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.

It's a good question that's hard to answer exhaustively. Here are some pointers. Maybe they can help to start a discussion.

1. Paul Zeitz writes the following advice in The Art and Craft of Problem Solving (emphasis added):
   
   > It isn't hard to acquire a modest amount of mental toughness. As a beginner, you most likely lack some confidence and powers of concentration, but you can increase both simultaneously. You may think that building up confidence is a difficult and subtle thing, but we are not talking here about self-esteem or sexuality or anything very deep in your psyche. Math problems are easier to deal with. You are already pretty confident about your math ability or you would not be reading this. You build upon your preexisting confidence by working at first on "easy" problems, where "easy" means that you can solve it after expending a modest effort. As long as you work on problems rather than exercises, your brain gets a workout, and your subconscious gets used to success. Your confidence automatically rises.
   
   
2. A useful term to know is self-efficacy, which means "one's belief in one's ability to succeed in specific situations or accomplish a task". The Wikipedia article mentions four factors affecting self-efficacy, which are worth looking at. "The experience of mastery is the most important factor determining a person's self-efficacy. Success raises self-efficacy, while failure lowers it." This is consistent with Zeitz' advice to start with easy problems and then gradually increase the level of difficulty.
   
   
3. Another factor affecting self-efficacy is "vicarious experience". This is "most effectual when we see ourselves as similar to the model", but I still think it helps to know that many eminent mathematicians have experienced feelings of self-doubt at some points in their careers. Here are some quotes that illustrate this; they're from Advice to a Young Mathematician, which is well worth reading.
   
   > One struggles unsuccessfully with small problems and one has serious doubts about one's ability to prove anything interesting. I went through such a period in my second year of research, and Jean-Pierre Serre, perhaps the outstanding mathematician of my generation, told me that he too had contemplated giving up at one stage. Only the mediocre are supremely confident of their ability. The better you are, the higher the standards you set yourself — you can see beyond your immediate reach. — Michael Atiyah
   
   > When I arrived in Moscow in my last year of graduate study, Gel’fand gave me a paper to read on the cohomology of the Lie algebra of vector fields on a manifold, and I did not know what cohomology was, what a manifold was, what a vector field was, or what a Lie algebra was. — Dusa McDuff
   
   
4. Finally, in my opinion there's nothing wrong with getting nervous in office hours and fumbling with easy questions. I've done this myself several times. It's just the nerves talking. As one gets more comfortable and relaxed with the situation (the fourth factor affecting self-efficacy), one gets better at keeping calm and tackling the questions one piece at a time.

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

hey nerdinthearena,

i too find this area to be fascinating and wish i knew more on the upper end myself. i'm just going to list off a few resources. in my opinion, graduate school will concentrate a lot on progressing your technical knowledge, but will likely not give you a lot of time to hone your intuition (at least in the first few years). so, the more time you spend in undergraduate school doing so, the better.

helpful for intuition and basic understanding

• geometrical vectors by gabriel weinreich - why there's more to vectors and how to draw them. this basically shows visual examples of differential forms and the like.
  
• advanced calculus: a differential forms approach by harold edwards - a fantastic approach to differential forms. it's a treasure, so just read it. :) it is actually well built for skimming and skipping around.
  
• vector calculus, linear algebra, and differential forms: a unified approach - this is a big book, but you could just jump ahead to the later sections covering differential forms for an interesting approach and technical skill building.
  
• div, grad, curl are dead by william burke - pure intuition and similar to geometrical vectors. don't look for anything else here, but definitely read it.
  
• applied differential geometry by william burke - a more complete work than the above and showcases a lot of the connections between geometry and physics.
  
  

  more advanced but still intuitive
  

• an introduction to manifolds - this is at the same level and covers essentially the same material as lee, but in my opinion, it is much better. it is much more concise and to the point with more focused exercises.
  
• differential forms in algebraic topology by raoul bott and loring tu - the sequel to the above, but i haven't read beyond the first chapter. it is well known as a must read classic, but having some topology background will help with this book.
  
• the laplacian on a riemannian manifold: an introduction to analysis on manifolds - an interesting introduction to geometric analysis.
  
• differentiable manifolds: forms, currents, harmonic forms by georges de rham - a classic by one of the greats and also one of the only sources i've found that covers de rham currents (besides research papers that just use them).
  
