to line up well with the needed knowledge to do well in the course.
Richard Hammack's Book of Proof also proved invaluable. Because so much of your success in the class relies on your ability to do proofs, strengthening those skills in advance will help.
Course Name: Discrete Structures
Course Link: http://uncw.edu/csc/courses-133.html
Text Book: http://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1289513076&amp;sr=1-1
Comments: It wasn't that book when I was there back in the 90's though. We were using a book written by a UNC-W professor and published in-house. But I took another course later, using the 3rd edition of the Susanna Epp book, and I have to say, I prefer Epp's book. But holy fuck would you look at the price for the current edition? WTF?!??
Also, Dr. Berman is nice enough, but is one of the most boring professors ever born... talk about the stereotypical dry, boring, dull, monotone delivery... this guy has it perfected.
FYI, Jaynes actually wrote a whole probability textbook that essentially put together all his thoughts about probability theory. I haven't read it, but many people say it got some good stuff.
Depends what your goal is. As you have a good background, I would not suggest any stats book or deep learning. First, read trough Probability theory - The logic of science and the go for Bishop's Pattern Recognition or Barbers's Bayesian Reasoning and ML. If you understand the first and one of the second books, I think you are ready for anything.
You may also enjoy Probability Theory: The Logic of Science by E.T. Jaynes, and Information Theory, Inference and Learning Algorithms by David McKay.
Probability Theory: The Logic of Science
Probability Theory: The Logic of Science. This is an online pdf, possibly of an older version of the book. Science covers knowledge of the natural world, and mathematics and logic covers knowledge of formal systems.
The deck corresponding to the intellectual property book has ~325 cards.
The deck corresponding to IEA has ~400 cards.
The deck corresponding to linear algebra has ~1000 cards. That seems weird to me, since I feel that I make fewer cards for math books; most of the extra time comes from doing a lot of scratch work. Weird. In addition to timing, I've more recently started keeping track of how many Ankis I've added each section, so maybe I'll have more insight there later. We'll see.
And please message me when you start doing math! If you're looking towards advanced mathematics (beyond calculus/linear algebra-for-engineers), I recommend starting with either Mathematics for Computer Science (review) or, if you really have no interest in doing that, How to Prove It.
ummarycoc has a good point. Snoop around his room and see if he already has How To Prove it: A Structured Approach. Someone bought this book for me and I return to it frequently.
Your mileage with certifications may vary depending on your geographical area and type of IT work you want to get into. No idea about Phoenix specifically.
For programming work, generally certifications aren't looked at highly, and so you should think about how much actual programming you want to do vs. something else, before investing in training that employers may not give a shit about at all.
The more your goals align with programming, the more you'll want to acquire practical skills and be able to demonstrate them.
I'd suggest reading the FAQ first, and then doing some digging to figure out what's out there that interests you. Then, consider trying to get in touch with professionals in the specific domain you're interested in, and/or ask more specific questions on here or elsewhere that pertain to what you're interested in. Then figure out a plan of attack and get to it.
A lot of programming work boils down to:
As a basic primer, you might want to look at Code for a big picture view of what's going with computers.
For basic logic skills, the first two chapters of How to Prove It are great. Being able to think about conditional expressions symbolically (and not get confused by your own code) is a useful skill. Sometimes business requirements change and require you to modify conditional statements. With an understanding of Boolean Algebra, you will make fewer mistakes and get past this common hurdle sooner. Lots of beginners struggle with logic early on while also learning a language, framework, and whatever else. Luckily, Boolean Algebra is a tiny topic. Those first two chapters pretty much cover the core concepts of logic that I saw over and over again in various courses in college (programming courses, algorithms, digital circuits, etc.)
Once you figure out a domain/industry you're interested in, I highly recommend focusing on one general purpose programming language that is popular in that domain. Learn about data structures and learn how to use the language to solve problems using data structures. Try not to spread yourself too thin with learning languages. It's more important to focus on learning how to get the computer to do your bidding via one set of tools - later on, once you have that context, you can experiment with other things. It's not a bad idea to learn multiple languages, since in some cases they push drastically different philosophies and practices, but give it time and stay focused early on.
As you gain confidence there, identify a simple project you can take on that uses that general purpose language, and perhaps a development framework that is popular in your target industry. Read up on best practices, and stick to a small set of features that helps you complete your mini project.
When learning, try to avoid haplessly jumping from tutorial to tutorial if it means that it's an opportunity to better understand something you really should understand from the ground up. Don't try to understand everything under the sun from the ground up, but don't shy away from 1st party sources of information when you need them. E.g. for iOS development, Apple has a lot of development guides that aren't too terrible. Sometimes these guides will clue you into patterns, best practices, pitfalls.
Imperfect solutions are fine while learning via small projects. Focus on completing tiny projects that are just barely outside your skill level. It can be hard to gauge this yourself, but if you ever went to college then you probably have an idea of what this means.
The feedback cycle in software development is long, so you want to be unafraid to make mistakes, and prioritize finishing stuff so that you can reflect on what to improve.
How to Prove It by Vellemen is a superb introduction to what proofs are, and how to make them.
Keep in mind certain proof based courses can be frustrating to some students (discrete math and real analysis) as these classes often make formal concepts students may understand intuitively. Abstract Algebra or Topology may give you a more accurate idea of your feelings towards math.
Depends on what you are looking for. You might not be aware that the concepts in that book are literally the foundations of math. All math is (or can be) essentially expressed in set theory, which is based on logic.
You want to improve math reasoning, you should study reasoning, which is logic. It's really not that hard. I mean, ok its hard sometimes but its not rocket science, its doable if you dedicate real time to it and go slowly.
Two other books you may be interested in instead, that teach the same kinds of things:
Introduction to Mathematical Thinking which he wrote to use in his Coursera course.
How to Prove It which is often given as the gold standard for exactly your question. I have it, it is fantastic, though I only got partway through it before starting my current class. Quite easy to follow.
Both books are very conversational -- I know the second one is and I'm pretty sure the first is as well.
What books like this do is teach you the fundamental logical reasoning and math structures used to do things like construct the real number system, define operations on the numbers, and then build up to algebra step by step. You literally start at the 1+1=2 type level and build up from there by following a few rules.
Also, I just googled "basic logic" and stumbled across this, it looks like a fantastic resource that teaches the basics without any freaky looking symbols, it uses nothing but plain-English sentences. But scanning over it, it teaches everything you get in the first chapter or two of books like those above. http://courses.umass.edu/phil110-gmh/text/c01_3-99.pdf
Honestly if I were starting out I would love that last link, it looks fantastic actually.
Read this book: http://www.amazon.com/How-Study-as-Mathematics-Major/dp/0199661316/ref=asap_bc?ie=UTF8
And work through this book in its entirety: http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1463201632&amp;sr=1-1
Like one of the other's suggested, learning more about proofs is probably what you're interested in since that's where these rules and equations come from. I've seen this book recommended a few times. It should give you a better understanding of how math is formed.
Probably something on coursera; however, I really recommend this gem of a book, http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1458411780&amp;sr=1-1&amp;keywords=how+to+prove+it .
Get the book [How to Prove It: A Structured Approach by Daniel J. Velleman] (http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995) it will teach you how to write, and I think more importantly, read proofs.
How comfortable are you with proofs? If you are not yet comfortable, then read this: How to Prove It: A Structured Approach
I may be in the minority here, but I think that high school students should be exposed to statistics and probability. I don't think that it would be possible to exposed them to full mathematical statistics (like the CLT, regression, multivariate etc) but they should have a basic understanding of descriptive statistics. I would emphasize things like the normal distribution, random variables, chance, averages and standard deviations. This could improve numerical literacy, and help people evaluate news reports and polls critically. It could also cut down on some issues like the gambler's fallacy, or causation vs correlation.
It would be nice if we could teach everyone mathematical statistics, the CLT, and programming in R. But for the majority of the population a basic understanding of the key concepts would be an improvement, and would be useful.
Edit At the other end of the spectrum, I would like to see more access to an elective class that covers the basics of mathematical thinking. I would target this at upperclassmen who are sincerely interested in mathematics, and feel that the standard trig-precalculus-calculus is not enough. It would be based off of a freshman math course at my university, that strives to teach the basics of proofs and mathematical thinking using examples from different fields of math, but mostly set theory and discrete math. Maybe use Velleman's book or something similar as a text.
How to Prove it by Velleman seems to be right up your alley.
Thanks for the answer!
Glad to hear about Spivak! I've heard good things about that textbook and am looking forward to going through it soon :). Are the course notes for advanced algebra available online? If so, could you link them?
Is SICP used only in the advanced CS course or the general stream one, too? (last year I actually worked my way through the first two chapters before getting distracted by something else - loved it though!) Also, am I correct in thinking that the two first year CS courses cover functional programming/abstraction/recursion in the first term and then data structures/algorithms in the second?
That's awesome to know about 3rd year math courses! I was under the impression that prerequisites were enforced very strongly at Waterloo, guess I was wrong :).
As for graduate studies in pure math, that's the plan, but I in no way have my heart set on anything. I've had a little exposure to graph theory and I loved it, I'm sure that with even more exposure I'd find it even more interesting. Right now I think the reason I'm leaning towards pure math is 'cause the book I'm going through deals with mathematical logic / set theory and I think it's really fascinating, but I realize that I've got 4/5 years before I will even start grad school so I'm not worrying about it too much!
Anyways, thanks a lot for your answer! I feel like I'm leaning a lot towards Waterloo now :)
How to Prove It is a nice introduction to writing proofs.
You should absolutely not give up.
None of this is groundbreaking, and a lot of it is pretty cliché, but it's true. Everyone struggles with math at some point. Einstein said something like "whatever your struggles with math are, I assure you that mine are greater."
As for specific recommendations, make the most of this summer. The most important factor in learning math in my experience is "time spent actively doing math." My favorite math quote is "you don't learn math, you get used to it." I might recommend a book like How to Prove It. I read it the summer before I entered college, and it helped immensely with proofs in real analysis and abstract algebra. Give that a read, and I bet you will be able to prove most lemmas in undergraduate algebra and topology books, and solve many of their problems. Just keep at it!
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
helps with the first part of the class. the stuff after that I would suggest just having good google-fu.
Math isn't going to be like the math classes you've already taken. It's a lot of writing and logic and very little calculating. If you go for mathematical sciences, you'll probably take more classes that involve calculations, but you won't make it that far if you can't handle proofs.
Check out this book: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
The full book is first-page googleable. If you find that material interesting, you'll probably enjoy being a math major.
To learn basic proof writing I highly recommend How to Prove It by Velleman.
I study topology and I can give you some tips based on what I've done. If you want extra info please PM me. I'd love to help someone discover the beautiful field of topology. TLDR at bottom.
If you want to study topology or knot theory in the long term (actually knot theory is a pretty complicated application of topology), it would be a great idea to start reading higher math ASAP. Higher math generally refers to anything proof-based, which is pretty much everything you study in college. It's not that much harder than high school math and it's indescribably beneficial to try and get into it as soon as you possibly can. Essentially, your math education really begins when you start getting into higher math.
If you don't know how to do proofs yet, read How to Prove It. This is the best intro to higher math, and is not hard. Absolutely essential going forward. Ask for it for the holidays.
Once you know how to prove things, read 1 or 2 "intro to topology" books (there are hundreds). I read this one and it was pretty good, but most are pretty much the same. They'll go over definitions and basic theorems that give you a rough idea of how topological spaces (what topologists study) work.
After reading an intro book, move on to this book by Sutherland. It is relatively simple and doesn't require a whole lot of knowledge, but it is definitely rigorous and is definitely necessary before moving on.
After that, there are kind of two camps you could subscribe to. Currently there are two "main" topology books, referred to by their author's names: Hatcher and Munkres. Both are available online for free, but the Munkres pdf isn't legally authorized to be. Reading either of these will make you a topology god. Hatcher is all what's called algebraic topology (relating topology and abstract algebra), which is super necessary for further studies. However, Hatcher is hella hard and you can't read it unless you've really paid attention up to this point. Munkres isn't necessarily "easier" but it moves a lot slower. The first half of it is essentially a recap of Sutherland but much more in-depth. The second half is like Hatcher but less in-depth. Both books are outstanding and it all depends on your skill in specific areas of topology.
Once you've read Hatcher or Munkres, you shouldn't have much trouble going forward into any more specified subfield of topology (be it knot theory or whatever).
If you actually do end up studying topology, please save my username as a resource for when you feel stuck. It really helps to have someone advanced in the subject to talk about tough topics. Good luck going forward. My biggest advice whatsoever, regardless of what you study, is read How to Prove It ASAP!!!
TLDR: How to Prove It (!!!) -> Mendelson -> Sutherland -> Hatcher or Munkres
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
This book was very helpful to me.
I would recommend the following two books:
The first book introduces most of the topics in the book that you linked, and was what was used in my Foundations of Mathematics class (essentially the same thing as your class).
Understanding Analysis, on the other hand, is probably the perfect book to follow up with, since it is such a well-motivated, yet rigorous book on the analysis of one real variable, that you may, in fact, become too accustomed to such lucid and entertaining prose for your own good.
Good advice, but I'd add that if you do revisit calc get an intro to analysis textbook to understand how we derived the rules that calc uses. For instance, an integral is not defined as an antiderivative, that had to be proven.
