(Part 3) Best products from r/math

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.

1. Mathematics: A very Short Introduction : A very good book, but also very short book about mathematics by Timothy Gowers, a Field medalist and overall awesome guy, gives you a feelling for what math is all about.
   
   
2. Concepts of Modern Mathematics: A really interesting book by Ian Stewart, it has more topics than the last book, it is also bigger though less formal than Gower's book. A gem.
   
   
3. What is Mathematics?: A classic that has aged well, it's more textbook like compared to the others, which is good because the best way to learn mathematics is by doing it. Read it.
   
   
4. An Infinitely Large Napkin: This is the most modern book in this list, it delves into a huge number of areas in mathematics and I don't think it should be read as a standalone, rather it should guide you through your studies.
   
   
5. The Princeton Companion to Mathematics: A humongous book detailing many areas of mathematics, its history and some interesting essays. Another book that should be read through your life.
   
   
6. Mathematical Discussions: Gowers taking a look at many interesting points along some mathematical fields.
   
   
7. Technion Linear Algebra Course - The first 14 lectures: Gets you wet in a few branches of maths.
   
   Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.
   
   
8. Linear Algebra Done Right: A pretty nice book to learn from, not as computational heavy as other Linear Algebra texts.
   
   
9. Linear Algebra: A book with a rather different approach compared to LADR, if you have time it would be interesting to use both. Also it delves into more topics than LADR.
   
   
10. Calculus Vol II : Apostols' beautiful book, deals with a lot of lin algebra and complements the other 2 books by having many exercises. Also it doubles as a advanced calculus book.
    
    
11. Khan Academy: Has a nice beginning LinAlg course.
    
    
12. Technion Linear Algebra Course: A really good linear algebra course, teaches it in a marvelous mathy way, instead of the engineering-driven things you find online.
    
    
13. 3Blue1Brown's Essence of Linear Algebra: Extra material, useful to get more intuition, beautifully done.
    
    Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.
    
    
14. Calculus: Tom Apostol's Calculus is a rigor-heavy book with an unorthodox order of topics and many exercises, so it is a baptism by fire. Really worth it if you have the time and energy to finish. It covers single variable and some multi-variable.
    
    
15. Calculus: Spivak's Calculus is also rigor-heavy by Calculus books standards, also worth it.
    
    
16. Calculus Vol II : Apostols' beautiful book, deals with many topics, finishing up the multivariable part, teaching a bunch of linalg and adding probability to the mix in the end.
    
    
17. MIT OCW: Many good lectures, including one course on single variable and another in multivariable calculus.
    
    Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.
    
    
18. Principle of Mathematical Analysis: Rudin's classic, still used by many. Has pretty much everything you will need to dive in.
    
    
19. Analysis I and Analysis II: Two marvelous books by Terence Tao, more problem-solving oriented.
    
    
20. Harvey Mudd's Analysis lectures: Some of the few lectures on Real Analysis you can find online.
    
    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.
    
    
21. Abstract Algebra: Dummit and Foote's book, recommended by many and used in lots of courses, is pretty much an encyclopedia, containing many facts and theorems about structures.
    
    
22. Harvard's Abstract Algebra Course: A great course on Abstract Algebra that uses D&F as its textbook, really worth your time.
    
    
23. Algebra: Chapter 0: I haven't used this book yet, though from what I gathered it is both a category theory book and an Algebra book, or rather it is a very different way of teaching Algebra. Many say it's worth it, others (half-jokingly I guess?) accuse it of being abstract nonsense. Probably better used after learning from the D&F and Harvard's course.
    
    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.

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.

• The Nature and Power of Mathematics, Donald M. Davis. This book seems to be a survey of some history of mathematics and various modern topics. Check out the table of contents to get an idea. You'll notice a few of the subjects in the list below. It seems like this would be a good buy if you want to taste a few different subjects to see what pleases your palate.
  
  
• Introduction to Graph Theory, Richard J. Trudeau. Check out the Wikipedia entry on graph theory and the one defining graphs to get an idea what the field is about and some history. The reviews on Amazon for this book lead me to believe it would be a perfect match for an interested high school student.
  
  
• Game Theory: A Nontechnical Introduction, Morton D. Davis. Game theory is a very interesting field with broad applications--check out the wiki. This book seems to be written at a level where you would find it very accessible. The actual field uses some heavy math but this seems to give a good introduction.
  
  
• An Introduction to Information Theory, John R. Pierce. This is a light-on-the-maths introduction to a relatively young field of mathematics/computer science which concerns itself with the problems of storing and communicating data. Check out the wiki for some background.
  
  
• Lady Luck: The Theory of Probability, Warren Weaver. This book seems to be a good introduction to probability and covers a lot of important ideas, especially in the later chapters. Seems to be a good match to a high school level.
  
  
• Elementary Number Theory, Underwood Dudley. Number theory is a rich field concerned with properties of numbers. Check out its Wikipedia entry. I own this book and am reading through it like a novel--I love it! The exposition is so clear and thorough you'd think you were sitting in a lecture with a great professor, and the exercises are incredible. The author asks questions in such a way that, after answering them, you can't help but generalize your answers to larger problems. This book really teaches you to think mathematically.
  
  
• A Book of Abstract Algebra, Charles C. Pinter. Abstract algebra formalizes and generalizes the basic rules you know about algebra: commutativity, associativity, inverses of numbers, the distributive law, etc. It turns out that considering these concepts from an abstract standpoint leads to complex structures with very interesting properties. The field is HUGE and seems to bleed into every other field of mathematics in one way or another, revealing its power. I also own this book and it is similarly awesome. The exposition sets you up to expect the definitions before they are given, so the material really does proceed naturally.
  
  
• Introduction to Analysis, Maxwell Rosenlicht. Analysis is essentially the foundations and expansion of calculus. It is an amazing subject which no math student should ignore. Its study generally requires a great deal of time and effort; some students would benefit more from a guided class than from self-study.
  
  
• Principles of Statistics, M. G. Bulmer. In a few words, statistics is the marriage between probability and analysis (calculus). The wiki article explains the context and interpretation of the subject but doesn't seem to give much information on what the math involved is like. This book seems like it would be best read after you are familiar with probability, say from Weaver's book linked above.
  
  
• I have to second sellphone's recommendation of Naive Set Theory by Paul Halmos. It's one of my favorite math books and gives an amazing introduction to the field. It's short and to the point--almost a haiku on the subject.
  
  
• Continued Fractions, A. Ya. Khinchin. Take a look at the wiki for continued fractions. The book is definitely terse at times but it is rewarding; Khinchin is a master of the subject. One review states that, "although the book is rich with insight and information, Khinchin stays one nautical mile ahead of the reader at all times." Another review recommends Carl D. Olds' book on the subject as a better introduction.
  
  Basically, don't limit yourself to the track you see before you. Explore and enjoy.

Hello! I'm interested in trying to cultivate a better understanding/interest/mastery of mathematics for myself. For some context:

 




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.

 




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 have a B.S. and M.S. in math, and am currently working on my PhD...here's my shot at your questions:

>1) At what point in your studies did you come to know about your limitations and abilities?

I didn't really have any struggles through my bachelor's, but as I got further into graduate studies I definitely had some hard classes and had to work much longer and harder to understand things than I ever had.

>I read about "Maryam Mirzakhani" two days ago. Do you think that you have a chance of producing worthy work in the future?

I don't think I'll ever win a Field's medal or be anywhere near the level of intelligence of someone like Maryam Mirzakhani, but I don't let that keep me from enjoying the journey. I know that I'll do something worthwhile, even if it's not groundbreaking.

>2) How did you choose your specific graduate program? I'm confused about what I should start with.