• riemannian manifolds: an introduction to curvature by john lee - the sequel to lee's smooth manifolds book.
  
  hopefully this helps. if i were to revisit geometric analysis, i would basically use the above books to help bone up my understanding, intuition, and technical skill before moving on. these are also mainly geometry books, so learning analysis (like functional analysis) would be good as well. i mainly have three suggestions there.
  
  

  three general analysis favorites
  

• introductory functional analysis with applications by erwin kreyszig - used everywhere.
  
• calculus of variations by gelfand and fomin
  
• linear differential operators by cornelius lanczos - a geometrical look at the subject. also take a look at the variational principles of mechanics by him as well.

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!

If you enjoy analysis, maybe you'd like to learn some more?

I really enjoyed learning introductory functional analysis, which is presented incredibly well in Kreyszig's book Introductory Functional Analysis with Applications. It's very easy to read, and covers a lot and assumes very little on the part of the reader (basic concepts from analysis and linear algebra). This will teach you about doing analysis on finite and infinite dimensional spaces and about operators between such spaces. It's incredibly interesting, and I highly recommend it if you enjoy analysis and linear algebra.

Another great analysis topic is Fourier Analysis and wavelets. I enjoyed the books by Folland Fourier Analysis and Its Applications. I don't believe that book has any wavelets in it, so if you're interested in learning Fourier analysis plus wavelet theory, then I highly recommend the very approachable and fun book by Boggess and Narcowich A First Course in Wavelets with Fourier Analysis. If you have any interest at all in applications (like signals processing), this subject is fundamental.

Real answers for real high school student interesting in the conventional path to a conventional first course in quantum physics. Much of this advice applies more to the American school system, as that's where I was educated.

You're right, the first job is to get to calculus. Khan Academy is a good place for that! It's a bit messy, but just follow the knowledge web they have set up until you get to the topic of limits, which is the front door to introductory calculus. Along the way you'll also learn algebra and geometry. As soon as you can and as soon as you're ready for it, try and take a proper calculus class in your school. If you're in the United States, try to take AP Calculus.

If possible, take a physics class at your high school. If it's a reasonably big school, they'll have an algebra-based physics class and may even offer a college-level physics course that uses calculus (if you're in the United States, this will be called AP Physics C). Take this as soon as possible! If it's not offered, you may be able to take the equivalent course at a nearby college before you leave high school.

If you've done all of this right, you should know how to calculate things called derivatives and integrals, manipulate things called sequences and series, and understand the the basic rules of mechanics (force, momentum, energy, etc) and the electric and magnetic fields phrased in terms of calculus. In the language of most American universities, you now know Physics I & II and Calculus I & II.

Physics-wise, the usual next step is to take a course on waves, vibrations, and oscillations (see this table or contents) and / or a survey course on modern physics (see this standard text).

Math-wise, the next step is to take classes usually called

• multivariable calculus / vector calculus
  
• ordinary differential equations
  
• linear algebra
  
  The simplest way to do this is just to take these courses in a college or university, but there are also great online resources. I can personally endorse MIT OCW and Paul's Online Notes. Many schools also offer surveys of applied math at this level (with names like "mathematical physics" or "mathematical methods") that cover the basics of partial differential equations, fourier series, and more linear algebra / multivariable calculus / ODEs. See this book by Boas for a good idea of the content.
  
  Once you know all of that, you're ready to ready a real textbook on quantum physics. Some of the usual standards for a physicists' first course are the books by Griffiths or Shankar.
  
  Edited for link formatting
  
  TL;DR To go the physicist route, learn the following through school if you can swing it, but independent learning is possible and good resources exist online:
  
  
• Algebra I and II
  
• Geometry
  
• Calculus I and II
  
  Then these three in any order:
  
  
• Multivariable calculus
  
• Ordinary differential equations
  
• Linear algebra
  
  Then this, if you're going the usual physicist route:
  
  
• Mathematical methods
  
  On the physics side, you should take
  
  
• Physics I (sometimes called 'mechanics') with calculus
  
• Physics II (sometimes called 'electromagnetism) with calculus
  
  and then one or both of
  
  
• Modern Physics (sometimes called Physics III)
  
• Vibrations, waves, and oscillations (also sometimes called Physics III)

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.

During my sophomore year I took an "intro to proofs" course (known formally at the institution as Foundations of Advanced Mathematics) and I found it to be extremely beneficial in my development as a mathematician. We used Chartrand's "Mathematical Proofs" textbook (here's the link for those who are interested).