Edit: My class used Principles of Mathematical Analysis by Rudin. It requires little to no initial knowledge and essentially builds multivariable calculus from the ground up.
Have you ever seen how much technical books cost?
For example, here's the standard text for mathematical analysis: Principles of Mathematical Analysis. That's $87 for a 325 page book.
Nobody's pretending that printing/binding/distributing is a significant fraction of that cost so an ebook would likely be similarly priced, maybe slightly less, possibly slightly more.
Manning, in particular, focuses on texts in computer science and programming for which such prices are pretty standard. The price difference between the ebook and print+ebook varies (I think it's proportional for most of their texts) but if the ebook is $35 then the physical+ebook is usually around $45. Again, this is very reasonable for a quality text in the field.
I guess I don't have a clear idea what an "elementary math degree" entails, so let me put it this way:
I learned about space-filling curves in my second semester of Real Analysis. First-semester Real Analysis was the first upper-division math class people take at my college; the second-semester is typically taken Junior or Senior year by those who are particularly passionate about the subject. It is not, as a general rule, a subject I recommend learning without the benefit of an instructor - at least, not from the book I used. To be clear: its a good reference book, and I developed a healthy respect for its approach to the subject in time, but its not the most user-friendly book as you're getting going.
To briefly paraphrase the argument: you basically construct a fractal via a sequence of functions, and then argue based on the convergence and continuity properties of the function family that a) the function they converge to is continuous and b) it passes through every point in the area to be covered.
I just bought this, and I'm waiting for it to be shipped. I heard it is life-changing.
This may not exactly be an answer to your question but I would recommend buying this book: https://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192
It's not quite a textbook nor it is a pop-sci book for the layperson. The blurb on the front says " "A lucid representation of the fundamental concepts and methods of the whole field of mathematics." - Albert Einstein"
In and of itself it is not a complete curriculum. It doesn't have anything about linear algebra for example but you could learn a lot of mathematics from it. It would be accessible to a reasonably intelligent and interested high-schooler, it touches on a variety of topics you may see in an undergraduate mathematics degree and it is a great introduction to thinking about mathematics in a slightly more creative and rigorous way. In fact I would say this book changed my life and I don't think I'm the only one. I'm not sure if i would be pursuing a degree in math if I had never encountered it. Also it's pretty cheap.
If you're still getting a handle on how to manipulate fractions and stuff like that you might not be ready for it but you will be soon enough.
Like justrasputin says, there usually is quite a lot of work to be done before you start to really see the beauty everyone refers to. I'd like to suggest a few book about mathematics, written by mathematicians that explicitly try to capture the beauty -
By Marcus Du Sautoy (A group theorist at oxford)
By G.H. Hardy,
Also, a good collection of seminal works -
God Created the Integers
And a nice starter -
What is Mathematics
Good luck and don't give up!
You might want to try "What is Mathematics?" by R.Courant and H.Robbins. The book is written for people new to the field of theoretical mathematics and is intended for those who wish to develop a solid foundation on the topic.
I had started college as an engineer, switched to English, and now work as an ESL instructor. However, my love of math never died (despite my university professors' best attempts). So, I picked up that book a little while ago. It's a good read (albeit a dense one), and it covers a little bit of what you have listed.
[Amazon link here] (http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192)
Edit: some words
This is a thing. I read a decent book with a lot of cool math tricks
https://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401
My favorite book that has a ton of these is this book. I remember seeing the author do all kinds of math tricks on talk shows. My favorite was determining what day of the week any date in history was (or at least, after the start of the Gregorian calendar)
The following easy to read book teaches kids (and adults) you how to do it. Its actually really easy:
Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks
Secrets of Mental Math
A lot of these tricks are very easy. He explains them all in his book Secrets of Mental Math
Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks
Secrets of Mental Math May be helpful for filling in some gaps. Also A Mind for Numbers gives helpful meta learning info: how to study, etc.
I began doing it in my head the same way. For clarity, my thought processes were based on the idea of "don't do something hard, do something easier instead and then fix it up afterwards", roughly:
(I stopped there because I just wasn't looking forward to adding 973 to 243250, but was pretty sure I could slog my way through it if I actually had to.)
But there are lots of tricks you can do to make mental math easier. I don't know them, but like the above, I know that I could* go and learn them. For example, here is a book by one of the world's best people at mental arithmetic, Arthur Benjamin; the book is filled with techniques you can use to make mental arithmetic easier. See him on TED here.
I'm only a high school sophomore, so I can't really help you with most of your questions, but if you want to improve your mental math, buy "Secrets of Mental Math" by Arthur Benjamin.
It's written in a way that makes sitting in your room doing mental calculations seem fun and it is very accessible. I have only gotten through 3 chapters (the addition/subtraction/multiplication chapters) and I can confidently add and subtract 3-digit numbers in seconds. I can even mentally cube two-digit numbers in a few minutes.
[Anyway, here's a link to the book] (http://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1381633585&amp;sr=1-1&amp;keywords=mental+math)
[If you don't want to buy it, you can use this PDF version of the book] (http://www.uowm.gr/mathslife/images/fbfiles/files/Secrets_of_Mental_Math___Michael_Shermer___Arthur_Benjamin.pdf)
[And here is the author, Arthur Benjamin, performing what he likes to call "Mathemagics"] (http://www.youtube.com/watch?v=e4PTvXtz4GM)
I hope this has been helpful and you succeed in whatever uni you go to :)
Check out Secrets of Mental Math by Arthur Benjamin. Benjamin is amazing, I've seen him at MAA meetings. He does lightning fast calculations in his head, and his book shows you how to do it. Your students may or may not think this is cool, but I do :) And the bonus is that they will never learn this kind of thing in school at any grade, so you won't be stepping on anybody's toes by teaching it to them now.
Also, the "third grade team" sucks. Screw those guys.
Meh, was a bit anti-climactic.
I preferred Fermat's Last Theorem. That took 350 years to solve, not just a quick google. Kids today etc...
Fermat's Last Theorem is a pretty good story. It's an easy to understand problem that was unsolved for 300 years until ~20 years ago.
There's a book about it and a PBS documentary you can watch for free.
When I was his age, I read a lot of books on the history of mathematics and biographies of great mathematicians. I remember reading Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem.
Any book by Martin Gardner would be great. No man has done as much to popularize mathematics as Martin Gardner.
The games 24 and Set are pretty mathematical but not cheesy. He might also like a book on game theory.
It's great that you're encouraging his love of math from an early age. Thanks to people like you, I now have my math degree.
Maybe a bit off topic, but I think that if you have a "math phobia" as you say, then maybe you need to find a way to become interested in the math for math's sake. I don't think you'll be motivated to study unless you can find it exciting.
For me, The Universal History of Numbers was a great book to get me interested in math. It's a vast history book that recounts the development of numbers and number systems all over the world. Maybe by studying numbers in their cultural context you'll find more motivation to study, say, the real number system (leading to analysis and so on). That's just an example and there are other popular math books you could try for motivation (Fermat's Enigma is good).
Edit: Also, there are numerous basic math books that are aimed at educated adults. Understanding Mathematics is one which I have read at one point and wasn't bad as far as I can remember. I am sure there are more modern, and actually for sale on Amazon, books on this topic though.
Fermat's Enigma by Simon Singh is an approachable history of Fermat's last theorem, various brilliant but failed proofs, and Wiles' ultimate conquest. While it's not technical, the book profiles the mathematicians tormented by Fermat's theorem and details the approaches they used. You may find it helpful as a map or a timeline. Certainly worth reading.
http://www.amazon.com/Fermats-Enigma-Greatest-Mathematical-Problem/dp/0385493622
Serge Lang's Basic Mathematics is probably the place to start if its been 8 years.
Basic Mathematics by Serge Lang is one. Not free, though.
I like to reccommend Basic Mathematics by Serge Lang. It will take you exactly from addition and subtraction to a prepared state for calculus and beyond. Don't let the name fool you though, it is a rigorous study, but with an honest effort you will do well.
hey i found something you need
Since you hope to study mathematics more seriously, I would look into this book link.
It's an excellent book that treats high school/basic college mathematics in an "adult" way. By adult I mean in the way that mathematicians think about it.
(The fun thing about Lang is that you can read only his books and get pretty much a high school through advanced graduate education).
Walter Rudin, principle of mathematical analysis
Hmm. I kept almost all my textbooks. I just looked through them and the most expensive one I could find cost $47.97 in 1987. That calculator says it would be $100.60 in 2014 dollars. I just checked Amazon, and it's now $109.15. Pretty close.
I seem to recall one book costing $80 or more, but I didn't write the prices on all my books. My books were math or statistics, and cost more than nonmathematical texts, but I always figured that was the cost of typesetting (which I'd guess is not as much a consideration as it once was).
I've found Rudin's Analysis useful. There's a lecture series on YouTube that roughly follows the book.
You could try Principles of mathematical analysis by Rudin. This is too much for me, so be warned.
I find Spivak's Calculus to be a lot more palatable, but I've read less of it than Rudin.
THIS WILL PUT HAIR ON HIS CHEST:
http://www.amazon.com/Principles-Mathematical-Analysis-Third-Walter/dp/007054235X
Baby Rudin, The Inferno, DPV
For the whys and hows, you're gonna need a full-blown analysis textbook like baby Rudin. Calc I and II at most universities don't even scratch the surface when it comes to understanding the whys of anything. Anyways, yeah. Engineering is cool.
RPCV checking in. This is a good idea... you're going to have a lot of downtime and it's a great opportunity to read all the things you've wanted to but haven't yet found the time for. That could mean math, or languages, or just old novels.
When I was learning functional analysis, if found this book by Bollobas to be incredibly helpful. Of course, the only real analysis reference you need is Baby Rudin, but if you want to learn measure theory you may want his Real & Complex Analysis instead.
For texts on the other subjects, take a look at this list. You should be able to find anything you need there.
If you have any questions about Peace Corps, feel free to PM me. Good luck!
Is this the book you are recommending?
Rudin.
This is super helpful, thank you!
And nothing against simulation, I know it's a powerful tool. I just don't want my foundations built on sand (I'm familiar with intro stats already).
Would Rubin's book on Real Analysis suffice: http://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X
Or are there even more advanced texts to pursue for Real Analysis?
A course I took previously used this book; it has a chapter on introductory real analysis, which is what you want to get at. I would not suggest going directly to a book like Rudin, as he (in my opinion) tends to amplify the "general route" problem that you mention.
Try Baby Rudin. I think the first chapter covers what you are looking for very thoroughly.
You might also find Analysis: With an Introduction to Proof to be rather helpful.
Oh. I'm sorry. I thought your name was in reference to the mathematician walter rudin. He wrote some popular upper undergraduate and graduate math books on analysis (baby rudin and papa rudin respectively). There are many math definitions and proofs in these books with very little background into what purpose they may serve in an applied mathematical field.
baby rudin
papa rudin
> You're not wrong, you're just an asshole. Anything else you'd like to say about how great you are? Tell me about me your thesis. I'll bet it's extremely groundbreaking stuff.
My thesis is on chaotic behavior of swarm traffic, swarm traffic analysis and using spectral graph theory to predict traffic patterns. Very fun, but something you really need schooling for.
You're right, I'm an asshole. And that may be so. Maybe you should put down the drugs and try to learn something that takes actual mental capacity like Real Analysis, and maybe I won't be such an asshole.
Edit: If you want to learn it on your own Rudin is the best.
I haven't read it yet, but Richard Courant's What is Mathematics? has been highly recommended to me.
Be brave, start with http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192
I recommend going through some of the lessons on Brilliant, and here is Brilliant's quick exposition on the set of complex numbers.
I don't know what a softer explanation would entail exactly, but I would offer you the alternative perspective that the representation of complex numbers as two real numbers a+ib for the real numbers a and b is extremely useful because of the interpretation of the extension of the one dimensional real number line into the two dimensional complex plane.
Also, I recommend reading on a simple exposition of complex numbers from Richard Courant's "What is Mathematics".
What Is Mathematics?: An Elementary Approach to Ideas and Methods
"Succeeds brilliantly in conveying the intellectual excitement of mathematical inquiry and in communicating the essential ideas and methods." Journal of Philosophy
https://www.amazon.ca/What-Mathematics-Elementary-Approach-Methods/dp/0195105192
Seriously this may be a great coming-of-age title for you: Infinite Jest.
Also since you got your first job check out The Wall Street Journal's Guide to Starting Your Financial Life. If you haven't yet appreciated math, I would suggest you do so as you're going to need it for any decent job these days. Detach yourself from Fallacious Thought.
Aside from The Princeton Companion to Mathematics, you might like to check out What Is Mathematics? An Elementary Approach to Ideas and Methods by Courant and Robbins, and Mathematics: Its Content, Methods and Meaning by three Russian authors including Kolmogorov.
Aleksandrov, Kolmogorov, Lavrent'ev. http://amzn.com/0486409163. Foundations to applicationsl.
Courant, Robbins, Stewart. http://amzn.com/0195105192. Tour of mathematics.
Honestly, I highly recommend this book, and pretty much anything else by Arthur Benjamin. He's the real deal when it comes to mental math. Take it seriously, and do tons of practice problems. Feel free to go "fast" through the book the first time through, but go super slow the second time through and get everything super solid.