I was confused about what area to work in also, until I began studying for my comprehensive examinations (we have to take 3, each in different areas). I found that I really enjoyed studying the logic material, and I wasn't even too worried about the exam because enjoying the preparation made me well prepared. I just wanted to keep learning more. Just pick something that you find really interesting. It doesn't have to be "your area" for the rest of your life...you can always try something else later.

>3) How did you develop your critical thinking skills that are needed in following through with proofs and ideas?

The only way to get better at proofs as to do a ton of them. I had to get reamed pretty bad on some proofs at the beginning of grad school before I really got it...and I still have a long way to go. There's is always something to be improved upon.

There's a great excerpt from The Number Devil that sums up my feelings about proofs exactly:

"Have you ever tried to cross a raging stream?" the number devil asked.

"Have I?" Robert cried. "I'll say I have!"

"You can't swim across: the current would sweep you into the rapids. But there are a few rocks in the middle. So what do you do?"

"I see which ones are close enough together so I can leap from one to the next. If I'm lucky, I make it; if I'm not, I don't."

"That's how it is with mathematical proofs," the number devil told Robert. "But since mathematicians have spent a few thousand years finding ways to cross the stream, you don't need to start from scratch. You've got all kinds of rocks to rely on. They've been tested millions of times and are guaranteed slip-resistant. When you have a new idea, a conjecture, you look for the nearest safe rock, and from there you keep leaping--with the greatest of caution, of course--until you reach the other side, the shore."
...
"Sometimes the rocks are so far apart that you can't make it from one to the next, and if you try jumping, you fall in. Then you have to take tricky detours, and even they may not help in the end. You may come up with an idea, but then you find that it doesn't lead anywhere. Or you may find that your brilliant idea wasn't so brilliant at all."

Ok then, I'm going to assume that you're comfortable with linear algebra, basic probability/statistics and have some experience with optimization.

• Check out Hastie, Tibshirani, & Friedman - Elements of Statistical Learning (ESLII): it's basically the data science bible, and it's free to read online or download.
  
• Andrew Gelman - Data Analysis Using Regression and Multilevel/Hierarchical Models has a narrower scope on GLMs and hierarchical models, but it does an amazing treatment and discusses model interpretation really well and also includes R and stan code examples (this book ain't free).
  
• Max Kuhn - Applied Predictive Modeling is also supposed to be really good and should strike a middle ground between those two books: it will discuss a lot of different modeling techniques and also show you how to apply them in R (this book is essentially a companion book for the caret package in R, but is also supposed to be a great textbook for modeling in general).
  
  I'd start with one of those three books. If you're feeling really ambitious, pick up a copy of either:
  
  
• Christopher Bishop - Pattern Recognition and Machine Learning - Bayes all the things.
  
• Kevin Murphy - Machine Learning: A Probabilistic Perspective - Also fairly bayesian perspective, but that's the direction the industry is moving lately. This book has (basically) EVERYTHING.
  
  Or get both of those books. They're both amazing, but they're not particularly easy reads.
  
  If these book recommendations are a bit intense for you:
  
  
• Pang Ning Tan - Introduction to Data Mining - This book is, as it's title suggests, a great and accessible introduction to data mining. The focus in this book is much less on constructing statistical models than it is on various classification and clustering techniques. Still a good book to get your feet wet. Not free
  
• James, Witten, Hastie & Tibshirani - Introduction to Statistical Learning - This book is supposed to be the more accessible version (i.e. less theoretical) version of ESLII. Comes with R code examples, also free.
  Additionally:
  
  
• If you don't already know SQL, learn it.
  
• If you don't already know python, R or SAS, learn one of those (I'd start with R or python). If you're proficient in some other programming language like java or C or fortran you'll probably be fine, but you'd find R/python in particular to be very useful.

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!

A few years ago I was in a similar situation to the students you describe and am now at one of the universities you mention, so these suggestions are bound on what I found useful, or would have liked in retrospect.

Do you know about nrich? They have some interesting puzzles, arranged by keystage. They used to have a forum 'Ask NRICH' which was great, but currently closed for renovation, so look out for its reopening.

If it doesn't already exist, encourage the students to set up a maths society, research into something they find interesting (you can give suggestions) and give a brief talk to their peers.

However, what most inspired me was my teachers talking about what they found interesting. At GCSE, my teacher told us about Cantor's infinities as a special treat one day; we had pictures of Escher drawings in the classroom. At A Level, my teacher used to come in with maths puzzles he'd been working on over the weekend, and programs he'd written to demonstrate them (in Processing & Mathematica). Encourage them to come to you with questions too!

You can recommend some books to get them hyped. Anything you've enjoyed. I'd recommend Gower's Introduction to Mathematics for an idea of what maths is really about (beyond crunching equations at GCSE & A Level). Singh's Fermat's Last Theorem and Hofstadter's Godel, Escher, Bach are classics (especially on uni application forms) - the former an easy read, the latter somewhat more challenging. I'm sure you can find some more ideas on /r/mathbooks.

For STEP preparation, Siklos has an unbelievably helpful booklet. For the older ones, this would be instructive to look through even if they're not planning to apply for Cambridge.

Also (topical), arrange a class trip to see The Imitation Game!

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.

​

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.

​

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 ).

​

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.

Mathematics and Its History by John Stilwell

This really is a great book. From a review by Richard Wilders, MAA Reviews

>The author’s goal for Mathematics and its History is to provide a “bird’s-eye view of undergraduate mathematics.” (p. vii) In that regard it succeeds admirably. ... Mathematics and its History is a joy to read. The writing is clear, concise and inviting. The style is very different from a traditional text. ... The author has done a wonderful job of tying together the dominant themes of undergraduate mathematics. ... While Stillwell does a wonderful job of tying together seemingly unrelated areas of mathematics, it is possible to read each chapter independently. I would recommend this fine book for anyone who has an interest in the history of mathematics. For those who teach mathematics, it provides lots of information which could easily be used to enrich an opening lecture in most any undergraduate course. It would be an ideal gift for a department’s outstanding major or for the math club president. Pick it up at your peril ― it is hard to put down!

There are a fair number of popular level books about mathematics that are definitely interesting and generally not too challenging mathematically. William Dunham is fantastic. His Journey through Genius goes over some of the most important and interesting theorems in the history of mathematics and does a great job of providing context, so you get a feel for the mathematicians involved as well as how the field advanced. His book on Euler is also interesting - though largely because the man is astounding.

The Man who Loved only Numbers is about Erdos, another character from recent history.

Recently I was looking for something that would give me a better perspective on what mathematics was all about and its various parts, and I stumbled on Mathematics by Jan Gullberg. Just got it in the mail today. Looks to be good so far.

Yea John Green certainly isn't for everyone, particularly outside of the YA target audience. I wouldn't say it's his strongest book either, but it might be useful to check out.

In terms of mathematical directions you could go, graph theory is actually a pretty solid field to work in. It's basics are easy to grasp, the open problems are easy to understand and explain, and there are many obscure open ones that are easily within reach of a talented high schooler. In fact a lot of combinatorics is like that as well. I would recommend the book Introduction to Graph theory by Trudeau (which was originally titled Dot's and Lines). It's a great introduction to mathematical proof while leading the reader to the forefront of graph theory.

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.

Yes, they do! On average at least. Intuitively, as you get bigger and bigger there are more and more primes with which to make numbers, so the need for them gets less and less. This is answered by the Prime Number Theorem which says that (on average) the number of primes less than the number x is approximately x/log(x). Proving this was a triumph of 19th century mathematics.

Now, this graph of x/log(x) is very smooth and nice, so it only approximates where primes will be. It's not a guarantee. Imagine the primes as a crowd of people in an airport terminal. The crowd is, in general, flowing nicely from the ticket agents to the gate and this appears to be very nice when we look at it from high above. But when we get closer, we see some people walking from the ticket agents to the coffee shop, against the flow. Some kids are running in circles, which is not in the "nice flow" prediction. These fluctuations were not predicted by our model.