The text covered set theory, logic, the various proof methods, and then dug into stuff like elementary number theory, equivalence relations, functions, cardinality (culminating in Cantor's two main results), abstract algebra, and analysis. Obviously the book only scratched the surface on a lot of these topics, but I felt it accomplished its goal.

Part of my satisfaction with the course is likely due to the fact that we had a brilliant professor who taught the course in the spirit of what u/Rtalbert235 spoke of. He was able to clearly articulate the distinction between computation and theory. The way I like to say it is that he taught us the difference between pounding a bunch of nails into a 2X4 (computation) and building a house (proving theorems).

I don't mean to universally praise "intro to proofs" courses, however. I can definitely see how they can be horrible wastes of time if not done properly, and I can also appreciate the idea of "throwing" students into proof-based courses (analysis, algebra, and so on). For me though, I think it's worth the effort to try and optimize these sorts of classes, which will ultimately serve a LOT of math students who need to understand proofs, but don't necessarily have a desire to pursue the subject beyond the undergraduate level.

tl;dr - Given the right combination of textbook and professor, an "intro to proofs" course can be just what the doctor ordered for developing mathematicians.

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.

LIST OF APPLICATIONS IN MY DIFF EQ PLAYLIST
Have you seen the first video in my series on differential equations?

I'm still working on the playlist, but the first video lists a bunch of applications that you might not have seen before. My goal was to provide a sample of the diversity of applications outside of mathematics, and I chose fairly concrete examples that include applications in engineering.

I don't go into any depth at all regarding any of the particular applications (it's just a short introductory video), but you might find the brief introduction to be helpful.

If you find any one of the applications interesting, then a Google search will reveal more detailed resources.

A COUPLE OF FREE OR INEXPENSIVE BOOKS
Also, off the top of my head, the books below have quite a few applications that you might not see in the more standard textbooks.

• Differential Equations and Their Applications: An Introduction to Applied Mathematics, Martin Braun (Amazon, PDF)
  
• Ordinary Differential Equations, Morris Tenenbaum and Harry Pollard (Amazon)
  
  I think you can find other legal PDFs of Braun's third edition, too. Pollard and Tenenbaum is an inexpensive paperback from Dover, and I actually found a copy at my local library.
  
  ENGINEERING BOOKS
  Of course, the books I listed are strictly devoted to differential equations, but you can find other applications if you look for books in engineering. For example, I used differential equations in a course on signals and systems that I tutored last semester (applications included electrical circuits and mass-spring-damper systems).
  
  NEAT VIDEO (SOFT BODY MODELING)
  By the way, here's a cool video of various soft body simulations based on mass-spring-damper systems modeled by differential equations.
  
  Here's a Wikipedia article on soft body dynamics. This belongs to the field of computer graphics, so I'm not sure if you're interested, but mass-spring-damper systems come up a fair amount in engineering courses, and this is an application of those ideas that might open your mind a bit to other possible applications.
  
  Edit: typo

Without knowing much about you, I can't tell how much you know about actual math, so apologies if it sounds like I'm talking down to you:

When you get further into mathematics, you'll find it's less and less about doing calculations and more about proving things, and you'll find that the two are actually quite different. One may enjoy both, neither, or one, but not the other. I'd say if you want to find out what higher level math is like, try finding a very basic book that involves a lot of writing proofs.

This one is aimed at high schoolers and I've heard good things about it, but never used it myself.

This one I have read (well, an earlier edition anyway) and think is a phenomenal way to get acquainted with higher math. You may protest that this is a computer science book, but I assure you, it has much more to do with higher math than any calculus text. Pure computer science essentially is mathematics.

Of course, you are free to dive into whatever subject interests you most. I picked these two because they're intended as introductions to higher math. Keep in mind though, most of us struggle at first with proofwriting, even with so-called "gentle" introductions.

One last thing: Don't think of your ability in terms of your age, it's great to learn young, but there's nothing wrong with people learning later on. Thinking of it as a race could lead to arrogance or, on the other side of the spectrum, unwarranted disappointment in yourself when life gets in the way. We want to enjoy the journey, not worry about if we're going fast enough.

Best of luck!

I have a variety of books to recommend.
Brushing up on your foundations:
http://www.amazon.com/Beginning-Functional-Analysis-Karen-Saxe/dp/0387952241
If you get this from your library or browse inside of it and it seems easy there are then three books to look at:

1. http://www.amazon.com/Functional-Analysis-Introduction-Princeton-Lectures/dp/0691113874/ref=sr_1_4?s=books&ie=UTF8&qid=1368475848&sr=1-4&keywords=functional+analysis challenging exercises for sure.
   