After completing the book you'll be able to do squares, multiplication, division, addition, subtraction pretty damn fast up to around 3-4 digits. With more practice you can eventually get as good as Prof. Benjamin (he doesn't leave anything out! Tells you the entire technique). By more, I mean years more, but hey, at least it's possible
Wow thanks. Also read Secrets of Mental Math. It provides lots of helpful tricks.
Aside from Khan, The Secrets of Mental Math was extremely helpful in this endeavor.
I used to be just like you, then really became fascinated by physics, which was very difficult given my deficiencies in math. I figured I would start with flash cards and what not, so I started browsing amazon and came across this. This guy is a genius, and teaches you a lot of tricks to do math quickly in your head. The next thing I did was checked out Khan Academy. I can not over-exaggerate how utterly fucking awesome this site is. Not only does he have like 2300+ videos on every topic, but he has something like 125 math modules that allow you to practice. It's completely free and all you need is a facebook or gmail account to log in...
I would enumerate on the various techniques I've used over the years, which drove my early math teachers somewhat mad, but, well, those little tricks and more are readily available in the book The Secrets Of Mental Math. I never finished the book, but it's got quite a few very useful tips, just in the opening couple of chapters, and it builds on them to add other neat things.
Benjamin Arthur is great at this. He wrote a book that may interest you.
I read this book in high school when it was originally published as "Mathemagics." https://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401/ref=pd_lpo_sbs_14_t_2?_encoding=UTF8&psc=1&refRID=WQYSFNW9WRJY77M30PZG
Its a collection of tips and shortcuts to make mental math easier. I really enjoyed it and found it very useful.
Asked myself the same question this morning. I found this book is supposed to be a good start.
http://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401
Read the book by Arthur Benjamin. He's one of my role models. :D The book has the most amazing mental math tricks ever, and I can square 2, 3, and even 4 digit numbers in my head. Getting to 5 digits soon. There are a lot of other cool tricks in there as well.
Here's another one that's pretty good
Little mental trick you can do to show off to some people:
any number * 11 is easy. Even in the 2 digits.
Let's do 32 again.
*32 11
Separate 32 into two digits, add them, and then put that number between those two digits. For example:
3 + 2 = 5
place between the two original digits:
352
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This works with three digits as well (but I have to go figure out how to do that one again). There is a book on the Apple Store that is an awesome read if you're into it. All of the things I am showing you are possible to do mentally. I can currently square 4 digit numbers in my head sorta reliably, and can square 3 and 2 digit numbers without fail. It is really fun and I enjoy doing it.
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EDIT:
PLEASE PLEASE PLEASE support this guy and do not download a pdf of the book. He is absolutely incredible with what he can do and is sharing it with people so they can do it too. Give him credit!
Book on Amazon
Book on iBooks
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Youtube video of this guy
This is one of the methods suggested in this book: https://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401
It’s a really nice read for doing mental math. The author, Arthur Benjamin, has some really impressive videos on YouTube IIRC
I'm premed, the most I know is just 2 semesters of calculus. However I am reading [Mental Math tricks] (http://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401). I don't know what good that'll do me other than make look more of a nerd than I do now. I am also learning how to program and work with computer. I'm starting small with PyScript and trying to get A+ certified.
Well, regarding Fermat's Last Theorem, it indeed was written by Aczel, as could easily be determined by following the link in the article. However, it looks like there are 100s of books with a similar name. The one your read by Simon Singh was called: Fermat's Enigma.
You weren't misled, you just "misremembered".
Initially I'd avoid books on areas of science that might challenge her (religious) beliefs. You friend is open to considering a new view point. Which is awesome but can be very difficult. So don't push it. Start slowly with less controversial topics. To be clear, I'm saying avoid books that touch on evolution! Other controversial topics might include vaccinations, dinosaurs, the big bang, climate change, etc. Picking a neutral topic will help her acclimate to science. Pick a book related to something that she is interested in.
I'd also start with a book that the tells a story centred around a science, instead of simply trying to explain that science. In telling the story their authors usually explain the science. (Biographies about interesting scientist are a good choice too). The idea is that if she enjoys reading the book, then chances are she will be more likely to accept the science behind it.
Here are some recommendations:
The Wave by Susan Casey: http://www.amazon.com/The-Wave-Pursuit-Rogues-Freaks/dp/0767928857
Fermat's Enigma by Simon Singh: http://www.amazon.com/Fermats-Enigma-Greatest-Mathematical-Problem/dp/0385493622
The Man who Loved Only Numbers by Paul Hoffman: http://www.amazon.com/Man-Who-Loved-Only-Numbers/dp/0786884061/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1405720480&amp;sr=1-1&amp;keywords=paul+erdos
I also recommend going to a book store with her, and peruse the science section. Pick out a book together. Get a copy for yourself and make it a small book club. Give her someone to discusses the book with.
After a few books, if she's still interested then you can try pushing her boundaries with something more controversial or something more technical.
You can start with "Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem" >> http://www.amazon.com/Fermats-Enigma-Greatest-Mathematical-Problem/dp/0385493622 . Is a great book, I read it several times.
Another vote for The Code Book, as a book targeted more towards the general public, I thought it was excellent. I read it in high school and it's one of the reasons I decided to go into math/CS in university!
Fermat's Enigma (also by Singh) is another one I enjoyed.
A great book on Andrew Wiles and Fermat's Last Theorem is Simon Singh's Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem.
This whole thing reminds me of a book I read a few years back about a guy who proved Fermat's Last Theorem. Fascinating stuff. Really gives one an insight into how beautiful the human mind is.
I've heard people recommend Kiselev's Geometry, on a physics forum. Warning, though; Kiselev's Geometry series(in English) is translated from Russian.
Here's the link to where I got all these resources(I also copy-pasted what's in the link down below; although, I did omit a few entries, as it would be too long for this reddit comment; click the link to see more resources):
https://www.physicsforums.com/insights/self-study-basic-high-school-mathematics/
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Note: Alternatively, you can order Kiselev's geometry series from http://www.sumizdat.org/
Geometry I and II by Kiselev
http://www.amazon.com/Kiselevs-Geometry-Book-I-Planimetry/dp/0977985202
http://www.amazon.com/Kiselevs-Geometry-Book-II-Stereometry/dp/0977985210
> If you do not remember much of your geometry classes (or never had such class), then you can hardly do better than Kiselev’s geometry books. This two-volume work covers a lot of synthetic (= little algebra is used) geometry. The first volume is all about plane geometry, the second volume is all about spatial geometry. The book even has a brief introduction to vectors and non-Euclidean geometry.
The first book covers:
The second book covers:
> This book should be good for people who have never had a geometry class, or people who wish to revisit it. This book does not cover analytic geometry (such as equations of lines and circles).
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Geometry by Lang, Murrow
http://www.amazon.com/Geometry-School-Course-Serge-Lang/dp/0387966544
> Lang is another very famous mathematician, and this shows in his book. The book covers a lot of what Kiselev covers, but with another point of view: namely the point of view of coordinates and algebra. While you can read this book when you’re new to geometry, I do not recommend it. If you’re already familiar with some Euclidean geometry (and algebra and trigonometry), then this book should be very nice.
The book covers:
> This book should be good for people new to analytic geometry or those who need a refresher.
> Finally, there are some topics that were not covered in this book but which are worth knowing nevertheless. Additionally, you might want to cover the topics again but this time somewhat more structured.
> For this reason, I end this list of books by the following excellent book:
Basic Mathematics by Lang
http://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/0387967877
> This book covers everything that you need to know of high school mathematics. As such, I highly advise people to read this book before starting on their journey to more advanced mathematics such as calculus. I do not however recommend it as a first exposure to algebra, geometry or trigonometry. But if you already know the basics, then this book should be ideal.
> I recommend this book to everybody who wants to solidify their basic knowledge, or who remembers relatively much of their high school education but wants to revisit the details nevertheless.
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More links:
https://math.stackexchange.com/questions/34442/book-recommendation-on-plane-euclidean-geometry
Note: oftentimes, you can find geometry book recommendations( as well as other math book recommendations) in stackexchange; just use the search bar.
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https://www.physicsforums.com/threads/geometry-book.727765/
https://www.physicsforums.com/threads/decent-books-for-high-school-algebra-and-geometry.701905/
https://www.physicsforums.com/threads/micromass-insights-on-how-to-self-study-mathematics.868968/
Here is a good book on trigonometry.
Here is one for algebra.
Here's another
The Serge Lang book looks to be pretty expensive on Amazon, is it worth it?
Thank you for the recommendations, the Gelfand books look like they're worth checking in to!
Okay. So..
You speak of this book I assume. Which is intended to be used by students in H.S. Yet you are familiar with abstract algebra? I understand abstract algebra has many levels to it. But how far did you go? Was it so close that you were touching on topographies or statements?
I'm very confused here. You're concerned about your math. But yet you're reading a calculus prep book?
What is an IT college exactly? Are you a freshman or sophomore at a Uni? And it happens that you are referring to your department? Or are you referring to a technical college / school?
These questions are to satisfy my assumptions. Optional at best.
As a math major with a CS minor in my uni, which is something I'm in the process of. I am required pre-algebra, algebra, pre-calc, calc, calc 1, calc 2, calc 3, abstract algebra, linear algebra, discreet math, some general programming classes involving these prerequisite math courses, and some other math classes I cannot remember.
Abstract algebra, in my opinion is something of a higher level language. So this should explain my confusion here.
First, please make sure everyone understands they are capable of teaching the entire subject without a textbook. "What am I to teach?" is answered by the Common Core standards. I think it's best to free teachers from the tyranny of textbooks and the entire educational system from the tyranny of textbook publishers. If teachers never address this, it'll likely never change.
Here are a few I think are capable to being used but are not part of a larger series to adopt beyond one course:
Most any book by Serge Lang, books written by mathematicians and without a host of co-writers and editors are more interesting, cover the same topics, more in depth, less bells, whistles, fluff, and unneeded pictures and other distracting things, and most of all, tell a coherent story and argument:
Geometry and solutions
Basic Mathematics is a precalculus book, but might work with some supplementary work for other classes.
A First Course in Calculus
For advanced students, and possibly just a good teacher with all students, the Art of Problem Solving series are very good books:
Middle & high school:
and elementary linked from their main page. I have seen the latter myself.
Some more very good books that should be used more, by Gelfand:
The Method of Coordinates
Functions and Graphs
Algebra
Trigonometry
Lines and Curves: A Practical Geometry Handbook
http://www.amazon.ca/Basic-Mathematics-Serge-Lang/dp/0387967877
My two cents
If you want to improve your skills you can do two things in the short term -- read and practice.
I would recommend Basic Mathematics by Lang (it gets mentioned a lot around here). Or if you are interested in higher math look at How to Prove It by Velleman
The great thing is that both include exercises.
The two books already mentioned sound awesome, but if you ever wanted a textbook with a formal approach to mathematics (written by a well-known and respected mathematician), check out Basic Mathematics by Serge Lang.
This is more for anyone reading who would like to continue on to a math or perhaps a physics major. The book takes you from elementary algebra and geometry all through pre-calculus; basically the only book you should need to prepare you for calculus and elementary linear algebra.
I've heard good things about Serge Lang's Basic Mathematics. It's pre-calculus geometry and algebra mainly I think, but it treats you like a grown-up.
Don't skip proofs and wrestle through them. That's the only way; to struggle. Learning mathematics is generally a bit of a fight.
It's also true that computation theory is essentially all proofs. (Specifically, constructive proofs by contradiction).
You could try a book like this: https://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472108/ref=sr_1_1?ie=UTF8&amp;qid=1537570440&amp;sr=8-1&amp;keywords=book+of+proof
But I think these books won't really make you proficient, just more familiar with the basics. To become proficient, you should write proofs in a proper rigorous setting for proper material.
Sheldon Axler's "Linear Algebra Done Right" is really what taught me to properly do a proof. Also, I'm sure you don't really understand Linear Algebra, as will become very apparent if you read his book. I believe it's also targeted towards students who have seen linear algebra in an applied setting, but never rigorous and are new to proof-writing. That is, it's meant just for people like you.
The book will surely benefit you in time. Both in better understanding linear algebra and computer science classics like isomorphisms and in becoming proficient at reading/understanding a mathematical texts and writing proofs to show it.
I strongly recommend the second addition over the third addition. You can also find a solutions PDF for it online. Try Library Genesis. You don't need to read the entire book, just the first half and you should be well-prepared.
As others mentioned, it is very hard to make progress learning programming without using a computer (think of reading about driving without ever driving a car). Instead, get yourself excited about science and computer science:
Science:
Computer Science (actually math, but this will help change the way you think to be more analytical, and will be useful for programming, vector graphics, etc.):
I think linear algebra is a much more interesting topic without getting bogged down in matrix computations, such as what Axler does with Linear Algebra Done Right. That's just my opinion I suppose.
Calc 3 was series for us, 4 was multivariable. We were quarters with summer quarter being optional so it was really trimesters for most people. Vector calc was basically taught from the book Div, Grad, Curl and All That. So it was useful prior to going into electrodynamics, which was also 4th year.
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EDIT: Added link.
There's a book called Div, Grad, Curl and All That, here is an Amazon link. It's an informal approach to vector mathematics for scientists and engineers and it's pretty readable. If you're struggling with the math, this is for you :) All their examples are EM too.