So even if primes obey the law x/log(x) overall, there are still fluctuations against this law. While the overall trend is for primes to get infinitely far apart we predict there are infinitely many primes that are right next to each other, totally against the flow. This is the Twin Prime Conjecture. We have recently proved that there are infinitely many pairs of primes, both of which are separated by only ~600 numbers. This was a huge deal and was done only within the last year or so, but we want to get that number down to 2.

We can also ask: "Do these fluctuations affect the overall flow in a significant way, or are they mostly isolated events that don't mess up the Prime Number Theorem approximation too much?" This is the content of the Riemann Hypothesis. If the Prime Number Theorem says that primes are somewhat ordered nicely, then the Riemann Hypothesis says that the primes are ordered as nicely as they can possibly get. That would mean that even though there are variations to the x/log(x) approximation, these fluctuations do not mess things up that bad.

Now, when looking for large primes, we generally look at expressions like 2^(n)-1 because we have fast algorithms to check if these guys are prime. But, in general, most primes do not look like that, they're just very nice numbers that we can check the primatlity of. We do not even know if there are infinitely many primes of the form 2^(n)-1, called Mersenne Primes so we could have already found them all. But we are pretty convinced there are infinitely many, so we're not too worried.

I don't know what your background is, but I've heard that the Prime Obsession is a good layperson book on this (though I haven't read it). If you have math background in complex analysis and abstract algebra, then you could look Apostol's Introduction to Analytic Number Theory.

OK, here's a suggestion. This won't help you with more modern algebraic tricks, but it'll give you a taste of what "real" math is while teaching you some geometry in the process. Buy this - there are cheaper versions, but the translation's great and the layout is easy to follow, with diagrams on two pages when proofs span two pages.

Read book 1, slowly, preferably with someone else to talk about it. Euclid is fascinating, and has inspired people to become mathematicians for over two thousand years.

Alternatively, there is a ton more material out there for algebra, problem solving, trig, whatever. But Euclid hits at the root of what's great about math - ingenuity, rigor, beauty.

Geometry is a beautiful subject and you can study it right now. Have you already read Euclid's Elements? It may take a while to understand but it's a very nice book. I'd also suggest you study more algebra and possibly trigonometry on your own so you may tackle Calculus earlier. Almost any text book or Khan Academy may help you there. Set theory can also be very nice but Wikipedia's articles are probably not the right place to go for a beginner. Wikipedia likes to focus on rigor rather than good explanations. I wish I could recommend a set theory book or web page but I do not experience with it. I learnt most of my set theory form college-level discrete math textbooks so I'm afraid I can't help you there.

EDIT: Although I have only skimmed through it, Mathematics: A very short introduction is an interesting an quite accessible book.

I’m not a mathematician, and my mathematical knowledge doesn’t extend beyond the undergraduate level in most fields, so I’m not sure what most professional geometers would recommend. Depends a lot on which specific “later topics” you’re interested in, I expect.

I don’t think Kiselev or the AoPS book is going to be quite what you’re asking for. In terms of content they’re both fairly typical high school geometry books, just a bit more rigorous than some watered down American textbooks I’ve seen, with better problems. You should take a look at the Coxeter and Greitzer book though. One other book you might look at is Felix Klein’s Elementary Mathematics from an Advanced Standpoint.

For use in physics, computer modeling (for graphics, games, robotics, computer vision, cartography, physical simulation, and the like), etc., I think Hestenes style “geometric algebra” (wikipedia) should be taught to bright high school students and most undergraduates in technical fields. Cf. “Reforming the Mathematical Language of Physics”, “Primer on Geometric Algebra”, “Grassmann’s Vision”.

In particular the “conformal model” is really powerful and pleasant to work with, cf. http://www.geometricalgebra.net (or if you don’t want to buy a book, try this Ph.D thesis.

• • *
    
    It sounds like you might be interested in a math history book though. I like Stillwell’s pretty well.

I have Mathematics:From the Birth of Numbers and it’s excellent.

Highly recommend

> This extraordinary work takes the reader on a long and fascinating journey--from the dual invention of numbers and language, through the major realms of arithmetic, algebra, geometry, trigonometry, and calculus, to the final destination of differential equations, with excursions into mathematical logic, set theory, topology, fractals, probability, and assorted other mathematical byways. The book is unique among popular books on mathematics in combining an engaging, easy-to-read history of the subject with a comprehensive mathematical survey text. Intended, in the author's words, "for the benefit of those who never studied the subject, those who think they have forgotten what they once learned, or those with a sincere desire for more knowledge," it links mathematics to the humanities, linguistics, the natural sciences, and technology.

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.

​

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 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 too love fun math[s] books! Here are some of my favorites.

The Number Devil: http://www.amazon.com/dp/0805062998

The Mathematical Magpie: http://www.amazon.com/dp/038794950X

I echo the GEB recommendation. http://www.amazon.com/dp/0465026567

The Magic of Math: http://www.amazon.com/dp/0465054722

Great Feuds in Mathematics: http://www.amazon.com/dp/B00DNL19JO

One Equals Zero (Paradoxes, Fallacies, Surprises): http://www.amazon.com/dp/1559533099

Genius at Play - Biography of J.H. Conway: http://www.amazon.com/dp/1620405938

Math Girls (any from this series are fun) http://www.amazon.com/dp/0983951306

Mathematical Amazements and Surprises: http://www.amazon.com/dp/1591027233

A Strange Wilderness: The Lives of the Great Mathematicians: http://www.amazon.com/dp/1402785844

Magnificent Mistakes in Mathematics: http://www.amazon.com/dp/1616147474

Enjoy!

Yes. This is a classic question and the typical answer is

f(x) = x^2 sin(1/x) if x != 0

f(x) = 0 if x = 0

The proof that f is continuous, and f' exists but is not continuous is left as an exercise for the reader. :-)

The book Counterexamples in Analysis has this and more. Having this book handy will do wonders for you and your class and I highly recommend it. Thank god Dover got hold of the copyright and re-printed it, it is a great book and the original is hard to find.

There's a couple options. You could pick up a basic elementary number theory book, which will have basically no prerequisites, so you'll be totally fine going into it. For instance Silverman has an elementary number theory book that I've heard great things about. I haven't read most of it myself, but I've read other things Silverman has written and they were really good.

There's a couple other books you might consider. Hardy and Wright wrote the classic text on it, which I've heard still holds up. I learned my first number theory from a book by Underwood Dudley which is by far the easiest introduction to number theory I've seen.

Another route you might take is that, since you have some background in calculus, you could learn a little basic analytic number theory. Much of this will still be out of your reach because you haven't taken a formal analysis class yet, but there's a book by Apostol whose first few chapters really only require knowledge of calculus.

If you decide you want to learn more number theory at that point, you're going to want to make sure you learn some basic algebra and analysis, but these are good places to start.

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

To answer your second question, KhanAcademy is always good for algebra/trig/basic calc stuff. Another good resource is Paul's online Math Notes, especially if you prefer reading to watching videos.

To answer your second question, here are some classic texts you could try (keep in mind that parts of them may not make all that much sense without knowing any calculus or abstract algebra):

Men of Mathematics by E.T. Bell

The History of Calculus by Carl Boyer

Some other well-received math history books:

An Intro to the History of Math by Howard Eves, Journey Through Genius by William Dunham, Morris Kline's monumental 3-part series (1, 2, 3) (best left until later), and another brilliant book by Dunham.

And the MacTutor History of Math site is a great resource.

Finally, some really great historical thrillers that deal with some really exciting stuff in number theory:

Fermat's Enigma by Simon Sigh

The Music of the Primes by Marcus DuSautoy

Also (I know this is a lot), this is a widely-renowned and cheap book for learning about modern/university-level math: Concepts of Modern Math by Ian Stewart.

