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Not a book, but I can share a few videos that I've found inspirational during some rough times with mathematics...

Fermat's Last Theorem -- This is a documentary on Andrew Wiles' proof of Fermat's last theorem. It's also probably the most emotional video I've ever watched about math. Highly recommended.

Fractals -- This is a neat NOVA documentary on fractals. In particular, it provides some inspiring history regarding Mandelbrot's discovery and journey with this subject.

Everything is relative, Mr. Poincare -- Another exceptional and inspiring documentary.

The only book I can recommend is Journey Through Genius by William Dunham, which provides an excellent treatise on the history of mathematics. From the book description

> Dunham places each theorem within its historical context and explores the very human and often turbulent life of the creator — from Archimedes, the absentminded theoretician whose absorption in his work often precluded eating or bathing, to Gerolamo Cardano, the sixteenth-century mathematician whose accomplishments flourished despite a bizarre array of misadventures, to the paranoid genius of modern times, Georg Cantor. He also provides step-by-step proofs for the theorems, each easily accessible to readers with no more than a knowledge of high school mathematics.

It's a very good read, and not too gigantic. Good wishes your way, mate.

I passed exam P and exam FM earlier this year.


The textbook that covered the exam P material was This One It did an ok job and covered all the exam P material. I assume any mathematical Statistics book would be similar.


I did use the Kellison Theory of Interest book a bit. However, an interest theory book will not cover all the material on the syllabus for exam FM. roughly 2/3 of the syllabus is interest theory while the other third (roughly) is financial derivatives covered in This Book which is also used in the syllabus for exam MFE/3F.


Now, these textbooks are great for learning the material but your goal is to be able to pass the exams. To do this I HIGHLY recommend you get a study manual and use it as your primary method of study.


Some good study manuals for exam P are either from ASM or ACTEX.


The best manual for Exam FM is ASM. Many people I have talked to have passed the exam in the first sitting using only this manual as their study material.


Also, if you are interested there is a site called The Infinate Actuary that has video lessons available.

I hope this was helpful.

I think it depends what kind of PDEs you're going to be doing really. If you're just looking at physicsy things like Laplaces equation, the heat equation and the wave equation then a methods book might be good. My personal choice would be this one but there is a lot of choice out there.

If it's a slightly higher level PDEs course (doing stuff like method of characteristics and conservation laws) then either this dover book or this book were the two recommended texts for my upper undergrad course on PDEs. The second is also recommended on a grad course I'm doing come september, and has loads of material in the book.

If you could give some more details of the course I could probably help you pick one of these easier. :)

I don't have any PDFs, but here is a good one you can get for pretty cheap. I used it as an undergrad and still refer back to it.

You might also try this Dover book.

Hope that helps.

There's no single book that's right for everyone: a suitable book will depend upon (1) your current background, (2) the material you want to study, (3) the level at which you want to study it (e.g., undergraduate- versus graduate-level), and (4) the "flavor" of book you prefer, so to speak. (E.g., do you want lots of worked-out examples? Plenty of exercises? Something which will be useful as a reference book later on?)

That said, here's a preliminary list of titles, many of which inevitably get recommended for requests like yours:

1. Undergraduate Algebra by Serge Lang
   
   
2. Topics in Algebra, 2nd edition, by I. N. Herstein
   
   
3. Algebra, 2nd edition, by Michael Artin
   
   
4. Algebra: Chapter 0 by Paolo Aluffi
   
   
5. Abstract Algebra, 3rd edition, by David S. Dummit and Richard M. Foote
   
   
6. Basic Algebra I and its sequel Basic Algebra II, both by Nathan Jacobson
   
   
7. Algebra by Thomas Hungerford
   
   
8. Algebra by Serge Lang
   
   Good luck finding something useful!

These are, in my opinion, some of the best books for learning high school level math:

• I.M Gelfand Algebra {[.pdf] (http://www.cimat.mx/ciencia_para_jovenes/bachillerato/libros/algebra_gelfand.pdf) | Amazon}
  
• I.M. Gelfand The Method of Coordinates {Amazon}
  
• I.M. Gelfand Functions and Graphs {.pdf | Amazon}
  
  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.