It's also a good idea to get a study group together. The blind leading the blind actually do get somewhere. :)
http://www.amazon.com/dp/0471725692/ref=wl_it_dp_o_pd_S_ttl?_encoding=UTF8&amp;colid=2UCFQZHNW5VVF&amp;coliid=I1RPWVCSMOOV09 is one good suggestion, I've seen around here. It's on my wishlist and the book that I intend to work from.
Now I always struggled with vector calculus and its motivations. So I have this one waiting for me as well http://www.amazon.com/dp/0393925161/ref=wl_it_dp_o_pC_nS_ttl?_encoding=UTF8&amp;colid=2UCFQZHNW5VVF&amp;coliid=I20JETA4TTSTJY since I think it covers a lot of the concepts that I had the most trouble with in calc 3
Div, Grad, Curl, and All That is a good way to shore up your knowledge of vector calc.
When I took EM in addition to Cheng the professor suggested getting Div, Grad, Curl and all of that. I found that to be alot of help in solidifying the math and intuition needed.
Whoa, great questions, but I think you want a textbook, not a reddit post response. I used Dummit & Foote but it is probably a bit "heavier" than what you want/need at this point.
Griffiths' Quantum Mechanics has a crash course in most of the linear algebra required to do a first course in quantum mechanics. It's not very complicated - you just need basic understanding of vector spaces, linear transformations and functionals, and inner products, with a little bit of practice using dual notation of vectors (not too much, just enough for the Dirac notation which the book explains). Griffiths' also has a good explanation of simple fourier series/transform.
The key thing is being able to do basic linear algebra without matrices since in most of the cases, the vector space is infinite dimensional. But spin is a good example where almost everything can be done with matrices.
Additionally, solving ordinary differential equations and using separation of variables for partial differential equations in 3-d quantum mechanics would help.
Group theory will be of help in more advanced classes. Dummit and Foote or Arton's books on algebra are decent introduction. They are a bit dense though. If you want a real challenge, try Lang's Algebra book. I don't know of any easier books though. My first algebra book was Dummit and Foote which can be done without any real prerequisites beyond matrix algebra, but isn't really well written.
Links to books: Griffiths, Dummit and Foote.
PS: I have ebooks of these two books in particular.
Dummit (or just D&F), Artin, [Lang] (https://www.amazon.com/Algebra-Graduate-Texts-Mathematics-Serge/dp/038795385X), [Hungerford] (https://www.amazon.com/Algebra-Graduate-Texts-Mathematics-v/dp/0387905189). The first two are undergraduate texts and the next two are graduate texts, those are the ones I've used and seen recommended, although some people suggest [Pinter] (https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178) and Aluffi. Please don't actually buy these books, you won't be able to feed yourself. There are free versions online and in many university libraries. Some of these books can get quite dry at times though. Feel free to stop by /r/learnmath whenever you have specific questions
Yes: Dummit and Foote. I used it in my freshman algebra class. It has excellent proofs and exercises. It will teach you the mathematical maturity faster than analysis and will most likely be more useful to you later on.
I'll throw out some of my favorite books from my book shelf when it comes to Computer Science, User Experience, and Mathematics - all will be essential as you begin your journey into app development:
Universal Principles of Design
Dieter Rams: As Little Design as Possible
Rework by 37signals
Clean Code
The Art of Programming
The Mythical Man-Month
The Pragmatic Programmer
Design Patterns - "Gang of Four"
Programming Language Pragmatics
Compilers - "The Dragon Book"
The Language of Mathematics
A Mathematician's Lament
The Joy of x
Mathematics: Its Content, Methods, and Meaning
Introduction to Algorithms (MIT)
If time isn't a factor, and you're not needing to steamroll into this to make money, then I'd highly encourage you to start by using a lower-level programming language like C first - or, start from the database side of things and begin learning SQL and playing around with database development.
I feel like truly understanding data structures from the lowest level is one of the most important things you can do as a budding developer.
If there is something close to an Encyclopaedia Mathematica, but you can read it like a novel, it is these three volumes from Aleksandrov/Kolmogorov/Laurentiev. Amazon
Edit: Ahem, but after reading carefully post0, I would recommend you simply to begin with the textbooks of secondary school or so.
Thanks for your reply. I read positive reviews about this , what do you think?
By Kolmogorov himself: http://www.amazon.ca/Mathematics-Its-Content-Methods-Meaning/dp/0486409163
Thanks! what do you think about Mathematics: Its Content, Methods and Meaning ? from what I searched it can teach a lot a novice like me and quite the wonderful book.
Mathematics Content Methods Meaning
I think this may be what you look for. I have read some chapters of it. It talks about meanings, where theories come from..
I also remembered it when I saw it in my bookshelve. Written by Roger Penrose. Penrose talks about math from numbers to modern physics application of math. Especially Einstein's math of space time can be understood in this book;
The Road to Reality
No worries for the timeliness!
For Measure and Integration Theory I recommend Elements of Integration and Measure by Bartle.
For Functional Analysis I recommend Introductory Functional Analysis with Applications by Kreyszig.
And for Topology, I think it depends on what flavor you're looking for. For General Topology, I recommend Munkres. For Algebraic Topology, I suggest Hatcher.
Most of these are free pdf's, but expensive ([;\approx \$200;]) to buy a physical copy. There are some good Dover books that work the same. Some good ones are this, this, and this.
Hello! I'm interested in trying to cultivate a better understanding/interest/mastery of mathematics for myself. For some context:
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To be frank, Math has always been my least favorite subject. I do love learning, and my primary interests are Animation, Literature, History, Philosophy, Politics, Ecology & Biology. (I'm a Digital Media Major with an Evolutionary Biology minor) Throughout highschool I started off in the "honors" section with Algebra I, Geometry, and Algebra II. (Although, it was a small school, most of the really "excelling" students either doubled up with Geometry early on or qualified to skip Algebra I, meaning that most of the students I was around - as per Honors English, Bio, etc - were taking Math courses a grade ahead of me, taking Algebra II while I took Geometry, Pre-Calc while I took Algebra II, and AP/BC Calc/Calc I while I took Pre-Calc)
By my senior year though, I took a level down, and took Pre-Calculus in the "advanced" level. Not the lowest, that would be "College Prep," (man, Honors, Advanced, and College Prep - those are some really condescending names lol - of course in Junior & Senior year the APs open up, so all the kids who were in Honors went on to APs, and Honors became a bit lower in standard from that point on) but since I had never been doing great in Math I decided to take it a bit easier as I focused on other things.
So my point is, throughout High School I never really grappled with Math outside of necessity for completing courses, I never did all that well (I mean, grade-wise I was fine, Cs, Bs and occasional As) and pretty much forgot much of it after I needed to.
Currently I'm a sophmore in University. For my first year I kinda skirted around taking Math, since I had never done that well & hadn't enjoyed it much, so I wound up taking Statistics second semester of freshman year. I did okay, I got a C+ which is one of my worse grades, but considering my skills in the subject was acceptable. My professor was well-meaning and helpful outside of classes, but she had a very thick accent & I was very distracted for much of that semester.
Now this semester I'm taking Applied Finite Mathematics, and am doing alright. Much of the content so far has been a retread, but that's fine for me since I forgot most of the stuff & the presentation is far better this time, it's sinking in quite a bit easier. So far we've been going over the basics of Set Theory, Probability, Permutations, and some other stuff - kinda slowly tbh.
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Well that was quite a bit of a preamble, tl;dr I was never all that good at or interested in math. However, I want to foster a healthier engagement with mathematics and so far have found entrance points of interest in discussions on the history and philosophy of mathematics. I think I could come to a better understanding and maybe even appreciation for math if I studied it on my own in some fashion.
So I've been looking into it, and I see that Dover publishes quite a range of affordable, slightly old math textbooks. Now, considering my background, (I am probably quite rusty but somewhat secure in Elementary Algebra, and to be honest I would not trust anything I could vaguely remember from 2 years ago in "Advanced" Pre-Calculus) what would be a good book to try and read/practice with/work through to make math 1) more approachable to me, 2) get a better and more rewarding understanding by attacking the stuff on my own, and/or 3) broaden my knowledge and ability in various math subjects?
Here are some interesting ones I've found via cursory search, I've so far just been looking at Dover's selections but feel free to recommend other stuff, just keep in mind I'd have to keep a rather small budget, especially since this is really on the side (considering my course of study, I really won't have to take any more math courses):
Prelude to Mathematics
A Book of Set Theory - More relevant to my current course & have heard good things about it
Linear Algebra
Number Theory
A Book of Abstract Algebra
Basic Algebra I
Calculus: An Intuitive and Physical Approach
Probability Theory: A Concise Course
A Course on Group Theory
Elementary Functional Analysis
I bought a copy of Dover's Linear Algebra (Border's Blowout) which I plan to go through after I finish A Book of Abstract Algebra.
I feel like I have a long way to go to get anywhere. :S
http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178
You can probably handle this book, and it's all of ten bucks anyways.
Refer to this book, it will most definitely have the answer for you. Refer pages 60-62, I think it is what you are looking for, if not that chapter should have the answer for you.
This image is used in one of my all time favorite textbook covers of all time: Introduction to Error Analysis
https://www.amazon.com/Introduction-Error-Analysis-Uncertainties-Measurements/dp/093570275X
If you're looking to apply basic error analysis, I recommend Taylor's book:
http://www.amazon.com/Introduction-Error-Analysis-Uncertainties-Measurements/dp/093570275X
It's pretty common to find this book on physics grad students' shelves. You may have already seen it, though, and you may be asking for something deeper.
A lack of this xD: http://www.amazon.com/Introduction-Error-Analysis-Uncertainties-Measurements/dp/093570275X
You need this book.
Until then - the general formula for error propagation in a function q(x, y, z, ....) with uncertainties &delta;x, &delta;y, &delta;z .... is equal to sqrt( (&delta;x&part;q/&part;x)^2 + (&delta;y&part;q/&part;y)^2 .....)
For your simple case where q = log10(x), &delta;q = &delta;x/(x*ln(10)).
Hope this helps.
We used this book in my intro level physics lab for error analysis.
I would agree that people can get lost in the illusion of what science can and cannot reasonably do. My course of study was very careful to make sure that people were not indoctrinated. The undergraduate courses were of course devoted to learning basic terminology and principles that have been around for decades if not centuries, but the upper division courses never presented you with "this is the answer spit it back" types of courses. It was all about teaching us how to design experiments and how to think critically. For example, one of my favorite courses was Advanced Molecular Genetics where our professor (who had a Nobel Prize and was just teaching for the hell of it- and because he loved tormenting students) would present us with papers that had been published and point us to the "further questions" section and say "design an experiment that would determine what is actually going on." We were judged based on the experiments we designed, and we actually had the equipment to run the experiments, which we did. It would have been a very good reality TV show called "So You Think You're a Scientist." He was brutal. Imagine Gordon Ramsey as a scientist. He would tear you a new one if your experiment was shit and he had nothing to lose. That class was awesome. You had to have balls to show up every day because he'd shit all over everything you did unless you had solid facts to back you up.
Come to think of it- I'd watch the shit out of that show.
But yeah- this book was required reading for all of us. It explicitly lays out what science can do and (more importantly) lays out what science can't do.
Relating to our conversation- people severely overestimate what science can and cannot do. GMOs (or any other technology for that matter) can potentially help or potentially harm. What we have to weigh is the potential harm of the new technology versus the actual harm of the current technology.
Here's an example for a though experiment: Horses vs automobiles. Automobiles emit greenhouse gases and require mining of minerals to make, among other things, catalytic converters. There are risks of using automobiles, but compare them to the hazards of using horses. Piles of manure attracting rats and spreading disease. Millions of acres of cropland being grown to provide fuel for the horses, etc.
Old vs new. Neither is perfect. If we wait for something perfect we'll never do anything and become stagnant.
But thanks for the conversation. And just so you know I have rather thick skin so your insults didn't phase me at all. Glad we could get to the point where we're having civil discourse.
This reminds me of a textbook I used for a physics class
Coincidentally, this is my favorite trainwreck of all time.
Since you are still in college, why not take a statistics class? Perhaps it can count as an elective for your major. You might also want to consider a statistics minor if you really enjoy it. If these are not options, then how about asking the professor if you can sit in on the lectures?
It sounds like you will be able to grasp programming in R, may I suggest trying out SAS? This book by Ron Cody is a good introduction to statistics with SAS programming examples. It does not emphasize theory though. For theory, I would recommend Casella & Berger, many consider this book to be a foundation for statisticians and is usually taught at a grad level.
Good luck!
Hrmh, given your background I guess I would go with a suggestion of Wasserman for Statistical Inference or Casella and Berger which isn't really applied. If those are too much for you (which I doubt with your background), there is also Wackerly's Mathematical Statistics with Applications :)
Maybe "too applied", depending on your fields, but there's always Casella and Berger, especially if you're in Economics.
If you are looking for something very calculus-based, this is the book I am familiar with that is most grounded in that. Though, you will need some serious probability knowledge, as well.
If you are looking for something somewhat less theoretical but still mathematical, I have to suggest my favorite. Statistics by William L. Hays is great. Look at the top couple of reviews on Amazon; they characterize it well. (And yes, the price is heavy for both books.... I think that is the cost of admission for such things. However, considering the comparable cost of much more vapid texts, it might be worth springing for it.)