That's not a bad idea at all - I used EM way back (like 2002) for natural language processing, still remember it a bit, but dang gonna have to brush up :) Thx for the pointer!

Edit: Haha just realized I have that book! Recognized it from the cover shot on amazon :)

For books that will help you appreciate math, I recommend Journey Through Genius by William Dunham for a general historical approach, and Love and Math by Edward Frenkel and Prime Obsession by John Derbyshire for specific focuses in "modern" mathematics (in these cases, the Langlands program and the Riemann Hypothesis).

There's a lot of mathematical lore that you'll find really interesting the first time you read it, but then it becomes more and more grating each subsequent time you come across it. (The example that springs most readily to mind is how the Pythagorean theorem rocked the Greeks' socks about their belief in numbers and what the brotherhood supposedly did to the guy who proved that irrational numbers exist). For that reason, I recommend reading only one or two books that summarize the historical developments in math up to the present, and then finding books that focus on one mathematician or one theorem that is relatively modern. In addition to the books I mentioned above, there are also some good ones on the Poincare Conjecture and Fermat's Last Theorem, and given that you're a computer science guy, I'm sure you can find a good one about P = NP.

What level of dynamical systems are we talking here? Graduate or undergraduate. In the former case I would recommend: http://www.amazon.com/Introduction-Dynamical-Encyclopedia-Mathematics-Applications/dp/0521575575
and for an undergraduate approach I would recommend:
http://www.amazon.com/Differential-Equations-Dynamical-Introduction-Mathematics/dp/0123497035
Both are pretty fun introductions to the subject. Good luck in your search

I have this version. One of the best features is that if a proof goes onto the backside of a page, they put another diagram on that page. That was you don't have to flip back and forth trying to follow the construction. I have only read the first book, but it was good, and the other reviews on amazon seem to say it was good throughout. As far as other good, understandable books with limited prerequisites, there is Axler's Linear Algebra Done Right. If you've already taken a Linear Algebra class this book will most likely give you a new and hopefully more intuitive perspective. Furthermore Linear Algebra might have more applications in your field.

if you want determinants, Shilov's is supposed to be "Determinants done right" I wouldn't recommend the other Dover LA book by Stoll

http://www.amazon.com/Linear-Algebra-Dover-Books-Mathematics/product-reviews/048663518X/

-----------

Anyway: Free!

http://www.math.ucdavis.edu/~anne/linear_algebra/

http://www.math.ucdavis.edu/~linear/linear.pdf

http://www.cs.cornell.edu/courses/cs485/2006sp/LinAlg_Complete.pdf (Dawkins notes that were recently pulled off lamar.edu site, gentle intro like Anton's)

http://joshua.smcvt.edu/linearalgebra/

http://www.ee.ucla.edu/~vandenbe/103/reader.pdf

http://www.math.brown.edu/%7Etreil/papers/LADW/LADW.pdf

https://math.byu.edu/~klkuttle/Linearalgebra.pdf

---------

Or, google "positive definite matrix" or "hermitian" or "hessian" or some term like that and it will show you lecture notes from dozens of universities after the inevitable wikipedia and Wolfram hits

I'm a general engineer myself, with a side interest in computer science. Szeliski's book is probably the big one in the computer vision field. Another you might be interested in is Computer Vision by Linda Shapiro.

You may also be interested in machine learning in general, for which I can give you two books:

• Heavily Theoretical: Christopher Bishop - Pattern Recognition and Machine Learning (Very much research oriented. Assumes you're decent at calculus.)
  
• Heavily Practical: Toby Segaran - Programming Collective Intelligence (A shallow look at the most common practical algorithms straightforwardly applied to real life examples.)
  
  I see you're interested in compilers. The Dragon book mainly focuses on parsing algorithms. I found learning about Forth implementations to be very instructive when learning about code generation. jonesforth is a good one if you understand x86 assembly.

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.

If you'd like an alternative to calculus, try learning linear and/or abstract algebra. Shilov's Linear Algebra is a good book on linear algebra. Linear algebra comes up everywhere, so it's definitely worth learning. The abstractions involved such as fields should also be a good introduction to higher mathematics. For even more abstraction, try A Book of Abstract Algebra by Charles Pinter which is one of my favorite books.

While calculus is also fundamental, personally I find linear and abstract algebra to be much more enjoyable subjects.

Here's three very good books:

1. De Morgan, On the Study and Difficulty of Mathematics. This is a free book available on the internet. Read the parts you find interesting.
   
   
2. Gelfand, Algebra.
   
   
3. Chrystal, Algebra: An Elementary Text-Book. This is available online for free. A lot of the greatest mathematicians and physicists of the last century lauded this (erdos, feynman...)

A true classic that will make you a beast at calculus:

Calculus: An Intuitive and Physical Approach by Morris Kline

It's old-school but totally awesome. Gives you great explanations for why we use what we use in the mathematical world.

Made me the man I am today.

http://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536

:D
http://betterexplained.com/

http://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536/ref=sr_1_1?ie=UTF8&qid=1422649729&sr=8-1&keywords=calculus+an+intuitive&pebp=1422649747330&peasin=486404536

The first site is fun, because it teaches you how to intuitively understand math. I love it. Second is a book that makes you think. Read the reviews for it. I really hope it helps because it's helped me, and I didn't even like math that much in the beginning, now I'm all excited for it :D

Think about basic high school math. You might have forgotten a few very specific ideas to solve a few very specific problems, but it's likely you remember almost all of it.

Why? Because you used it exhaustively in your basic undergrad math courses. Setting a derivative = 0 often demanded you factor, even if your directions never specifically said "find all solutions to this equation."

So, maybe you forgot a specific application of something like finding the principal value given blah blah compounded continuously, but you certainly know how to rearrange equations to solve for a variable.

Using something was practice, and so it was ingrained into your head (plus, after years of doing it, it's simple and downright monotonous).

But what about now? Were you extensively using Weierstrass's M-test on series in later classes? If you say yes, I won't believe you. Can you still find the integral of an obnoxious complex-valued function using residue theorems? Did you use these extensively in other classes? Doubtful, but possible.

This is the problem you are facing. I STRONGLY DOUBT you've been underexposed, but I HIGHLY AGREE with the possibility that you've forgotten.

So here's the important question: CAN you go back and relearn things? You say "progress is slow," but this is not a real answer to my question. Given one hour each day, can you, in 3 days, Mon/Wed/Fri, reteach yourself to determine if a metric space is compact? If you say Yes, then you are in a great position! There are many who sit through the class in one week and still have no clue! If you say No, then you're not necessarily in a BAD position (though you might be), you're just possibly in NO position.

So, here's the idea: you can't get good at upper level math (which will be considered lower level MATH math when you're going through grad school) by simply figuring it out. You got good at lower level math through practice; this is how you will get good at upper level math.

So what if progress is "slow"? Speed is subjective, but it's far more important that you CAN solve abstract problems rather than being able to blast through them--speed will develop later, and I know many PhD students at great schools who don't always remember what the subgroups of some strange group are or even how to find them.

So, let's answer, now, your REAL question: are you in for a rude awakening?

Yes, you are. But not for the reason you suspect. When you are in grad school, your faculty will (or it BETTER) have higher expectations of what you know vs. what you can do, and they're more concerned with what you can do than they are with what you know (forget something? Look it up. Forget how to do something? Looking it up may not help you...).

The fact that you are making ANY progress at ALL is enough to show that you are capable of doing things, even if you don't know things.

But are you in for a rude awakening because things are going to be hard because you've forgotten so much knowledge and thus you might have made a mistake because you'll never get up to speed? No. Most of my graduate level courses redefined things defined for me back as an undergrad, since at that level it gets difficult to figure out what students know and what they don't know based on where they came from.