2. http://www.amazon.com/Introductory-Functional-Analysis-Applications-Kreyszig/dp/0471504599/ref=sr_1_2?s=books&ie=UTF8&qid=1368475848&sr=1-2&keywords=functional+analysis (A great expositor)
   
3. Rudin's Functional Analysis (A challenging book for sure)
   
   More advanced level:
   
4. http://www.amazon.com/Functional-Analysis-Introduction-Graduate-Mathematics/dp/0821836463/ref=cm_cr_pr_product_top
   (An awesome book with exercise solutions that will really get you thinking)
   
   Working on this book and Rudin's (which has many exercise solutions available online is very helpful) would be a very strong advanced treatment before you go into the more specialized topics.
   
   The key to learning this sort of subject is to not delude yourself into thinking you understand things that you really don't. Leave your pride at the door and accept that the SUMS book may be the best starting point. Also remember to use the library at your institution, don't just buy all these books.

Several good books have already been mentioned in this thread, but some good books are hard to get into as a beginner.

I recommend Elementary Number Theory by Underwood Dudley as a good starting point for a beginner, as well as something like Apostol or Ireland-Rosen if you want more details.

I think it makes sense to start with something like Dudley to get an overall framework, and then rely on more detailed books to flesh out the details of whatever topics you're interested in more.

In particular, I think Dudley's book has an approach to Chebyshev's theorem (i.e. there is always a prime between n and 2n) that's great for beginners, even if someone with a bit more experience can streamline that proof a little.

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.

Competitive math problems are the best way to improve pure math skills. Although quite different from what would be considered mathematical research, they build core problem solving intuition. Look into the following:

http://www.artofproblemsolving.com/ - Website and forum dedicated to pure and competitive math problems, as well as the skills needed to solve them. They also keep archives of past competition problems. I recommend that you start looking at the AMC and AIME competitions, and see if you can participate in the AMC before you graduate. It's offered in February for high schoolers.

http://209.222.159.179/amc/a-activities/a7-problems/putnamindex.shtml - Website to find past problems from the Putnam Competition, a collegiate undergraduate math competition. These are very difficult for someone who is just starting off but are still very good problems to look at. If you are planning on majoring in math then I recommend that you take the Putnam exam all 4 years.

http://www.usamts.org/ - Website for USAMTS, a high school mathematics competition sponsored by the NSA. USAMTS is a lot different from typical competitions in that instead of a strict time limit, the problems are put out a month before any solutions are due. They also provide resources on their site for those who are unfamiliar with proof writing. The first round has already finished and the second round is due in about 2 weeks. However, you can jump in at any time, and if you score nicely you can get a lot of free stuff. I recommend that you do this, even if you don't plan on submitting your solutions.

As for books, I recommend Paul Zeitz's Art and Craft of Problem solving:

http://www.amazon.com/Art-Craft-Problem-Solving/dp/0471789011

This is one of the best books to get acquainted with the mental process for solving difficult math problems, as well as some of the more advanced techniques which you probably didn't pick up if you are just starting to become interested in more difficult math problems. If you don't want to spend the money, I'm sure you can find it online somewhere, but it's worth every penny.

For algebra, I'd recommend Mac Lane/Birkhoff. They may not be as comprehensive as some other texts but to me, they are more motivating, and will probably provide a better introduction to categorical thinking.

For linear algebra, I'm going to suggest something slightly unusual: Kreyszig's Introductory Functional analysis with applications. Functional analysis is essentially linear algebra on infinite dimensional spaces, and it generalizes a lot of the results in finite dimensions. Kreyszig does a good job motivating the reader. I can definitely sit down and read it for hours, much longer than I can for other books, and I definitely don't consider myself an analyst. However, it could be difficult if you are not familiar with basic topology and never seen linear algebra before.

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.

You could try Book of Proof by Richard Hammack. I've never read Velleman so I can't directly compare, but it's free for pdf (link to author's site above) and quite cheap in paperback (~$15). I found the explanations quite clear, the examples well worked and the exercises plentiful and helpful. Amazon reviewers seem to like it as well.

There are a few options. Firstly, if you are more familiar using infinity in the context of Calculus, then you might want to look into Real Analysis. These subjects view infinity in the context of limits on the real line and this is probably the treatment you are probably most familiar with. For an introductory book on the subject, check out Baby Rudin (Warning: Proofs! But who doesn't like proofs, that's what math is!)