We use Casella and Berger. It glosses over the measure theory somewhat but it appropriately develops the concept of "a probability". If you haven't had much background in proper math stats, then this is a good place to start (even if you've done the more applied courses).
I’m finishing up my stats degree this summer. For math, I took 5 courses: single variable calculus , multi variable calculus, and linear algebra.
My stat courses are divided into three blocks.
First block, intro to probability, mathematical stats, and linear models.
Second block, computational stats with R, computation & optimization with R, and Monte Carlo Methods.
Third block, intro to regression analysis, design and analysis of experiments, and regression and data mining.
And two electives of my choice: survey sampling & statistical models in finance.
Here’s a book for intro to probability. There’s also lectures available on YouTube: search MIT intro to probability.
For a first course in calculus search on YouTube: UCLA Math 31A. You should also search for Berkeley’s calculus lectures; the professor is so good. Here’s the calc book I used.
For linear algebra, search MIT linear algebra. Here’s the book.
The probability book I listed covers two courses in probability. You’ll also want to check out this book.
If you want to go deeper into stats, for example, measure theory, you’re going to have to take real analysis & a more advanced course on linear algebra.
Since you're an applied math PhD, maybe the following are good. They are not applied though.
This is the book for first year statistics grad students at OSU.
http://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126/ref=sr_1_1?ie=UTF8&amp;qid=1368662972&amp;sr=8-1&amp;keywords=casella+berger
But, I like Hogg/Craig much more.
http://www.amazon.com/Introduction-Mathematical-Statistics-7th-Edition/dp/0321795431/ref=pd_sim_b_2
I believe each can be found in international editions, and for download on the interwebs.
Also endorse this book as a primer on mathematical thinking. No background necessary: http://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X
Practice, practice, practice, practice. Getting good at maths is 90% equal to the practice you put in. People who seem "naturally" good at maths, most of the time, are just used to trying everything in their head and thus get more practice. Also, they may have done more in the past, and gotten used to using the smaller concepts they need to solve a bigger problem.
2 good books about learning: Waitzkin, The Art of Learning and Polya, How to Solve It.
Yes. Read this book, regardless of your major.
Putnam comp http://www.math.rutgers.edu/~saks/PUTNAM/
Also look at "Customers Also Bought" for books by Devlin, Mason etc
http://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X/
Www.lpthw.org
Www.hackerrank.com
lots of www.google.com
And when he's not in front of a computer he should be reading
http://www.amazon.com/How-Solve-It-Mathematical-Princeton/dp/069111966X (don't let the math scare him away if that's not his thing...at its core it's a book about how to solve any type of problem)
You might want a book like How to Solve It which will give you a general toolkit of problem solving techniques. It's not a textbook per se, but if you're struggling with how to even approach math problems then it might be a good first step.
Azcel wrote a good book on Fermat's Last Theorem and Wiles' solution. Amazon
Simon Singh's book on the same subject is also good, but Amazon has it at $10.17 whereas Azcel's is $0.71 better at $10.88.
Either way you get an enjoyable read of one man's dedication to solve a notoriously tricky problem and just enough of the mathematical landscape to get a sense of what was involved.
Another fun & light holiday read is Polya's 'How To Solve it' - read the glowing reviews over at Amazon
I recommend maybe doing more math instead. Or pick up a book called how to solve it . Alot of the things are easily translatable to programming and computer science really is mathematics as well. They're both related.
This is just my perspective, but . . .
I think there are two separate concerns here: 1) the "process" of mathematics, or mathematical thinking; and 2) specific mathematical systems which are fundamental and help frame much of the world of mathematics.
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Abstract algebra is one of those specific mathematical systems, and is very important to understand in order to really understand things like analysis (e.g. the real numbers are a field), linear algebra (e.g. vector spaces), topology (e.g. the fundamental group), etc.
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I'd recommend these books, which are for the most part short and easy to read, on mathematical thinking:
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How to Solve It, Polya ( https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X ) covers basic strategies for problem solving in mathematics
Mathematics and Plausible Reasoning Vol 1 & 2, Polya ( https://www.amazon.com/Mathematics-Plausible-Reasoning-Induction-Analogy/dp/0691025096 ) does a great job of teaching you how to find/frame good mathematical conjectures that you can then attempt to prove or disprove.
Mathematical Proof, Chartrand ( https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094 ) does a good job of teaching how to prove mathematical conjectures.
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As for really understanding the foundations of modern mathematics, I would start with Concepts of Modern Mathematics by Ian Steward ( https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Books/dp/0486284247 ) . It will help conceptually relate the major branches of modern mathematics and build the motivation and intuition of the ideas behind these branches.
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Abstract algebra and analysis are very fundamental to mathematics. There are books on each that I found gave a good conceptual introduction as well as still provided rigor (sometimes at the expense of full coverage of the topics). They are:
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A Book of Abstract Algebra, Pinter ( https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178 )
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Understanding Analysis, Abbott ( https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ).
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If you read through these books in the order listed here, it might provide you with that level of understanding of mathematics you talked about.
When I first went through it, I found it very verbose and too abstract for me. I was clearly not prepared for it.
Then I happened to read Gödel's proof, by Nagel and Newman, with an updated commentary by Hofstader. What a terrific book! Having gone through it, I began enjoying GEB.
There's tremendous depth in both books, and I look forward to iterating through these two alternately and getting more and more insights.
Godel's Proof is the original inspiration for Hofstadter. I find it a shorter but no less interesting read.
http://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371
Those are/were my interests and I enjoyed [Logical Dilemmas] (http://www.amazon.com/Logical-Dilemmas-Life-Work-Godel/dp/1568812566/ref=sr_1_1?ie=UTF8&amp;qid=1324312359&amp;sr=8-1), a thorough biography of Kurt Godel. [Godel's Proof] (http://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1324312476&amp;sr=1-1) might be too basic but is a good read.
I am a Strange Loop is about the theorem
Another book I recommend is David Foster-Wallace's Everything and More. It's a creative book all about infinity, which is a very important philosophical concept and relates to mind and machines, and even God. Infinity exists within all integers and within all points in space. Another thing the human mind can't empirically experience but yet bears axiomatic, essential reality. How does the big bang give rise to such ordered structure? Is math invented or discovered? Well, if math doesn't change across time and culture, then it has essential existence in reality itself, and thus is discovered, and is not a construct of the human mind. Again, how does logic come out of the big bang? How does such order and beauty emerge in a system of pure flux and chaos? In my view, logic itself presupposes the existence of God. A metaphysical analysis of reality seems to require that base reality is mind, and our ability to perceive and understand the world requires that base reality be the omniscient, omnipresent mind of God.
Anyway these books are both accessible. Maybe at some point you'd want to dive into Godel himself. It's best to listen to talks or read books about deep philosophical concepts first. Jay Dyer does a great job on that
https://www.youtube.com/watch?v=c-L9EOTsb1c&amp;t=11s
BTW, if you want a relatively easy description of Godel's work, this book may be useful.
There's a great introduction to Gödel's Incompleteness Theorems, it's called and Gödel's Proof by Nagel & Newman. Hofstadter has wrote it's foreword. It's a very short book, 160 pages in total.
Amazon Link!
Here's three very good books:
https://www.amazon.com/dp/0817636773/
https://www.amazon.com/dp/B000HMQ9VU/
I like this book. https://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773
I second the recommendation to find someone more experienced to help you one-on-one. Is there any way you could hire a private tutor? A big benefit of a tutor is that they'll be able to point out the gaps in your knowledge and point you to relevant resources. This can be tough to do on your own or through web discussions. For example, let's say one thing that's holding you back is that you haven't memorized your times table. This would be a major problem and a blind spot for you that would be immediately obvious to me if we were working face to face, but it would be impossible to see from reading your reddit comments.
Let me make a few more concrete suggestions. First, experiment with different study techniques. Take a look at this comment and the linked video. Try the "Feynman Technique" (video) -- this is not easy but it's the only way to really get a solid understanding. Don't expect to be spoon-fed knowledge when you're watching videos: you need to be spending most of your study time with a pen and paper, puzzling out for yourself why things work.
Second, for algebra, I can recommend two textbooks:
Khan Academy is a good supplement, but in my opinion it's too passive to be used as your main resource. It doesn't encourage independent thinking and it has no problems (easy drill exercises don't count as problems.) You need to do lots of problems. In particular, you need to struggle through problems that you're not explicitly told how to solve ahead of time.
Finally, mechanical knowledge is incredibly important, but of course it does need to be built upon a conceptual foundation. For every technique you learn (like solving 2/3 = 3R) you should first be able to explain why the technique works in simple, obvious terms, and then practice it (invent your own problems!) and add it to your collection of techniques. Math is (arguably) simply a grab bag of such techniques together with explanations of why they work. It's often not obvious which technique to apply in a specific case: this can only be learned through experience. Avoid problem sets with ten variants of one specific problem -- they don't teach this skill! Instead look for varied problems which require creativity (Rusczyk's book is a good start.)
You might also want to check out /r/learnmath and #math on freenode if you have more specific questions.
As an introduction to Algebra I can recommend https://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773/ref=sr_1_1?ie=UTF8&amp;qid=1486044711&amp;sr=8-1&amp;keywords=Shen+Algebra You may be also interested in https://www.amazon.com/Prime-Obsession-Bernhard-Greatest-Mathematics/dp/0452285259/ref=sr_1_1?rps=1&amp;ie=UTF8&amp;qid=1486042204&amp;sr=8-1&amp;keywords=prime+obsession+derbyshire
Moreover, it would be nice to watch these 'Youtube' channels: https://www.youtube.com/user/Vihart https://www.youtube.com/user/njwildberger
Gelfand's Algebra is interesting, encourages mathematical thinking, and has the additional advantage of being much more approachable than the books you've listed.
This is probably a much better place to start for someone who's interested in "starting from the basics."
It's just called "algebra" by I.M. Gelfand and another dude.
For highschool level math I reccomend i.m.gelfands books, one of which is Algebra.
They're excellent for self-study, and provide you with many insights not found elsewhere afaik.
Most schools just use 1 textbook for calc 1-3 : http://www.amazon.com/Calculus-James-Stewart/dp/0538497815
Doesn't really matter which edition you get, you're still going to suffer through it.
A popular other book recommended by math majors/professors is
http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918
You can get the pdf on "certain websites."
Videos will make you lazy and you will likely lose focus and turn to reddit or games or whatever because the professors can be really boring. Just stay focused on the text.
"Just do it."
If your Calculus is rusty before Rudin read Spivak Calculus it is great intro to analysis and you will get your calculus in order. Rudin is going to be overkill for you. Also before trying to do proofs read How to prove it It is a great crash course to naive set theory and proof strategies. And i promise i won't bore you with math any more.:D
No, he wrote a book on single-variable calculus, too: https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918
None of the questions you asked is “silly” or “simple.” There’s a whole lot going on in calculus, most of which is typically explained in a real analysis course. Rigorous proofs of things like the mean value theorem or various forms of integration are challenging, but they will provide the clarity you’re looking for.
I recommend that you check out something like Spivak’s Calculus, which is going to give a more rigorous intro to the subject. Alternately, you can just find a good analysis or intro to proofs class somewhere. It’s a fascinating subject, so good luck!
I would highly advise going with the 31/37 route. As both of the above courses are proof based, they will be play an integral role in upper year courses. Please be warned that they are extremely challenging but worthwhile courses. I would highly recommend you start preparing for the above two courses. For A37, I would suggest starting with Spivak:
https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918
I strongly suggest you take your time learning calculus, because anything you don't grasp completely will come back to haunt you.
But the good news is that there are lots of great resources you can use. MIT OCW has a full course with lectures, notes, and exams. Here are three free online books. If you're looking to buy a textbook, some good choices are Thomas, Stewart, and Spivak. (You can find dirt-cheap copies of older editions at abebooks.com.)
If you want more guidance, another great place to find it is at /r/learnmath.
You can start with Calculus by Spivak. If you're going to buy it then wait until after the Fall semester begins; the price is inflated right now because students need it for school.
This is a PDF of the third edition of the above book.
This is an excellent introduction to logic and proofs. You will want a strong understanding of how mathematicians communicate via proof and that book will really help.
The math subreddit is primarily undergrads talking about various topics. Make a point of just hanging out and reading stuff. If you don't understand something just tell us and they'll do their best to help out.
Hang out on the math stack exchange and ask questions about things you do not understand while trying to help with things you do understand.
Hope that helps!
You'll remember and forget formulae as you use them. It's the using them that makes things concrete in your head.
Once you're comfortable with algebra, trig. I'm assuming you've had geometry, since you were taking algebra 2; if not, geometry as well.
Once you're comfortable with those topics, you'll have enough of the basics to start branching out. Calculus is one obvious direction; a lot people have recommended Spivak's book for that. Introductory statistics is another (far too few people are even basically statistically literate.) Discrete math is yet another possibility. You can also start playing with "problem math", like the Green Book or Red Book. Algebraic structures is yet another possibility (I found Herstein's abstract algebra book pretty easy to read when we used it in school).
Edit: added Amazon links.
Question about Spivak's Calculus and Ross' Elementary Classical Analysis:
Are they books treating mathematics on the same level? Do they treat the rigorous theoretical foundation and computational techniques equally well? Can each one be an alternative to the other? Could someone please give brief comparative reviews/comments on them?