But let's not build false hope and try and stay grounded in reality by this--

Check out this book: http://www.amazon.com/All-Mathematics-You-Missed-Graduate/dp/0521797071

Tinker through it and, when you're done, retake the MGRE. If all goes well, you're fine. If not, then you may very well not be. Don't rely entirely on that book to fill in gaps: use it for the TOPICS it presents, read through it, and when you're confused go find ANOTHER source relevant to the current chapter to fill in the gap.

But don't be crazy: I specifically never went through chapters 5,6,7,8,12,13,15,16 until I was in grad school. So, rather, figure out what you did as an undergrad, and go through THOSE relevant chapters in this book to get you up to speed with the ideas, and maybe dabble in some other chapters as time allows.

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.

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.

The BEST way to study history of math is to read classic math texts from history such as the Elements, the Conics, the Principia, etc.

Original texts aside, I recommend The World of Mathematics by James Newman.

I also recommend Newton's Principia: The Central Argument by Densmore.

I found 'Understanding Analysis' by Stephen Abbott ( https://www.amazon.co.uk/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ) to be super helpful/enlightening post Real Analysis insofar that it helped me build an intuition and understanding for some of the key ideas. Earlier today someone highly recommended this book as well: 'A Story of Real Analysis'
http://textbooks.opensuny.org/how-we-got-from-there-to-here-a-story-of-real-analysis/ (download link on the right). I had a quick glance through it and it seems pretty good.

take a look at Pattern Recognition an Machine Learning by Bishop,

http://www.amazon.com/Pattern-Recognition-Learning-Information-Statistics/dp/0387310738

it's an excellent text, though not for the faint of heart. just the first chapter should provide you with a great answer to your question.

Mathematics: A Very Short Introduction by Timothy Gowers. From the product description:

> The aim of this book is to explain, carefully but not technically, the differences between advanced, research-level mathematics, and the sort of mathematics we learn at school. The most fundamental differences are philosophical, and readers of this book will emerge with a clearer understanding of paradoxical-sounding concepts such as infinity, curved space, and imaginary numbers. The first few chapters are about general aspects of mathematical thought. These are followed by discussions of more specific topics, and the book closes with a chapter answering common sociological questions about the mathematical community (such as "Is it true that mathematicians burn out at the age of 25?") It is the ideal introduction for anyone who wishes to deepen their understanding of mathematics.

Correct, thats the bachelor's I took... And continued on the Mathematical Modelling and Computing masters.

​

In regards to books, if you intend on going with Machine Learning Christopher Bishop - Pattern Recognition and Machine Learning (2006 pdf version) is pretty much the bible. Its a bit expensive, but well worth the investment and goes through everything you will ever need to know.

It wont be able to replace course books, but will be just about the best supplement you can find.

And if going with Neural Networks (Deep Learning), basically look up George Hinton and start reading all his stuff (And Yann LeCun, but they often wrote together)

This free pdf book should help you: Proof, Logic, and Conjecture - The Mathematician's Toolbox

It's really well written (I like it better than Velleman's How to Prove It.) After this you should go through something easier than Rudin, like Spivak Calculus. Then you can try a real analysis book, but try using Abbott or Pugh instead; I hear those books are much better than Rudin.

I wouldn't call it a "branch" exactly, but pathological functions are pretty much the definition of "weird." Things like Weierstrass functions, the Cantor function, the Conway base 13 function. There's a good book with a lot of this stuff in it called Counterexamples in Analysis. There's another one on topology I haven't read yet.

They may not be the best books for complete self-learning, but I have a whole bookshelf of the small introductory topic books published by Dover- books like An Introduction to Graph Theory, Number Theory, An Introduction to Information Theory, etc. The book are very cheap, usually $4-$14. The books are written in various ways, for instance the Number Theory book is highly proof and problem based if I remember correctly... whereas the Information Theory book is more of a straightforward natural-language summary of work by Claude Shannon et al. I still find them all great value and great to blast through in a weekend to brush up to a new topic. I'd pair each one with a real learning text with problem sets etc, and read the Dover book first quickly which introduces the reader to any unfamiliar terminology that may be needed before jumping into other step by step learning texts.

That reminds me of a book that could be perfect for a course like this:

http://www.amazon.com/The-Symmetries-Things-John-Conway/dp/1568812205/

It discusses the idea of symmetry in great mathematical depth, but in a way that is much less formal and pedantic than a traditional math text. For me, there is something beautiful in the extraordinary variety available in the forms of symmetry explored in this book.

The Turing Omnibus has a bit of that sort of thing. It is mainly focused on computer science, and features some anecdotes about the uses of the techniques explained. This book has a lot of contributors, so the tone varies a bit from chapter to chapter, but it introduces a lot of topics.

In Code examines the RSA (and goes into a bit of depth about Modular Arithmetic) as well as the author's exploration of an alternative encryption.

Aha! Insight and The Number Devil are good books too. They're both aimed at younger readers, and feature lots of illustrations but focus more on thinking about numbers (and problems) than the mechanics of doing calculations.

Jan Gullberg's Mathematics: From the birth of numbers is a great book I'd recommend: https://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X

It introduces a lot of mathematical topics starting from the "simplest" (numbers you asked about) and advances to common stuff found in university studies (although not going extremely far), but what might be the biggest feat and useful to your case is that tells as a non-fictional story while at it, explaining mathematical tools, their history and how they relate to each other extremely well in a way a normal college textbook doesn't, and it doesn't assume you already know everything from school.

Introduction to Analytic Number Theory by Apostol is a great introduction to analytic number theory. This would be a great way to tie together the number theory, combinatorics, and calculus that you've seen so far.

His love of math is the most important thing to preserve. Do look for local math circles and places he can play with math, rather than simply doing it. It is not simply about going to the next level of the school progression. Get him math toys if you can. I have some suggestions for resources.

For your son's age a couple of things that might also be useful are the books Math Circles for 3-7 year olds and The Number devil.

(I am a math professor, but have worked with bright kids in this age group in a variety of ways)

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&qid=1346855198&sr=8-5&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&qid=1346860077&sr=8-1&keywords=rudin

http://www.amazon.com/Invitation-Linear-Operators-Matrices-Bounded/dp/0415267994/ref=sr_1_4?ie=UTF8&qid=1346860052&sr=8-4&keywords=from+matrix+to+bounded+linear+operators

http://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753/ref=sr_1_5?ie=UTF8&qid=1346860077&sr=8-5&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&ie=UTF8&qid=1346860356&sr=1-2&keywords=chaos+and+dynamics

http://www.amazon.com/Numerical-Analysis-Richard-L-Burden/dp/0534392008/ref=sr_1_5?s=books&ie=UTF8&qid=1346860179&sr=1-5&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.

Consider getting and working through Thomas Garrity's wonderful All the Mathematics You Missed But Need to Know for Graduate School. It's quite dense, but the goal is to help you develop intuition for all of the fields you listed and more. You won't really be able to learn a semester's worth of knowledge over the summer, but if you come into your coursework in mathematics with some intuition for what you're learning, you will have a huge leg up.

There's a nice little book, All the Mathematics You Missed: But Need to Know for Graduate School, that serves well as an answer to your question. It's pretty well-written, and lives up to the title.

In my opinion, the ideal undergraduate has had introductory courses in real analysis/advanced calculus, algebra, general topology, differential geometry of curves and surfaces, complex analysis, and combinatorics. Furthermore, more than one semester of linear algebra would be preferred.

I was in a similar position to you, until I had to read Euclid's Elements for my freshman math course in college. The difference between what I'd been taught in school before that and what I came to understand after reading source texts was huge.