Secondly, you might want to look at Projective Geometry. This is essentially the type of geometry you get when you add a single point "at infinity". Many things benefit from a projective treatment, the most obvious being Complex Analysis, one of its main objects of study is the Riemann Sphere, which is just the Projective Complex Plane. This treatment is related to the treatment given in Real Analysis, but with a different flavor. I don't have any particular introductory book to recommend, but searching "Introductory Projective Geometry" in Amazon will give you some books, but I have no idea if they're good. Also, look in your university library. Again: Many Proofs!

The previous two treatments of infinity give a geometric treatment of the thing, it's nothing but a point that seems far away when we are looking at things locally, but globally it changes the geometry of an object (it turns the real line into a circle, or a closed line depending on what you're doing, and the complex plane into a sphere, it gets more complicated after that). But you could also look at infinity as a quantitative thing, look at how many things it takes to get an infinite number of things. This is the treatment of it in Set Theory. Here things get really wild, so wild Set Theory is mostly just the study of infinite sets. For example, there is more than one type of infinity. Intuitively we have countable infinity (like the integers) and we have uncountable infinity (like the reals), but there are even more than that. In fact, there are more types of infinities than any of the infinities can count! The collection of all infinities is "too big" to even be a set! For an introduction into this treatment I recommend Suppes and Halmos. Set Theory, when you actually study it, is a very abstract subject, so there will be more proofs here than in the previous ones and it may be over your head if you haven't taken any proof-based courses (I don't know your background, so I'm just assuming you've taken Calc 1-3, Diff Eq and maybe some kind of Matrix Algebra course), so patience will be a major virtue if you wish to tackle Set Theory. Maybe ask some professors for help!

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 :)

Usual hierarchy of what comes after what is simply artificial. They like to teach Linear Algebra before Abstract Algebra, but it doesn't mean that it is all there's to Linear Algebra especially because Linear Algebra is a part of Abstract Algebra.

Example,

Linear Algebra for freshmen: some books that talk about manipulating matrices at length.

Linear Algebra for 2nd/3rd year undergrads: Linear Algebra Done Right by Axler

Linear Algebra for grad students(aka overkill): Advanced Linear Algebra by Roman

Basically, math is all interconnected and it doesn't matter where exactly you enter it.

Coming in cold might be a bit of a shocker, so studying up on foundational stuff before plunging into modern math is probably great.

Books you might like:

Discrete Mathematics with Applications by Susanna Epp

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Building Proofs: A Practical Guide by Oliveira/Stewart

Book Of Proof by Hammack

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al

How to Prove It: A Structured Approach by Velleman

The Nuts and Bolts of Proofs by Antonella Cupillary

How To Think About Analysis by Alcock

Principles and Techniques in Combinatorics by Khee-Meng Koh , Chuan Chong Chen

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) by Carol Ash

Problems and Proofs in Numbers and Algebra by Millman et al

Theorems, Corollaries, Lemmas, and Methods of Proof by Rossi

Mathematical Concepts by Jost - can't wait to start reading this

Proof Patterns by Joshi

...and about a billion other books like that I can't remember right now.

Good Luck.

Oh sorry, I forgot to reply to you. There's two textbooks I typically reference for ODEs that will be more than sufficient by the sounds of it.

There is Braun - Differential Equations and their Applications. The author tries to motivate each problem by going to some interesting applications. I think he fails in this as the applications are quite standard at the end of the day, he just goes into more detail to give the context. But the book is overall great and quite well explained.

The other one is kind of tough to track down as it seems to be out of print now Differential Equations with applications and historical notes by Simmons. Again the book has a bit of a gimmick where it puts each of the big contributors, and methods, in historical context. It's an excellent resource I find, it gets pretty in depth on Laplace transforms.

It's really your call which one you go with, if you have some extra time once you make it through your course content I would recommend taking a look at the sections on qualitative theory, systems of ODEs, and numerical methods. Those topics are both fascinating and I think tend to have more active work associated with them.

For light, math-related reading I've always enjoyed semi-biographical books about mathematicians, because these books usually include a summary of their mathematical contributions without getting too technical or dry. And they always get me pumped to do more math. Here's books I'd recommend in that vein:

The Man Who Loved Only Numbers An Autobiography of Paul Erdos. This one's a really great read with lots of Number Theory and Graph Theory in it. Not to mention a heartwarming bio of Erdos. If you haven't read this yet, give it a go.