This question is also on r/learnmath: HERE.
This is good advice. Source: I flunked a private engineering school at age 17, in spite of of being 99th percentile in the ACT. Reason? Besides socialization issues, poor mathematics and academic preparation at my rural high school, where few went to college, let alone out-of-state.
I'm a strong believer in self-education (and self-employment) and am currently rectifying the above-stated issues.
Came here to plug Spivak's Calculus. It's a bit harder and more detailed than most calculus texts used today, but that's because he actually explains all the tricky bits, rather than just using hand-waving to finish those tricky bits. (It was the hand-waving that always left me confused in classroom teaching.) Spivak's Calculus might not be the place to start, but it's where you want to end up, so I want you to know about it.
Peace out, bro, and keep working. We'll make it. ME/EE is a great combo, btw. ME is the first branch of engineering, though it was called something else, when "engines of war", catapaults and whatnot, was the only game in town. But, all machines need sensors, controls, and power, which is the EE bit. Put it together, and you get mechatronics, which is part of the future.
One piece of added advice: stick to one of the main-line branches of engineering: mechanical, electrical, chemical, maybe civil, instead of one of the new, hybrid branches, like biomedical, etc. The jobs are more plentiful, you'll get a sounder foundation in engineering principles, and specializing is still possible.
Ed: Do you already know about MIT's Open Course Ware site? Most MIT courses are online with videoed lectures, recommended textbooks, homework and tests. It's a great resource. They also have edX, a co-operative venture with a bunch of fancy schools.
Calculus - Michael Spivak
> There are some philosophical reasons and some practical reasons that being a "pure" Bayesian isn't really a thing as much as it used to be. But to get there, you first have to understand what a "pure" Bayesian is: you develop reasonable prior information based on your current state of knowledge about a parameter / research question. You codify that in terms of probability, and then you proceed with your analysis based on the data. When you look at the posterior distributions (or posterior predictive distribution), it should then correctly correspond to the rational "new" state of information about a problem because you've coded your prior information and the data, right?
Sounds good. I'm with you here.
> However, suppose you define a "prior" whereby a parameter must be greater than zero, but it turns out that your state of knowledge is wrong?
Isn't that prior then just an error like any other, like assuming that 2 + 2 = 5 and making calculations based on that?
> What if you cannot codify your state of knowledge as a prior?
Do you mean a state of knowledge that is impossible to encode as a prior, or one that we just don't know how to encode?
> What if your state of knowledge is correctly codified but makes up an "improper" prior distribution so that your posterior isn't defined?
Good question. Is it settled how one should construct the strictly correct priors? Do we know that the correct procedure ever leads to improper distributions? Personally, I'm not sure I know how to create priors for any problem other than the one the prior is spread evenly over a finite set of indistinguishable hypotheses.
The thing about trying different priors, to see if it makes much of a difference, seems like a legitimate approximation technique that needn't shake any philosophical underpinnings. As far as I can see, it's akin to plugging in different values of an unknown parameter in a formula, to see if one needs to figure out the unknown parameter, or if the formula produces approximately the same result anyway.
> read this book. I promise it will only try to brainwash you a LITTLE.
I read it and I loved it so much for its uncompromising attitude. Jaynes made me a militant radical. ;-)
I have an uncomfortable feeling that Gelman sometimes strays from the straight and narrow. Nevertheless, I looked forward to reading the page about Prior Choice Recommendations that he links to in one of the posts you mention. In it, though, I find the puzzling "Some principles we don't like: invariance, Jeffreys, entropy". Do you know why they write that?
This one? Damn, it's £40-ish. Any highlights or is it just a case of this book is the highlight?
It's on my wishlist anyway. Thanks.
>All I'm saying is that the origin of a claim contains zero evidence as to that claim's truth.
I had a look back though your other posts and found this, which explains a lot, for me anyway. Most people would put some more options in there - yes, no, im pretty sure, its extremely unlikely etc..
Heres what I think is the problem, and why I think you need to change the way you are thinking - Your whole concept of what is "logical" or what is "using reason" seems to be constrained to what is formally known as deductive logic. You seem to have a really thorough understanding of this type of logic and have really latched on to it. Deductive logic is just a subset of logic. There is more to it than that.
I was searching for something to show you on other forms of logic and came across this book - "Probability Theory - The Logic of Science" Which looks awesome, Im going to read it myself, it gets great reviews. Ive only skimmed the first chapter... but that seems to be a good summary of how science works- why it does not use just deductive logic. Science draws most of its conclusions from probability, deductive logic is only appropriate in specific cases.
Conclusions based on probability - "Im pretty sure", "This is likely/unlikely" are extremely valid - and rational. Your forcing yourself to use deductive logic, and only deductive logic, where its inappropriate.
>You have no way of knowing, and finding out that this person regularly hallucinates them tells you nothing about their actual existence.
Yeah I think with the info you've said we have it would be to little to draw a conclusion or even start to draw one. Agreed. It wouldnt take much more info for us to start having a conversation about probabilities though - Say we had another person from the planet and he says its actually the red striped jagerwappas that are actually taking over - and that these two creatures are fundamentally incompatible. ie. if x exists y can't and vice-versa.
I'd suggest MATP 4600, Probability Theory & Applications. Only prerequisite is Calc if I remember right.
Or if you're confident in your time management, maybe read this textbook on your own; it's pretty accessible: https://www.amazon.com/gp/aw/d/0521592712/
(Neither of these will teach you a bunch of statistical tests, but those are easy to abuse if you don't understand the fundamentals ... and very easy to look up if you do understand the fundamentals.)
> For one, you need a categorical definition by which to justify your "probability" with. What, does each time you tell a god to speak deduct 1%? That's absurdly vague, stupid, and unheard of, so no wonder I never thought you'd actually be arguing this.
I don't happen to know the appropriate decibel-values to assign to E and not-E in this case. But I know the fucking SIGNS of the values.
No, I don't know how many times god needs to appear for me to believe that I wasn't drugged or dreaming or just going crazy. But god appearing is evidence for the existence of god, and him not appearing is evidence against.
Does it really matter if we are talking intervals of 5-seconds versus lifetimes?
3 pages, and you don't even have to go to a library! Check it out:
http://www.amazon.com/reader/0521592712?_encoding=UTF8&amp;ref_=sib%5Fdp%5Fpt#reader
Click on "First Pages" to get to the front.
You can lead a horse to water...
"Bayesian" is a very very vague term, and this article isn't talking about Bayesian networks (I prefer the more general term graphical models), or Bayesian spam filtering, but rather a mode of "logic" that people use in everyday thinking. Thus the better comparison would be not to neural nets, but to propositional logic, which I think we can agree doesn't happen very often in people unless they've had lots of training. My favorite text on Bayesian reasoning is the Jaynes book..
Still, I'm less than convinced by the representation of the data in this article. Secondly, the article isn't even published yet to allow anyone to review it. Thirdly, I'm suspicious of any researcher that talks to the press before their data is published. So in short, the Economist really shouldn't have published this, and should have waited. Yet another example of atrocious science reporting.
I've heard the book How To Prove it is pretty good. Also I'd recommend the Art of Problem Solving books as well for algebra and the likes. (It seems to go over stuff you'd learn in 7th grade, but written at a level adequate for adults).
I would also recommend sites like www.expii.com and www.brilliant.org
Khan academy also has a problem generator iirc.
Get used to proof based mathematics. How to Prove It: A Structured Approach, by Daniel J. Velleman, would be a great start.
EDIT: Ok math that's useful for a STEM major, maybe forget about the proof based math unless you're considering mathematical physics. It's still a good book though.
"I'm also sure that due to my limited educational resources, self-directed study will be a huge part. Any suggestions on which books are must reads to gain competency in CS?"
Here are a few good choices for the more theoretical areas of computing:
http://www.amazon.com/Algorithms-4th-Edition-Robert-Sedgewick/dp/032157351X/ref=sr_1_1?ie=UTF8&amp;qid=1408406629&amp;sr=8-1&amp;keywords=algorithms+4th+edition
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/ref=sr_1_1?ie=UTF8&amp;qid=1408406673&amp;sr=8-1&amp;keywords=how+to+prove+it
You'll also want to look at a decent discrete mathematics book. Sadly the book I used as an undergrad was rubbish, so I don't have a good recommendation.
Don't pay too much attention to the other replies - if you really want to take Math 145/146 it's possible, it will just be a lot of work.
My marks were good in high school (but not 95+) and my score on the Euclid was terrible (in order to enrol without an override you need 80+ on the Euclid). The thing to know is these courses have heavy emphasis on proofs, so the summer before coming I worked my way through the first half of a book on proofs and ended up doing relatively well in these courses.
You can certainly do it, but you have to be really dedicated.
If you do decide on it, definitely read this beforehand:
http://www.amazon.ca/How-Prove-It-Structured-Approach/dp/0521675995
This book helped me out a bit: http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521675995 -- However, even though I have a background in programming, I felt it moved rather quickly, especially after about halfway through the book.
Oh man 2011 was probably the hardest MATA31 revision. Don't worry, about that midterm though, the course content is really different now, that was when CSC/MATA67 used to be merged with MATA31, so they did a lot more set theory/number theory in MATA31 than they do now. I doubt most people who took MATA31 (and did well) could even pass that midterm just because we don't learn that stuff in MATA31 anymore. If you're trying to get started on studying for MATA31 now, I actually recommend you don't learn MATA31 material. Instead, improve on your critical thinking skills which your high school has definitely not given you. "Find" a book called how to prove it and go through maybe the first two or so chapters which just introduce proofs, and start to build up your proof skills. Becoming comfortable with proofs will come in handy immensely for CSCA67, MATA37, and in a big chunk of MATA31.
You might want to check out Stein and Shakarchi's book Complex Analysis http://press.princeton.edu/titles/7563.html. This book is a bit hard but iirc doesn't require you to have had real analysis before hand. I would highly recommend that you work through a proof based book before hand though. Often times this will be a course book but something like https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995?ie=UTF8&amp;*Version*=1&amp;*entries*=0 that should also get the job done.
Or you can go the traditional route like other people mentioned of getting about a semester's worth of real analysis under your belt. The reason why this is usually the suggested path is because it's not expected that you are 100% competent at writing proofs in the beginning of real but you are in complex.
For whichever professor you have for Math 42, I highly recommend you get this book: https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
It definitely saved me a ton. It’s straight to the point, and not as dry as most textbooks can be. Math 32 will be a bit more work, but in my experience just start homework early and don’t be afraid to go to professor office hours and ask questions. Even if they seem distant during class, most professors do appreciate students who make the effort to ask questions. If you need free tutoring in any of your classes, contact Peer Connections. Specifically for math, I believe MacQuarrie Hall room 221 offers drop-in tutoring for free as well! And for physics, Science building room 319 has free drop-in tutoring.
Not to pile on, but as has been previously stated what you wrote is not a proof. I'm not going to focus on whether or not what you said is true or false because the larger problem is that it's not written as a proof structure-wise. By this I mean, proofs are written using logic. If you're really interested in proof writing and basic analysis I suggest this book: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1331568877&amp;sr=1-1
If you happen to have the UCLA edition of Friedberg's Linear Algebra (the one you'll likely use for 115A) already, there's a section at the end with an intro to proofs. This book is pretty popular at universities with a dedicated intro to proofs class, so it might be worth checking out; I read a bit of it before taking the upper divs. Hope that helps!
Hmm...sorry but a lot of your post shows a lack of mathematical rigor and philosophical understanding of the terms you say. Not trying to offend you, but you really want to practice on proofs.
> Let me see if I understand you OP. You are asserting that by adopting a position where a positive claim (and BTW a claim that something does not exist or does not work is still a positive claim even though the claim involves a negative) must be justified and supported, such as the position of non-belief in the existence of Gods (for or against), or a person is innocent until proven guilty, "harms discourse and is dishonest"? Really?
Except, this is exactly what the burden of proof is? Any claim, positive or negative, must be proven. Yes, even unicorns existing. This has been discussed at length throughout math and philosophy so I don't see how you think (unless you're ignorant) otherwise. Atheist conflict the burden of proof as a legal tenant and one from an epistemological essence. Legal wise, this is more as "innocent until proven guilty" but in no way does that mean x person didn't do it.
Deeper discussion here: https://www.reddit.com/r/philosophy/comments/72o984/the_natural_world_is_all_there_is_as_far_as_we/
>Any claim that purports to be of knowledge has a burden of proof.
>
>Any claim that limits itself merely to belief does not have a burden of proof.
>
>It makes no difference if the claim is theistic (gnostic or agnostic) or naturalistic (strong or weak), nor does it make any difference if it's a claim that a particular thing exists or is true, or that a particular thing does not exist or is not true, or anything else really for that matter. If it's a claim that purports to be of knowledge, it has a burden of proof, and if it's merely a belief, it does not.
Your version of the burden of proof (taken from rational wiki) has no basis in math nor philosophy. Do not get information from rational wiki. Get a copy of many proofs based mathematical books and start from there by actually proving problems.
Again from stack: https://philosophy.stackexchange.com/questions/678/does-a-negative-claimant-have-a-burden-of-proof
>I would say that generally, the burden of proof falls on whomever is making a claim, regardless of the positive or negative nature of that claim. It's fairly easy to imagine how any positive claim could be rephrased so as to be a negative one, and it's difficult to imagine that this should reasonably remove the asserter's burden of proof.