While I somewhat understand why the modern American education system focuses almost exclusively on applied math, I can't help but feel that something significant was lost when we moved away from using texts like the Elements as primary textbooks; it lasted in that role for more than 2,000 years for a reason.

If you're looking to really begin understanding math, I'd start there.

http://www.amazon.com/Euclids-Elements-Euclid/dp/1888009195/

Is an excellent edition of Euclid.

How about The Number Devil? It might be a bit below the reading (and mathematical) level of a 15-year-old, but it brings up some really insightful ideas that highlight how basic principles can lead to really exciting results.

Check out this list. It includes most if not all books mentioned here.

From more to less advanced, generally:

• Geometry by Gray, Esplen, and Brannan
  
  
• The Infinite in the Finite
  
  
• A Geometrical Picture Book
  
  
• 99 Points of Intersection: Examples-Pictures-Proofs
  
  
• Lines and Curves: A Practical Geometry Handbook
  
  
• The Heart of Mathematics
  
  
• The Symmetries of Things
  
  
• A Mathematical Gift, 1: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World) (v. 1)
  
  A Mathematical Gift, II
  
  A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World) (v. 3)
  
  Note the first three volumes are cheaper as a set.
  
  Intuitive Topology (Mathematical World, Vol 4)
  
  Groups and Symmetry: A Guide to Discovering Mathematics (Mathematical World, Vol. 5)
  
  Knots and Surfaces: A Guide to Discovering Mathematics (Mathematical World, Vol. 6)
  
  
• A Beginner's Guide to Constructing the Universe
  
  and 5 activity books in the same style
  
  
• Going beyond books, but visual, manipulative, and tactile, the Zome Tools

You might find this book to be a good place to start: Algebra, by Gelfand and Shen.

Another book in a similar vein might be Basic Mathematics by Serge Lang.

I haven't used either of these books myself, but I came across them recently, and it looks like they might be among the few titles that cover high-school math in the way that you describe (they were written by prominent research mathematicians).

You might consider using the materials on Khan Academy (articles, videos, and exercises) to structure your studies, since these may be more closely aligned with current standards in the U.S. Then, as you go along, you can use these books as supplements (e.g. if you feel that a different perspective on a particular topic might be helpful).

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.

For visual beauty, it's hard to beat The Symmetries of Things (2008) by Conway, Burgiel & Goodman-Strauss.

The MAA review says, "The first thing one notices when one picks up a copy of The Symmetries of Things is that it is a beautiful book."

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.

Counterexamples in Analysis is a wonderful menagerie of mathematical oddities—it's full of pathological examples. It's the most fun math book I know of.

My favourite used to be Calculus on Manifolds until I started reading Munkres' Analysis on Manifolds. It covers the same material and then some and does a better job at explaining it. Spivak's purpose was a graduate reference book, and I think it does a good job at that. But in terms of learning Multivariable Analysis from it, it is very dense, and leaves out some stuff which I feel hinders it.

In terms of DE you could look at this one by Hirsh. It has some humour like Spivak, and is very theoretical, it has some applications in it but we skipped them when we took DE at my uni. There's also the dover book Advanced Ordinary Differential Equations (I think) which was used for the same course. However DE/Dynamical systems/chaos isn't a really concrete subject as opposed to analysis, so there are many ways of approaching it.

Find a copy of 101 Careers in Mathematics and look through it.

You may also be interested that a math major is among the best for taking the MCAT and LSAT (for medical school and law school, respectively).

Specific to your situation, I would concur with the other posters that say that upper-division mathematics is quite different from lower-division, and this difference scares some people away. You should try some courses and see for yourself!

A good answer to the third question is to compare Hilbert's millennium address with the book Mathematics: Frontiers and Perspectives.

Hilbert's address and list of problems did a fairly good job of capturing the mathematical zeitgeist at the turn of the previous century. It took a book, and there will certainly be those who feel the book has missed something vital, to make an attempt at capturing the mathematical zeitgeist at the turn of this one. The book, and a few articles in particular, make attempts at sketching the next 100 years, and the personal accounts of the process of doing mathematics should give you an idea of an answer to your first.

Unfortunately, this book isn't very accessible to a non-math person. To really understand the answers to your question you'll first have to learn some math. A very good first step, especially if you want just enough of a taste to figure out why Frontiers and Perspectives was written, is Gowers' Mathematics: A Very Short Introduction (Tim Gowers also contributed to F&P). From the introduction,

> ... I do presuppose some interest on the part of the reader rather than trying to drum it up myself. For this reason I have done without anecdotes, cartoons, exclamation marks, jokey chapter titles, or pictures of the Mandelbrot set. I have also avoided topics such as chaos theory and Gödel's theorem, which have a hold on the public imagination out of proportion to their impact on current mathematical research, and which are in any case well treated in many other books.

What text are you using?

Edit: Most calc II or multivariable textbooks that I've encountered (e.g.: this one, this one, this one, or this one) are full of examples, problems, and sections dealing with physical applications, if that's what you mean by outside the classroom.

From what I recollect, Calc II was mostly about developing facility with integration techniques, with some extensions of the concept of integration to boot. Although some of the material may seem to be of little relevance, think of it as an important stepping stone. It is preparing you for some super interesting subjects (like line integrals on vector fields!) that are used to model physical systems.

The Symmetries of Things is a wonderful book about geometric group theory, which in particular leads the reader through a classification of wallpaper patterns. You could try to work through it with him. Also, Burnside's lemma lets you answer some nice counting questions (such as the one on the Wikipedia page).

My general advice is to avoid formalism like the plague. Come up with questions that a lay person can understand without introducing any notation or definitions.

Perhaps you might find Shilov's Linear Algebra or Roman's Advanced Linear Algebra to be useful. Both of them treat bilinear and quadratic forms.

I think Shilov does actually discuss Gram-Schmidt orthonormalization, but he doesn't call it that, and it seems to be spread over several sections in chapters 7 and 8. Roman might be better for that. Anyway, you can peruse both of these at libgen.

That's great, it reminds me a lot of Calculus by Kline. He takes a similar approach and his introduction perfectly foresaw 60 years ago the problems with math education now.

https://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536

This is meant for younger children, probably, but The Number Devil is still an excellent children's book on many mathematical topics.

I picked up [Mathematics: A Very Short Introduction by Tim Gowers] (https://www.amazon.com/Mathematics-Short-Introduction-Timothy-Gowers/dp/0192853619) a few years ago. It talked about the types of problems mathematicians worked on, had some puzzles, and I think talked about the job of a math professor (I have not read it in a while). It was an interesting read and seemed like it would be accessible to most people (also, it's super short, which is always a plus when trying to get people into things).

How much depth do you need? For the basics of linear algebra, the text on Wikibooks should suffice. Make sure you read about eigenvalues. I like the coverage of PCA in section 12.1 of Bishop's book. As for differential equations, I'm not familiar enough with them to recommend a textbook on the topic.

http://www.mathman.biz/

http://wheelof.com/whitney/index.php?var=v1

http://vihart.com/

http://www.go.kestrel.nu/

http://www.amazon.com/Escher%C2%AE-Pop-Ups-Courtney-Watson-McCarthy/dp/0500515905

http://www.amazon.com/Alexs-Adventures-Numberland-Alex-Bellos/dp/1408809591

http://www.amazon.co.uk/You-Can-Cube-Patrick-Bossert/dp/0141326506

http://www.amazon.co.uk/Godel-Escher-Bach-Eternal-Golden/dp/0465026567

http://www.amazon.com/Mathematics-Very-Short-Introduction-Introductions/dp/0192853619

http://www.amazon.com/Project-Origami-Activities-Exploring-Mathematics/dp/1568812582

http://www.amazon.com/Origami-Tessellations-Awe-inspiring-Geometric-Designs/dp/1568814518

http://www.amazon.com/Proofs-without-Words-Exercises-Classroom/dp/0883857006

Once I got started ....whew. good luck!