The Equation That Couldn't Be Solved - A book about the history of Group Theory and how Galois was able to prove the Abel-Ruffini Theorem. At times it's a little simple, but it's fun to read, and it gives some insight into elementary group theory.

Of Men and Mathematics - not too much actual math in this one, but it's a very solid cheap, quick read. Well worth the money/time.

I agree that Arnold's ODE book is the best on the subject, but as you said it's not for beginners. For an intro to the subject, I think Martin Braun's Differential Equations and Their Applications is the best.

I've always thought that Fritz John's Partial Differential Equations is the best PDE book.

Start with fun things. Write down some of your favorite fun proofs. Think about and prove weird but true things (series that converge in one order but not in another, the fing central limit theorem, etc). Check out Proofs without Words, which is extra fun because you have to figure out how the proof works solely from the picture. Learn about and prove that there are only 17 possible types of wallpaper patterns, and do a hunt to find them all irl.

Once you remember why you started doing math in the first place, it will be easier to swim back into deeper waters.

edit:fixed links

Foolproof is a good example of this. Lots of self-contained chapters on random fun problems. (My only large critique is that the first chapter is very out of place; being basically a history schpiel. Mischaracterizes the book.)

Then there’s math adjacent stuff like Zero: the history of a dangerous idea that look at the history of math development.
(Side note: the first chapter of Pinter’s A Book of Abstract Algebra is a top knotch example of that. And very much in place, unlike the foolproof chapter I mentioned.

Then there are things that aren’t quite “pop”, but make themselves more accessible. Like An Illustrated Guide to Number Theory, which is both a legitimate intro to number theory and a reasonably sexy coffee table book that guests can leaf through. (Though I’d like to see a book that pushes the coffee table style accessibility further.)

Assuming by "numbers" you mean the set of numbers {0,1,2,.....} (commonly known as the "Natural" numbers) you should first know that there is some debate as to whether or not to include 0.

Whether you decide to include 0 or not, no one knows where/when the concept of the Natural numbers originated. In some cultures, notably the Piraha, they appear to have never been developed so if you believe Kronecker's "God gave us the integers...", God appears to have forgotten to tell to the Piraha.

My own view on the origin of the natural numbers is that they probably arose from trade. A scenario, which may or may not be true but I find particularly appealing, is given in Eugenia Cheng's book "How to Bake Pi". As an example suppose that I want to trade 1 salt cake for each sheep you have. I could line up all the sheep and parade them by one by one. As each sheep passed I could hand over 1 salt cake. This involves lining up the sheep, which, having lived with sheep as a kid, I can tell you is not the easiest thing to do. So instead you could just point at each sheep and hand over a salt cake, perhaps, as Cheng proposes in a likely nod to her musical background, singing a song while doing so. Then the song itself becomes the counting mechanism. The reason I like this so much is that it fits well with Eeny-Meeny-Miny-Mo

If you want to learn more about the historical origins of zero you might try: Zero:The Biography of a Dangerous Number be forewarned that this is a pop-sci book and it's tone is a fairly hyperbolic, here's a review that I think sums up this up pretty well.

An Introduction to Manifolds by Tu is a very approachable book that will get you up to Stokes. Might as well get the full version of Stokes on manifolds not just in analysis. From here you can go on to books by Ramanan, Michor, or Sharpe.

A Guide to Distribution Theory and Fourier Transforms by Strichartz was my introduction to Fourier analysis in undergrad. Probably helps to have some prior Fourier experience in a complex analysis or PDE course.

Bartle's Elements of Integration and Legesgue Measure is great for measure theory. Pretty short too.

Intro to Functional Analysis by Kreysig is an amazing introduction to functional analysis. Don't know why you'd learn it from any other book. Afterwards you can go on to functional books by Brezis, Lax, or Helemskii.

Possible path:

Learn to think like mathematicians because you'll need it. For example, Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al is a good book for that. When you got the basics of math argumentation down, it's time for abstract algebra with emphasis on vector spaces(you really need good working knowledge of linear algebra). People like Axler's Linear Algebra Done Right. Maybe, study that. Or maybe work through Maclane's Algebra or Chapter 0 by Aluffi.

After that you want to get familiar with more or less rigorous calculus. One possibility is to study Spivak's Calculus, then pick up Munkres Analysis on Manifolds.

Up next: differential geometry which is your main goal. At this point your mathematical sophistication will have matured to the level of a grad student of math.

Good luck.

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.