The book Ideals, Varieties and Algorithms by Cox, Litle and O'Shea is a very good undergraduate level algebraic geometry book. It has the benefit of teaching you the commutative algebra you need along the way instead of assuming you know it.

I'm not really aware of any algebraic topology books I'd consider undergraduate, but most of them are accessible to first year grad students anyway, which isn't too far away from senior undergrad. Some of my favorite sources for that are Munkres' book and Fulton's Book.

For knot theory, I haven't really studied it myself, but I've heard that The Knot Book is quite good and quite accessible.

Is your objective to build a comprehensive understanding of the underlying topics of Calculus or is your objective to master quick problem solving, tricks, etc? If it is the latter I would suggest you pick this up as an auxiliary resource; Stuart is good but mastery of the mechanics of solving the problems will come only through ardent practice. You will need to see, and solve, a wider set of examples than is typically found in Stewart.

If your objective is the former I would grab this instead. Probably look for it on a used book seller's site like abebooks.com, though.

A few Suggestions:

• Kostrikin - Linear algebra and Geometry: The best there is. This book is solid and you can go far with it, it's a book for life. But beware that some of it's exercises are really difficult.
  
  
• Hoffman - Linear Algebra: This one has lots of examples and is particularly good on the eingenvectors part you want, but sometimes the notation is bad.
  
  
• Lang - Linear Algebra
  
  All this books can easily be found. Good luck.

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

As an introduction to Algebra I can recommend https://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773/ref=sr_1_1?ie=UTF8&qid=1486044711&sr=8-1&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&ie=UTF8&qid=1486042204&sr=8-1&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

I'd recommend Mathematics: A Very Short Introduction by Tim Gowers if you'd like a fairly serious but informative book describing what mathematics is really about.

For a more fun, coffee-table kind of book, have a look at The Math Book by Clifford Pickover.

You might look at Michael Spivak's Calculus ( http://www.amazon.com/Calculus-Michael-Spivak/dp/0914098896 ). In the preface to the second edition, Spivak writes:

>I have often been told that the title of this book should really be something like "Introduction to Analysis", because the book is usually used in courses where the students have already learned the mechanical aspects of calculus--such courses are standard in Europe.

The book starts by developing the real and complex number systems and later goes into proofs that pi is irrational, e is transcendental, etc.

Please note that I'm not a math major and have only just started working through the Spivak book myself, so I'm far from an authority on the subject. But it's the book I stumbled onto when I was looking for a similarly non-numeric perspective on calculus and basic analysis and so far I've been pleased with it.

I recommend Stewart's book for calculus. There are many computational problems and examples.
http://www.amazon.com/Calculus-6th-Edition-Stewarts-Series/dp/0495011606/ref=sr_1_2?ie=UTF8&qid=1371868204&sr=8-2&keywords=calculus+stewart
It's not too expensive if you buy it from an Amazon user. And the 6th edition will be fine. I actually like it more than the 7th edition.

http://www.amazon.com/gp/aw/d/0306812835/ref=mp_s_a_1_1?qid=1449385131&sr=8-1&pi=SY200_QL40&keywords=structures+or+why+things+don%27t+fall+down&dpPl=1&dpID=516rxXknhsL&ref=plSrch

I just ordered a copy for me and my just off to college wannabe engineer brother in law. Thanks!

I liked this one by Munkres

http://www.amazon.co.uk/Topology-Featured-Titles-James-Munkres/dp/0131816292

Hey, a probability and geometry book is on sale as well.

Then there's "Mathematics for Non-mathematicians"

As an undergrad physics major, I would recommend this as well. If you're going to continue and do graduate PDE work, I would just jump into Evans after that.

Graduate or undergraduate level?


If graduate, this is THE book to get.

This is much more applied.

You can find Michael Spivak's Calculus, which everyone tells me ought to be titled "Introduction to Analysis" on libgen.

Apostol's classic calculus textbook, used at Caltech and MIT. The Art of Problem Solving textbook for calculus. The Stanford and Harvard-MIT Math Tournaments have calculus subject tests. The college-level Putnam competition has calculus problems, in addition to linear algebra, abstract algebra, etc.