Great video! Keep going!

Also, for those who love math, but are not mathematicians (like myself) I could recommend to read the book Prime Obsession by John Derbyshire. It is gonna blow your mind!

https://www.amazon.com/Prime-Obsession-Bernhard-Greatest-Mathematics/dp/0452285259

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

Here is the book I always recommend for people who want an introduction to graph theory:

• Introduction to Graph Theory, Richard J. Trudeau
  
  It's super cheap (only $3.99 on Amazon) and I think it's really a good introduction to the subject. It doesn't go as far in depth as more advanced books, but Kuratowski's theorem is covered in Chapter 3.

I read Mathematics: From the Birth of Numbers in high school / early college. It's a long book, but it's definitely worth checking out.

Definitely agree with the people recommending Calculus Made Easy by Silvanus P. Thompson. Often imitated, never equalled.

Another book similar to that is The Calculus for the Practical Man by J.E. Thompson. Besides its fame for being the book that Richard Feynman used to teach himself calculus, it has a completely nonstandard proof that the derivative of sin(x) is cos(x), using an argument based on arc length, which I haven't seen in any other book.

For more modern books I'd recommend Kline's book, which is underrated in my opinion. I'd avoid Spivak's book, which I feel is vastly overrated; it makes calculus even drier than the standard books do.

Proceeding with knowledge gaps is something everyone has to do. In your case, you're going to have to improvise a lot. What I tend to do is put a black box on any confusing detail and write it off as
"this blackbox let's me do X." If a definition is confusing for me, I replace the definition with an example that I understand and leave it be. (Some definitions have this sort of infinite regression to it; to understand this definition, you need to understand these other 3 definitions, which requires you to understand these 9 definitions and so on.)

Normally people have to take classes and pass an exam so you have at least 1 year to build that knowledge.

I don't recommend trying to learn what took others years to learn in 1 month, that's just unrealistic. Talk to your advisor; a lot of times you don't need to know the subject 100%, just some parts of it.

For analysis, you might not need to know everything about it, just maybe what a Hilbert space is and some standard results. For complex, honestly I think that class was more to teach people how to do analysis (the proofs are very elegant and it really give you experience on how one ought to go about proving something in classical analysis), as far as results goes, I only know the residue theorem and Riemann mapping theorem. For algebra, I guess I know what all the structures are... but don't remember much else.

Oh, there's this book that supposedly give a good outline on math you need to know.

Prime Obsession

http://www.amazon.com/Prime-Obsession-Bernhard-Greatest-Mathematics/dp/0452285259/ref=sr_1_1?ie=UTF8&s=books&qid=1261458431&sr=1-1

It explains the Riemann Hypothesis mathematically and historically, alternating every chapter. It explains it at a level that people with a decent ability to understand math can follow.

I've not finished it, but I have gotten 2/3rd through and I've really enjoyed it.

These were the most enlightening for me on their subjects:

• Differential and Integral Calculus, Vol. 1&2 by Richard Courant
  
• Linear Algebra by Georgi Shilov
  
• Differential Equations and Their Applications by Martin Braun
  
• Introduction to Geometry by H.S.M. Coxeter
  
• A Course in Mathematical Analysis, Vol. 1&2 by Edouard Goursat
  
• Introduction to Topology and Modern Analysis by George Simmons
  
• The Theory of Functions of a Complex Variable by A.I. Markushevich
  
• Basic Topology by M.A. Armstrong
  
• Topology by James Dugundji
  
• Ordinary Differential Equations by V.I. Arnold
  
• Partial Differential Equations by Fritz John
  
• Introduction to Probability Theory by Paul Hoel, Sidney Port and Charles Stone
  
• Linear Programming by Katta Murty
  
• Nonlinear Programming: Theory and Algorithms by M. Bazaraa, H. Sherali and C. Shetty
  
• A First Course in Numerical Analysis by Anthony Ralston and Philip Rabinowitz
  
• A First Course in Stochastic Processes by Samuel Karlin and Howard Taylor
  
• Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields by John Guckenheimer and Philip Holmes
  
• Homology Theory by James Vick

For a general overview of everything to do with the history of math, which might be what you're looking for, I recommend Mathematics: From the Birth of Numbers. Very inspiring with a little bit of "how to do everything."

The Nature of Computation

(I don't care for people who say this is computer science, not real math. It's math. And it's the greatest textbook ever written at that.)

Concrete Mathematics

Understanding Analysis

An Introduction to Statistical Learning

Numerical Linear Algebra

Introduction to Probability

I know this is removed, so I can recommend my tool which builds a graph of products that are often bought together at Amazon.

http://www.yasiv.com/#/Search?q=graph%20theory&category=Books&lang=US - this is a network of books related to graph theory. Finding the most connected product usually yields a good recommendation. In this case it recommends to take a deeper look at https://www.amazon.com/Introduction-Graph-Theory-Dover-Mathematics/dp/0486678709

These might interest you:

Poincare's Prize


Prime Obsession


Fermat's Enigma

The Code Book

There is an excellent series of Counterexamples in ... books which might be relevant to this thread:

counterexamples in...

• calculus
  
• analysis
  
• topology
  
• probability & statistics
  
• several complex variables
  
  Or just run a search on amazon for "Counterexamples in".

Mathematics: From the Birth of Numbers

It's gigantic, but really entertaining to flip around in.

John Stillwell, Mathematics and its History. https://www.amazon.com/Mathematics-Its-History-Undergraduate-Texts/dp/144196052X

One of the best math books I've read.

First saw this in John Derbyshire's Prime Obsession book. Quite beautiful.

I had a copy of The Number Devil when I was a kid and it was wonderful.

OCW

Single Variable Calculus

Multivariable Calculus

Differential Equasions

Linear Algebra

Helpful books

Mathematical Proofs: A Transition to Advanced Mathematics

Understanding Probability: Chance Rules in Everyday Life

Linear Algebra

Other Resources

Kahn Academy

Probability (keithpeterb on youtube)

Probability (Kahn)

Linear Algebra (Kahn)


There, Fixed up your links.

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:

• How to Think Like a Mathematician: A Companion to Undergraduate Mathematics
  
  
• Thinking Mathematically
  
  
• How to Solve It: A New Aspect of Mathematical Method
  
  
• The Art of Problem Posing
  
  
• How to Study as a Mathematics Major
  
  
• How to Read and Do Proofs: An Introduction to Mathematical Thought Processes

two excellent books by John Derbyshire:

Prime Obsession regarding the Riemann Hypothesis

Unknown Quantity which is about the history of Algebra

This was an excellent primer I used the summer before grad school, it's all undergraduate level math. There's an analysis section and algebra section. https://www.amazon.com/All-Mathematics-You-Missed-Graduate/dp/0521797071

I've already made a comment but I just remembered that this book exists:

https://www.amazon.com/All-Mathematics-You-Missed-Graduate/dp/0521797071

You might find it helpful

I highly recommend Polya's How to Solve it too.

Try Stillwell. It covers a lot, and some may not be accessible to those with less than an undergraduate math education, but there is enough elementary material to make it interesting for anyone of any background.

Unfortunately, many books like Spivak or Thomas are going to be very expensive, although you can find scans of them online if you look hard enough.

Dover books are cheap and are often classics, for example Calculus by Kline.

Spivak would be worth it if you plan to go on to study mathematics. It's going to have the rigor (and interesting stuff from a mathematical standpoint) that are omitted or hidden in other texts.

Not a scholarly article, but I like this book https://www.amazon.com/Prime-Obsession-Bernhard-Greatest-Mathematics/dp/0452285259 and think it does a decent job going into the history and attempting to explain the math in a way that doesn't require a grad degree.