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

First, consistently solving A1 and B1 is a great start! Puts you well above the typical. Be sure to pay attention to how you write it up: Putnam graders are very strict and solutions most often get 0, 1, 9, or 10 points. Be also aware of what your goals are and don’t get anxious, you’re not looking to solve everything, so it's good to fully solve one problem before moving on. Putnam problems in particular often have short clean solutions that are really satisfying to find.

You also can't beat just working through problems. Putnam 1985-2000 by Vakil, Kedlaya, Poonen is fantastic as it gives many ways of solving or approaching each of the problems. It also gives just the right level of hints. This way you can learn both by working through the problem and by seeing the different perspectives. For example, with a single problem there may be a long brute-force solution, a quick but hard to discover solution, and a quick solution based on advanced math (you can use most things that come up in an undergrad math curriculum, even elliptic curves).

The Art and Craft of Problem Solving is a great read for general strategies and practice, and will remain relevant throughout any later work.

Mathematical Olympiad Challenges by Andreescu and Gelca shows off a few major problem solving styles and has a great selection of problems. I studied it in high school and it ended up being very important for me getting Putnam Fellow.

Earlier I had also studied Problem-Solving Strategies but that may be too big and not as focused on Putnam type of problems

: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

Similarly, McQuarrie Physical Chemistry may be helpful.

At my school, pchem was divided into a first semester which covered the quantum chemistry of individual atoms/molecules, and a second semester which used some of these quantum ideas (but mostly statistics and thermo) to talk about the statistical mechanics of collections of particles. I believe that McQuarrie's Physical Chemistry covers both, but note that the "mathematical review" sections are just brief interludes. For a more thorough treatment of math methods for physical scientists, consider the Mary Boas book. This book mostly focuses on physics applications, but from my experience in pchem, I would argue that it's just a very "applied" or "specific" version of quantum (or thermal, E&M, etc.) physics.

Also, for quantum chem, it is of utmost importance to be familiar with matrices, vectors, and ideally some of the more fancy portions of a first course in linear algebra, like bases and diagonalization. Although the relative importance of calculus/DE vs. linear algebra might depend on whether your course follows a "Schrodinger" vs. "Heisenberg" (not the Walter White one) approach, respectively.

I don't claim to know Category Theory, but I came across it when doing exercises in the beginning part of Chapter 0 by Aluffi. It was very terse, but still understandable. The video seems to be much more relaxed in comparison. It is even more relaxed than Awodey's book which is a much better intro to CT than Aluffi's Chapter 0. In short, it reminds me of Conceptual Mathematics: A First Introduction to Categories by Lawvere/Schnauel a little.

Sure, there are lots of cool websites that don't ask for crazy prerequisites. One which I share with all of my friends who are starting out in math is the Fun Facts site, hosted by Harvey Mudd College.

As far as learning specific materials, you can try Khan Academy for what are perhaps some of the more elementary topics (it goes up to differential equations and linear algebra). If you want to learn more about number systems and algebra I think that either picking up a good, cheap book on number theory, or even checking out the University of Reddit's Group Theory course (presented by Math Doctor Bob) are both very strong options. Otherwise, you can check out YouTube for other lecture series that people are more and more frequently putting up.

I know nearly nothing about this topic, but I find it extremely interesting and I've done some searching.

Two more lectures:

• Dimensionality Reduction by Neil D. Lawrence.
  
  
• Neighbourhood Components Analysis by Sam Roweis.
  
  Supposedly a nice overview:
  
  
• Dimension Reduction: A Guided Tour by Christopher J.C. Burges.
  
  Very readable papers:
  
  
• A global geometric framework for nonlinear dimensionality reduction by J. B. Tenenbaum, V. De Silva, J. C. Langford, about Isomap.
  
  
• Nonlinear dimensionality reduction by locally linear embedding by S. T. Roweis and L. K. Saul, about LLE.
  
  A paper that I haven't tried reading yet:
  
  
• Visualizing Data with t-SNE by L.J.P van der Maaten and Geoffrey E. Hinton, about t-SNE.
  
  A book chapter:
  
  
• Pattern Recognition and Machine Learning by Christopher M. Bishop, chapter 12 (Continuous Latent Variables).

Bayes is the way to go: Ed Jayne's text Probability Theory is fundamental and a great read. Free chapter samples are here. Slightly off topic, David Mackay's free text is also wonderfully engaging.

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.

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.

The Man Who Loved Only Numbers is a beautiful telling of Paul Erdos' life.