>
>Now, the problem lies in the fact that it's often thought to be extremely difficult, if not actually impossible, to prove a negative. It's easy to imagine (in theory) how one would go about proving a positive statement, but things become much more difficult when your task is to prove the absence of something.
>
>But many philosophers and logicians actually disagree with the catchphrase "you can't prove a negative". Steven Hales argues that this is merely a principle of "folk logic", and that a fundamental law of logic, the law of non-contradiction, makes it relatively straightforward to prove a negative.
Any claim, false or positive requires to be proven. Whether I say for all natural numbers in set N there exists no element such that N\^N <= N\^2. Or I state the inverse "for all natural numbers in set N there exists an element such that N\^N <= N\^2. The burden of proof is on me.
> Or OP, would you just accept that the grobbuggereater exists because I give witness to this existence?
I truly wish my professors were as simple-minded....so many hours could have been saved by proving negative statements in Mathematics and theoretical computer science. However, yes. Philosophically speaking, to claim grobbugereater does not exist requires proof. Grobbugereater is an idea x, where the probability is x / |r| where r is the set of all ideas. as r tends to infinity the probability of grobbugereater existing tends to 0. Thusly, since grobbugereater has no epistemological evidence then we can conclude his probability of existing is infinitely small. This is how you prove grobbugereater does not exist.
One of your claims (presumably) is that induction is better than deduction. That somehow science is far better than math, philosophy, theism, or any other deductive method. Such a claim is metaphysical and cannot be proven via induction thusly a contradiction.
I find it odd, that so many people who use rational claims lack mathematical rigor. Honestly dilutes the topic into a mindless debate and petty insults. Here is a good read to strengthen your skills:
https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
For compsci you need to study tons and tons and tons of discrete math. That means you don't need much of analysis business(too continuous). Instead you want to study combinatorics, graph theory, number theory, abstract algebra and the like.
Intro to math language(several of several million existing books on the topic). You want to study several books because what's overlooked by one author will be covered by another:
Discrete Mathematics with Applications by Susanna Epp
Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand, Albert D. Polimeni, Ping Zhang
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Numbers and Proofs by Allenby
Mathematics: A Discrete Introduction by Edward Scheinerman
How to Prove It: A Structured Approach by Daniel Velleman
Theorems, Corollaries, Lemmas, and Methods of Proof by Richard Rossi
Some special topics(elementary treatment):
Rings, Fields and Groups: An Introduction to Abstract Algebra by R. B. J. T. Allenby
A Friendly Introduction to Number Theory Joseph Silverman
Elements of Number Theory by John Stillwell
A Primer in Combinatorics by Kheyfits
Counting by Khee Meng Koh
Combinatorics: A Guided Tour by David Mazur
Just a nice bunch of related books great to have read:
generatingfunctionology by Herbert Wilf
The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates by by Manuel Kauers, Peter Paule
A = B by Marko Petkovsek, Herbert S Wilf, Doron Zeilberger
If you wanna do graphics stuff, you wanna do some applied Linear Algebra:
Linear Algebra by Allenby
Linear Algebra Through Geometry by Thomas Banchoff, John Wermer
Linear Algebra by Richard Bronson, Gabriel B. Costa, John T. Saccoman
Best of Luck.
https://www.amazon.ca/How-Prove-Structured-Daniel-Velleman/dp/0521675995/ref=sr_1_1?ie=UTF8&amp;qid=1501343615&amp;sr=8-1&amp;keywords=how+to+prove+it
Learn the information of chapter 4 in this book, in particular the stuff from page 90 to the end of the chapter
I also tried to learn calculus through spivak and found it very difficult; I stopped at then 4th chapter and switched to an easier textbook. If it's your first time learning calculus choosing an easier and verbose text like Stewart may suite you better. It's important to remember Spivak's Calculus is more like a textbook on Analysis (the theory of calculus), which is what often comes junior or senior year for math majors/minors.
If you have already learned calculus I'd suggest the bookHow to Prove It which helps think of math in a more concrete way that can help with proofs, even though no calculus is presented. Also, remember that Spivak likely didn't intend for people to find his questions easy, so don't feel like you are unprepared if it takes a while to do a single question.
To lean from a book like that of Artin's, you need to get a few basics down:
How to Prove It: A Structured Approach by Daniel Velleman
Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand et al
Depends what kind of math you're interested in. If you're looking for an introduction to higher (college) math, then How to Prove It is probably your best bet. It generally goes over how proofs work, different ways of proving stuff, and then some.
If you already know about proofs (i.e. you are comfortable with at least direct proofs, induction, and contradiction) then the world is kind of your oyster. Almost anything you pick up is at least accessible. I don't really know what to recommend in this case since it's highly dependent on what you like.
If you don't really know the basics about proofs and don't care enough to yet, then anything by Dover is around your speed. My favorites are Excursions in Number Theory and Excursions in Geometry. Those two books use pretty simple high school math to give a relatively broad look at each of those fields (both are very interesting, but the number theory one is much easier to understand).
If you're looking for high school math, then /u/ben1996123 is probably right that /r/learnmath is best for that.
If you want more specific suggestions, tell me what you have enjoyed learning about the most and I'd be happy to oblige.
> I assume I ought to check it out after my discrete math class? Or does CLRS teach the proofs as if the reader has no background knowledge about proofs?
Sadly it does not teach proofs. You will need to substitute this on your own. You don't need deep proof knowledge, but just the ability to follow a proof, even if it means you have to sit there for 2-3 minutes on one sentence just to understand it (which becomes much easier as you do more of this).
> We didn't do proof by induction, though I have learned a small (very small) amount of it through reading a book called Essentials of Computer Programs by Haynes, Wand, and Friedman. But I don't really count that as "learning it," more so being exposed to the idea of it.
This is better than nothing, however I recommend you get very comfortable with it because it's a cornerstone of proofs. For example, can you prove that there are less than 2 ^ (h+1) nodes in any perfect binary tree of height h? Things like that.
> We did go over Delta Epsilon, but nothing in great detail (unless you count things like finding the delta or epsilon in a certain equation). If it helps give you a better understanding, the curriculum consisted of things like derivatives, integrals, optimization, related rates, rotating a graph around the x/y-axis or a line, linearization, Newton's Method, and a few others I'm forgetting right now. Though we never proved why any of it could work, we were just taught the material. Which I don't disagree with since, given the fact that it's a general Calc 1 course, so some if not most students aren't going to be using the proofs for such topics later in life.
That's okay, you will need to be able to do calculations too. There are people who spend all their time doing proofs and then for some odd reason can't even do basic integration. Being able to do both is important. Plus this knowledge will make dealing with other math concepts easier. It's good.
> I can completely understand that. I myself want to be as prepared as possible, even if it means going out and learning about proofs of Calc 1 topics if it helps me become a better computer scientist. I just hope that's a last resort, and my uni can at least provide foundation for such areas.
In my honest opinion, a lot of people put too much weight on calculus. Computer science is very much in line with discrete math. The areas where it gets more 'real numbery' is when you get into numerical methods, machine learning, graphics, etc. Anything related to theory of computation will probably be discrete math. If your goal is to get good at data structures and algorithms, most of your time will be spent on discrete topics. You don't need to be a discrete math genius to do this stuff, all you need is some discrete math, some calc (which you already have), induction, and the rest you can pick up as you go.
If you want to be the best you can be, I recommend trying that book I linked first to get your feet wet. After that, try CLRS. Then try TAOCP.
Do not however throw away the practical side of CS if you want to get into industry. Reading TAOCP would make you really good but it doesn't mean shit if you can't program. Even the author of TAOCP, Knuth, says being polarized completely one way (all theory, or all programming, and none of the other) is not good.
> From reading ahead in your post, is Skiena's Manual something worth investing to hone my skills in topics like proof skills? I'll probably pick it up eventually since I've heard nothing but good things about it, but still. Does Skiena's Manual teach proofing skills to those without them/are not good at them? Or is there a separate book for that?
You could, at worst you will get a deeper understanding of the data structure and how to implement them if the proof goes over your head... which is okay, no one on this planet starts off good at this stuff. After you do this for a year you will be able to probably sit down and casually read the proofs in these books (or that is how long it took me).
Overall his book is the best because it's the most fun to read (CLRS is sadly dry), and TAOCP may be overkill right now. There are probably other good books too.
> I guess going off of that, does one need a certain background to be able to do proofs correctly/successfully, such as having completed a certain level of math or having a certain mindset?
This is developed over time. You will struggle... trust me. There will be days where you feel like you're useless but it continues growing over a month. Try to do a proof a day and give yourself 20-30 minutes to think about things. Don't try insane stuff cause you'll only demoralize yourself. If you want a good start, this is a book a lot of myself and my classmates started on. If you've never done formal proofs before, you will experience exactly what I said about choking on these problems. Don't give up. I don't know anyone who had never done proofs before and didn't struggle like mad for the first and second chapter.
> I mean, I like the material I'm learning and doing programming, and I think I'd like to do at least be above average (as evident by the fact that I'm going out of my way to study ahead and read in my free time). But I have no clue if I'll like discrete math/proving things, or if TAOCP will be right for me.
Most people end up having to do proofs and are forced to because of their curriculum. They would struggle and quit otherwise, but because they have to know it they go ahead with it anyways. After their hard work they realize how important it is, but this is not something you can experience until you get there.
I would say if you have classes coming up that deal with proofs, let them teach you it and enjoy the vacation. If you really want to get a head start, learning proofs will put you on par with top university courses. For example at mine, you were doing proofs from the very beginning, and pretty much all the core courses are proofs. I realized you can tell the quality of a a university by how much proofs are in their curriculum. Any that is about programming or just doing number crunching is literally missing the whole point of Computer Science.
Because of all the proofs I have done, eventually you learn forever how a data structure works and why, and can use it to solve other problems. This is something that my non-CS programmers do not understand and I will always absolutely crush them on (novel thinking) because its what a proper CS degree teaches you how to do.
There is a lot I could talk about here, but maybe such discussions are better left for PM.
check out this mahfukka http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/ref=tmm_pap_title_0
Start with a book like this:
http://www.amazon.com/books/dp/0521597188
or this:
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
or the one teuthid recommended. When you're doing self-study, it's doubly important to be able to read and follow most of the material.
Advanced math is subjective. Discrete math is a lot of topics mixed together into one class. A little bit of logic, graph theory, set theory, number theory, modular arithmetic, combinatorics, introduction to proofs, algorithm analysis and some other stuff I might be missing. The only prerequisite for it is pre-calculus. The difficulty of the class is subjective some people find it hard and some people find it easy. If you can remember definitions and theorems and string them together to construct a proof you should be fine. How to prove it is recommended a lot as an intro to writing proofs.
I've been studying How To Prove It by Daniel Velleman for a few months now and I don't know if it's the best book, but it's really good and it has opened my mind in so many ways. Plus, it's really cheap for a textbook.
Logic, Number theory, Graph Theory and Algebra are all too much for you to handle on your own without first learning the basics. In fact, most of those books will probably expect you to have some mathematical maturity (that is, reading and writing proofs).
I don't know how theoretical your CS program is going to be, but I would recommend working on your discrete math, basic set theory and logic.
This book will teach you how to write proofs, basic logic and set theory that you will need: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
I can't really recommend a good Discrete Math textbook as most of them are "meh", and "How to Prove It" does contain a lot of the material usually taught in a Discrete Math course. The extra topics you will find in discrete maths books is: basic probability, some graph theory, some number theory and combinatorics, and in some books even some basic algebra and algorithm analysis. If I were you I would focus mostly on the combinatorics and probability.
Anyway, here's a list of discrete math books. Pick the one you like the most judging from the reviews:
Don't bother trying to learn too much too soon, as you really do need to let time for the math to sink in.
I will second this. I used this book for my year of undergrad foundations of probably and stats.
I also really like Casella and Berger's 'Statistical Inference.'
https://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126https://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126
What do you want to do, though? Is your goal to read math textbooks and later, maybe, math papers or is it for science/engineering? If it's the former, I'd simply ditch all that calc business and get started with "actual" math. There are about a million books designed to get you in the game. For one, try Book of Proof by Richard Hammack. It's free and designed to get your feet wet. Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand/Polimeni/Zhang is my favorite when it comes to books of this kind. You'll also pick up a lot of math from Discrete Math by Susanna Epp. These books assume no math background and will give you the coveted "math maturity".
There is also absolutely no shortage of subject books that will nurse you into maturity. For example, check out [The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Grinberg](https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Princeton/dp/0691172935/ref=sr_1_1?ie=UTF8&amp;qid=1486754571&amp;sr=8-1&amp;keywords=real+analysis+lifesaver() and Book of Abstract Algebra by Pinter. There's also Linear Algebra by Singh. It's roughly at the level of more famous LADR by Axler, but doesn't require you have done time with lower level LA book first. The reason I recommend this book is because every theorem/lemma/proposition is illustrated with a concrete example. Sort of uncommon in a proof based math book. Its only drawback is its solution manual. Some of its proofs are sloppy, messy. But there's mathstackexchange for that. In short, every subject of math has dozens and dozens of intro books designed to be as gentle as possible. Heck, these days even grad level subjects are ungrad-ized: The Lebesgue Integral for Undergraduates by Johnson. I am sure there are such books even on subjects like differential geometry and algebraic geometry. Basically, you have choice. Good Luck!