Define "being a mathematician" because the job market is fantastic at the moment for people with MAs, Ph.Ds, and even BA/BSs in math. Data science, quantitative analysis, actuarial science, or algorithmic trading, to name a few, are all jobs that if you have the chops and maybe a bit of coding experience are available. I'd consider anybody working in those positions a mathematician, as their daily work is going to involve a good bit of mathematical machinery. Maybe take a look at this book if you need inspiration.

The job market in academia on the other hand is extremely competitive, and if you haven't started on grad school yet, I don't have any hard evidence to back this up but I think you may be running out of time to achieve that, especially if you want to start a family, etc. So if you define being a mathematician as being a researcher in academia, you're right to be scared, and taking the risk is going to be a tough call. But if you feel like the inability to be a research mathematician means you have to work "crap jobs," rise above that - there are plenty of fine jobs out there that use math that are a hell of a lot easier to attain than academia. Even if you make it through grad school and find yourself not able to enter academia, a Ph.D is well respected in industry and unless you studied a very esoteric topic, you'll be easily employable.

Not gonna tell you how to live life, or what your current situation is though. It's your call in the end.

Set Theory:

Naive Set Theory

Number Theory:

Elementary Number Theory

Introduction to Analytic Number Theory

A Classical Introduction to Modern Number Theory

Topology:

Topology

Introduction to Topological Manifolds

On a more serious note, this book by Polya is wonderful.

if I can offer a suggestion, this is one that I really liked, if I remember the author is out of Microsoft's research and dev.

http://www.amazon.com/Pattern-Recognition-Learning-Information-Statistics/dp/0387310738?ie=UTF8&keywords=pattern%20recognition%20and%20machine%20learning&qid=1464626674&ref_=sr_1_1&s=books&sr=1-1

In addition to Baby Rudin, I really liked this book when I first start learning analysis.

Some good readings from the University of Cambridge Mathematical reading list and p11 from the Studying Mathematics at Oxford Booklet both aimed at undergraduate admissions.

I'd add:

Prime obsession by Derbyshire. (Excellent)

The unfinished game by Devlin.

Letters to a young mathematician by Stewart.

The code book by Singh

Imagining numbers by Mazur (so, so)

and a little off topic:

The annotated turing by Petzold (not so light reading, but excellent)

Complexity by Waldrop

He really should be starting with the Trudeau, much better bed side reading.

Here's a great reference that I didn't know about before grad school: "All the Mathematics You Missed: But Need to Know for Graduate School" https://www.amazon.com/dp/0521797071/ref=cm_sw_r_cp_api_-Zqryb5W5PQEA
It's not for learning new subjects, rather it's useful for seeing context and figuring out where your weak points are so that you can brush up using more thorough references.

All The Mathematics You Missed is a good overview of topics that are good to know for graduate school. Not all of them are on the GRE, but the summaries of the GRE topics hit most of the key points.

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

The Number Devil is for kids, but I read it as adult and it was fun.The first part of Playing with Infinity could be accessible as well.

I really like this book.

https://www.amazon.com/Symmetries-Things-John-H-Conway/dp/1568812205/ref=sr_1_1?s=books&ie=UTF8&qid=1493910460&sr=1-1&keywords=symmetries+of+things

Does he have a DS? You can get him the Professor Layton games. There are some that are pure math. Others are puzzles.

I'd also look for the book "The Number Devil"

http://www.amazon.com/Number-Devil-Mathematical-Adventure/dp/0805062998/ref=sr_1_1?ie=UTF8&qid=1289355197&sr=8-1

It presents some pretty sophisticated mathematical concepts (for a 13 year old) in an easy to understand way. Don't let the title scare off the religious folk

What are you majoring in?

What you're describing could just be a personality issue that's unrelated to maths, that maths is just be an example of. That being said, I find the way people are taught maths to be a form of abuse. It's like the way someone who was molested as a child might have weird issues with sex, so do most people have issues with maths who have had to go through maths in high school.

Just so that you know, what you think maths is, is actually almost not at all what maths really is. I would recommend, after you finish your exams and have nothing better to do, read this book about graph theory. It's $4 + shipping from amazon, or you may have it in the library wherever you're studying. It's kind of pointless, but there are a few nice bits about the philosophy of maths.

I'm reading Mathematics: From the Birth of Numbers right now, I recommend it for your needs.

Maybe start with a book like this and when you hit a wall, you get the relevant textbook and do exercises.

One of my undergrad profs said that "The only way to learn mathematics is to DO mathematics."

I like this book. https://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773

They all fit in about 500 or so pages. Here's the copy I own: http://www.amazon.com/gp/product/1888009195/ref=oh_aui_detailpage_o01_s00?ie=UTF8&psc=1

This Mathematics: A Very Short Introduction should be her first book. It's short, easily digestible and has insights on math thinking.

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?"

The rate of your learning is defined by your determination. If you don't give up then you will learn the material.

Look at the book that is required and only learn what you need in the class. Don't learn everything in the book either. Just learn what you need to do well and refer to the books when you get confused.

Note don't try to learn everything that's below. Only use it to learn what you actually need. This can be overwhelming at first but just set aside a set time to study this.
EDIT I added more books and courses.
OCW
http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/
http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/index.htm
http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/
http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/
Helpful books
http://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321390539/ref=sr_1_3?s=books&ie=UTF8&qid=1312542911&sr=1-3
http://www.amazon.com/Understanding-Probability-Chance-Rules-Everyday/dp/0521540364
http://www.amazon.com/gp/product/048663518X/ref=pd_lpo_k2_dp_sr_1?pf_rd_p=486539851&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=0155510053&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=0YXJR9EVHCH9PCBDN372

Khan Academy
http://khan-academy.appspot.com/#calculus
http://www.youtube.com/user/keithpeterb#p/u/19/dS2p_APpcnI
http://khan-academy.appspot.com/video/probability--part-1?playlist=Old%20Algebra
http://www.youtube.com/user/keithpeterb#p/u/19/dS2p_APpcnI
http://khan-academy.appspot.com/video/linear-algebra--introduction-to-vectors?playlist=Linear%20Algebra

EDIT: I knew nothing about topological quantum computation about 1.5 years ago but then I took a independent study in college and I was assigned 1-3 papers a week to read. Eventually I got it a few months ago. What got me through it was not giving up...

God Created the Integers, edited by Stephen Hawking. Includes selected works of various big names in mathematics with a brief biography of each preceding the math. The wiki article on the book has a list of all mathematicians included.

Prime Obsession, about Riemann and his famous hypothesis.

The Man Who Knew Infinity, about Ramanujan.

Shilov gives a rigorous, determinant-heavy treatment of LA in his $10 book. All the nice properties of determinants are verified in the first chapter

The course I took as an undergraduate used Friedberg, Insel and Spence. I remember liking it fine, but it's insultingly expensive. Find it in a library or get a used copy if you can. If you're looking for a bargain, it can't hurt to try Shilov. He's Russian, so the book is very terse, but covers a lot of ground.

You might try checking out the book, "Mathematics: From the Birth of Numbers"

Read a history, e.g. Gullberg.

I read somewhere (I think it was Derybshire?) that it really should be called a conjecture, but it's been labeled a hypothesis because of its importance/fame.

Edit: nope, here it is.

Here's another.

I found Prime Obsession really captivating.

he means this

If you're on a budget, check out Calculus: An Intuitive and Physical Approach by Morris Kline.

Calculus: An Intuitive and Physical Approach (Second Edition) (Dover Books on Mathematics) https://www.amazon.com/dp/0486404536/ref=cm_sw_r_cp_apip_qmMduBiBKxeqD


This book currently. I learned precalculus using Kahn academy over the year along with trig.

Counterexamples in topology

Counterexamples in analysis

Take your pick from this book: https://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753