All the books listed can be found on libgen.io
If interest is theoretical mathematics:
Become adept at writing proofs.
I recommend
https://www.amazon.com/Discrete-Transition-Advanced-Mathematics-Undergraduate/dp/0821847899
Do some exercises in the first chapter, and go around the book doing whatever is of interest. I suggest learning about proofs/truth tables, functions, infinite sets, and number theory. This book will have chapters approaching all of these.
After this, you have some choice. I would take a beginners book in any of the following fields
Abstract algebra: https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178
Linear algebra: Linear Algebra Done Right by axler
Analysis: foundations of mathematical analysis by rudin (this will be hard but don’t be afraid!)
Approach each of these books slowly. Do not rush. Self-studying math is HARD. You might only get through 3 pages in a week, but I guarantee that you will get the ropes, and a few weeks later, look back and wonder how it was difficult at all.
In making the choice of what to study first, go to the subjects Wikipedia page or google “should I study x or y first” and you’ll likely find good resources
I highly recommend Pinter's "A Book of Abstract Algebra" for a quick course and handy refresher book.
Many functions don't take real numbers or integers as their arguments. Consider the multiplication of an MxN matrix and an NxM matrix where M != N. The result of which is an NxN matrix. In this context, matrix addition doesn't even have a relation to matrix multiplication.
If you're interested, these relationships are what group theory tries to explore. My favorite book on the subject is A Book of Abstract Algebra
Sure. https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178
How about some nice, inexpensive classics from Dover Publications?
For number theory, Andrews, Number Theory or Leveque, Elementary theory of numbers or the more advanced Leveque, Fundamentals of Number Theory
For linear algebra, Cullen, Matrices and Linear Transformations.
I bet you haven't read Edwards, Riemann's Zeta Function.
Edit: Oops! Now I see that you wanted to avoid linear algebra. Cullen might still be good as a second source. Maybe Pinter, A book of Abstract Algebra would appeal to you for a taste of field theory. However, vector spaces just naturally go with fields, so you may want to wait until after you have studied linear algebra.
There are these videos and there is also this book. The book is better if you struggled the first time, and it includes a short section on number theory.
AH HA, one of the few times I will link a dover book in good heart!
http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178
Pinter offers a fine introduction to abstract algebra.
I haven't yet started practicing for the GRE, but does it include Linear Algebra or Modern/Abstract Algebra? Also is there Calculus on it? I'm taking (or have taken, or will take by the time of the GRE) all of those classes and they're all very interesting. I just bought this book on Abstract Algebra, if you're interested.
I just bought this for $10. Not all textbook companies are jokes. Just most.
General Advice
Specific Advice
Dover's ODE textbook.
What about the second example that triggers the same domain problem for 0. Do you get the same result?
https://www.amazon.ca/Ordinary-Differential-Equations-Morris-Tenenbaum/dp/0486649407/ref=mp_s_a_1_1?keywords=ordinary+differential+equations&amp;qid=1567534646&amp;s=gateway&amp;sprefix=ordinary+differ&amp;sr=8-1
This is the link for the textbook.
Tenenbaum and Pollard's book is fine. It is cheap too (published by Dover methinks)
https://www.amazon.com/Ordinary-Differential-Equations-Dover-Mathematics/dp/0486649407
That helps a little. I'm not too familiar with that world (I'm a physics major), but I took a look at a sample civil engineering course curriculum. If you like learning but the material in high school is boring, you could try self-teaching yourself basic physics, basic applied mathematics, or some chemistry, that way you could focus more on engineering in college. I don't know much about engineering literature, but this book is good for learning ODE methods (I own it) and this book is good for introductory classical mechanics (I bought and looked over it for a family member). The last one will definitely challenge you. Linear Algebra is also incredibly useful knowledge, in case you want to do virtually anything. Considering you like engineering, a book less focused on proofs and more focused on applications would be better for you. I looked around on Amazon, and I found this book that focuses on applications in computer science, and I found this book focusing on applications in general. I don't own any of those books, but they seem to be fine. You should do your own personal vetting though. Considering you are in high school, most of those books should be relatively affordable. I would personally go for the ODE or classical mechanics book first. They should both be very accessible to you. Reading through them and doing exercises that you find interesting would definitely give you an edge over other people in your class. I don't know if this applies to engineering, but using LaTeX is an essential skill for physicists and mathematicians. I don't feel confident in recommending any engineering texts, since I could easily send you down the wrong road due to my lack of knowledge. If you look at an engineering stack exchange, they could help you with that.
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You may also want to invest some time into learning a computer language. Doing some casual googling, I arrived at the conclusion that programming is useful in civil engineering today. There are a multitude of ways to go about learning programming. You can try to teach yourself, or you can try and find a class outside of school. I learned to program in such a class that my parents thankfully paid for. If you are fortunate enough to be in a similar situation, that might be a fun use of your time as well. To save you the trouble, any of these languages would be suitable: Python, C#, or VB.NET. Learning C# first will give you a more rigorous understanding of programming as compared to learning Python, but Python might be easier. I chose these three candidates based off of quick application potential rather than furthering knowledge in programming. This is its own separate topic, but my personal two cents are you will spend more time deliberating between programming languages rather than programming if you don't choose one quickly.
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What might be the best option is contacting a professor at the college you will be attending and asking for advice. You could email said professor with something along the lines of, "Hi Professor X! I'm a recently accepted student to Y college, and I'm really excited to study engineering. I want to do some rigorous learning about Z subject, but I don't know where to start. Could you help me?" Your message would be more formal than that, but I suspect you get the gist. Being known by your professors in college is especially good, and starting in high school is even better. These are the people who will write you recommendations for a job, write you recommendations for graduate school (if you plan on it), put you in contact with potential employers, help you in office hours, or end up as a friend. At my school at least, we are on a first name basis with professors, and I have had dinner with a few of mine. If your professors like you, that's excellent. Don't stress it though; it's not a game you have to psychopathically play. A lot of these relationships will develop naturally.
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That more or less covers educational things. If your laziness stems from material boredom, everything related to engineering I can advise on should be covered up there. Your laziness may also just originate from general apathy due to high school not having much impact on your life anymore. You've submitted college applications, and provided you don't fail your classes, your second semester will probably not have much bearing on your life. This general line of thought is what develops classic second semester senioritis. The common response is to blow off school, hang out with your friends, go to parties, and in general waste your time. I'm not saying don't go to parties, hang out with friends, etc., but what I am saying is you will feel regret eventually about doing only frivolous and passing things. This could be material to guilt trip yourself back into caring.
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For something more positive, try to think about some of your fun days at school before this semester. What made those days enjoyable? You could try to reproduce those underlying conditions. You could also go to school with the thought "today I'm going to accomplish X goal, and X goal will make me happy because of Y and Z." It always feels good to accomplish goals. If you think about it, second semester senioritis tends to make school boring because there are no more goals to accomplish. As an analogy, think about your favorite video game. If you have already completed the story, acquired the best items, played the interesting types of characters/party combinations, then why play the game? That's a deep question I won't fully unpack, but the simple answer is not playing the game because all of the goals have been completed. In a way, this is a lot like second semester of senior year. In the case of real life, you can think of second semester high school as the waiting period between the release of the first title and its sequel. Just because you are waiting doesn't mean you do nothing. You play another game, and in this case it's up to you to decide exactly what game you play.
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Alternatively, you could just skip the more elegant analysis from the last few paragraphs and tell yourself, "If I am not studying, then someone else is." This type of thinking is very risky, and most likely, it will make you unhappy, but it is a possibility. Fair warning, you will be miserable in college and misuse your 4 years if the only thing you do is study. I guarantee that you will have excellent grades, but I don't think the price you pay is worth it.
For DEs try:
Ordinary Differential Equations by Tennenbaum
Its a great book with a TON of worked examples and solutions to all the exercises. This text was my holy book during my undergrad engineering courses.
I had a hard time getting through dif eq also, because the book was unreadable (to me). I also hate reading anything by Hibbler. The Munson fluid mechanics book is... barely tolerable. When that happens, I tend to look, with more vigor than usual, for other sources. Dif eq: I was lucky, and our tutoring center has dif eq tutors. Fluids: I found a wonderful lecture series done by UC Irvine OpenCourseWare. Hibbler... well, I've been S.O.L. on that so far. Generally, I also try to find a solutions manual. If I'm having a terrible time with a problem, I work through it and check myself each step of the way. I often try to find a different book, too. The only reason you need the required book is so you know what to look for in your chosen book.
I recently discovered there is a very highly-rated dif eq book available used on Amazon for about $13, so I ordered it in the hopes that it will be readable, as I now need to brush up on dif eq and can't stand the book we used in class.
Ebay. You have to be careful you're getting the right edition, though.
Here's an international paperback version of a 4th Edition book (about $50 includes shipping) that has a very different cover from the U.S. 4th Edition of the same book (about $270 on Amazon).
The book is incredibly good, it's really very well written, so I'm glad the author got paid their share for my purchase but at $270 there's no way I was going to buy the book for full price. I'd have gone without the book and just googled the topics every week, at that price.
I've started with this course, then I dropped because it touches on things too superficially and there are very few exercises. I find that I learn best doing exercises so I picked up a book (different from the on on the website).
The book which most seem to agree is great for self-study of Discrete Math is Discrete mathematics with applications by Susanna S. Epps. There is a similar book with a similar name - Dicrete Mathematics and its Applications by Rosen - but a lot of people seem to dislike this book. They're frustrated with the fact that proofs are not well explained and the book doesn't prepare the reader for the exercises.
Discrete math gives you a good foundation for learning higher math.
This one is a good book:
Discrete Mathematics with Applications by Susanna Epp.
It deals with logic, sets, relations, counting, proofs etc...you know the stuff you need in higher math. Seeing as you're a CS student, I am guessing Discrete Math would be much more important to you than Calculus.
I'm finishing up a discrete math course using Susanna Epp's textbook, and while it's pretty good, I just see it as a preparation for Concrete Mathematics.
As a side note, I never know how to go about math in how it relates to my major. It takes a lot of effort on my part to really learn the concepts in depth, and I have a hard time justifying that kind of investment before I really know what I'm eventually going to do for work.
The easiest one that I know of is the one by Epp. She doesn't go into the history as much, but her writing style is extremely easy on the brain.
The only "math" you don't learn in secondary education that I would say is truly important to all students of computer science is discrete mathematics. One of my old instructors told me he was trying to get permission to use Discrete Mathematics With Ducks in his curriculum. I flipped through it, it seemed like a pretty good discrete math book, but maybe only if you already had some basic understanding. The more traditional choice might be a better option, but it's much drier.
Every other field of math that has applications in computer science has a narrower breadth of use than discrete mathematics, in my opinion. Learning discrete math well will probably do you more good than learning lots of subjects poorly.
You need Susanna Epp's Discrete Math. Her stuff can be ever so slightly rough at certain spots, but absolutely phenomenal everywhere else. But there's Book of Proof by Richard Hammack[FREE] to help you overcome these difficulties.
For a slightly different take on essentially the same material you could also try Mathematics: A Discrete Introduction by Scheinerman. Also a very gentle book.
Can you get Discrete Mathematics with Applications https://www.amazon.com/dp/0495391328/ref=cm_sw_r_cp_apa_i_lf2vCbG0MVRXJ
> How do you think I'll struggle?
The reason I think you'll struggle is because you can't reverse engineer grad school exams, assuming your program is a decent stats program. You have to be more open to learning more when you need to learn it, whether it be for school or for your professional life. (Yes, I do learn in my full-time job. I've learned around 4-5 different programming languages in the two years I've been in this job.)
One thing I should emphasize is that although you did well in Statistical Methods, the grad-school version of this (in my experience) is much more theoretical and proof-oriented. I got an A in the grad-school version, but it was by no means an easy A, and I tutored the calculus-based probability and statistics material for about 3 years in my undergrad. The class definitely put me on edge at times.
> Any tips to study technique? Should I really never look up any solution? Even if it takes weeks to solve a problem? I don't think my University has any introduction to proof courses, I think you just have to know proof or learn it on your own.
Get your hands on a Discrete Mathematics book - my favorite one is the text by Epp (see https://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1503020282&amp;sr=1-1&amp;keywords=discrete+mathematics+with+applications). You can read the index of this to get an idea of what it covers. But basically, Discrete Math goes through the logic of how a proof is constructed to guide you through proofs.
> Any tips on that? Never taken notes before, except for in lectures to not fall asleep, but I never look back on the notes I take.
Notes are essential. Our minds only have a finite amount of memory space, and we forget things. You need to take notes with the mindset that you'll probably be using this stuff later, and if you need to re-learn it again, you can read your notes and brush up on it. In my case, I've kept all of my grad school notes.
Thank you. I really want to go into the 'Curse the entire field of CS and maths' level. This is the book I was suggested via PM. I'll try the first approach you suggest, then I'll scale up since we are a small group of people trying to learn together so I think that a couple of heads tackling into a problem might solve it.
I enjoyed
Introduction to Probability Theory, Hoel et. al
Also,
Probability Theory, Jaynes
is essential. For probabilistic programming I would also look into
Bayesian Methods for Hackers