Someone else mentioned The Man Who Knew Infinity which I also love.

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)

The exercises in Spivak’s Calculus (amzn) are the best part of the book.

• • 
    
    /u/WelpMathFanatic You’ll probably have a better (more efficient, more enjoyable) time if you take a course, or otherwise find someone to help you face to face. But if you’re studying by yourself you might want to look at a book about writing proofs, such as Velleman’s [How to Prove It](https://amzn.com/0521675995) or Hammack’s [Book of Proof*](https://amzn.com/0989472108). (Disclaimer: I haven’t read either of these.)
    
    

Book of proof is a more gentle introduction to proofs then How to Prove it.

​

No bullshit guide to linear algebra is a gentle introduction to linear algebra when compared to the popular Linear Algebra Done Right.

​

An Illustrated Theory of Numbers is a fantastic introduction book to number theory in a similar style to the popular Visual Complex Analysis.

At the moment, psychology is largely ad-hoc, and not a modicum of progress would've been made without the extensive utilization of statistical methods. To consider the human condition does not require us to simply extrapolate from our severely limited experiences, if not from the biases of limited datasets, datasets for which we can't even be certain of their various e.g. parameters etc..

For example, depending on the culture, the set of phenotypical traits with which increases the sexual attraction of an organism may be different - to state this is meaningless and ad-hoc, and we can only attempt to consider the validity of what was stated with statistical methods. Still, there comes along social scientists who would proclaim arbitrary sets of phenotypical features as being universal for all humans in all conditions simply because they were convinced by limited and biased datasets (e.g. making extreme generalizations based on the United States' demographic while ignoring the entire world etc.).

In fact, the author(s) of "Probability Theory: The Logic of Science" will let you know what they think of the shaky sciences of the 20th and 21st century, social science and econometrics included, the shaky sciences for which their only justifications are statistical methods.
_

With increasing mathematical depth and the increasing quality of applied mathematicians into such fields of science, we will begin to gradually see a significant improvement in the validity of said respective fields. Otherwise, currently, psychology, for example, holds no depth, but the field itself is very entertaining to me; doesn't stop me from enjoying Michael's "Mind Field" series.

For mathematicians, physics itself lacks rigour, let alone psychology.
_

Note, the founder of "psychoanalysis", Sigmund Freud, is essentially a pseudo-scientist. Like many social scientists, he made the major error of extreme extrapolation based on his very limited and personal life experiences, and that of extremely limited, biased datasets. Sigmund Freud "proclaimed" a lot of truths about the human condition, for example, Sigmund Fraud is the genius responsible for the notion of "Penis Envy".

In the same century, Einstein would change the face of physics forever after having published the four papers in his miracle year before producing the masterpiece of General Relativity. And, in that same century, incredible progress such that of Gödel's Incompleteness Theorem, Quantum Electrodynamics, the discovery of various biological reaction pathways (e.g. citric acid cycle etc.), and so on and so on would be produced while Sigmund Fraud can be proud of his Penis Envy hypothesis.

Khan Academy and Professor Leonard on YouTube will cover up to Calculus 3. From there you can use this Mathematical Methods book to cover the rest of what you would need for an undergraduate physics major. Then you can start learning the physics.

For a brief overview of the scope of math and physics, look at these two videos.

I want to emphasize that learning the math and the physics up to and especially including the theory of relativity is very difficult and time consuming. General Relativity itself is quite beyond undergraduate level physics.

I suggest if you are curious about topics like relativity that you check out Paul Sutter's Ask a Spaceman! podcast. He breaks down what the math says and explains complex subjects in a way that is easy to understand.

I also recommend watching Richard A. Muller's physics for presidents course, which is another great resource for learning about physics without the math getting in the way of understanding the concepts.

This is good, with respect to learning tips and tricks for competitions, I think you're best off getting a book.

http://www.amazon.co.uk/Art-Craft-Problem-Solving/dp/0471789011

Is good

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.

In a math course I recently took that was basically an introduction to math proofs we used Mathematical Proofs: A Transition to Advanced Mathematics which I found to be a great text. It begins by going through the language and syntax used in proofs and slowly progresses through theory, different types of proofs, and eventually proofs from advanced calculus. There are so many examples that are very well laid out and explained. I would highly recommend it for learning proofs from scratch.

I'll be that guy. There are two types of Calculus: the Micky Mouse calculus and Real Analysis. If you go to Khan Academy you're gonna study the first version. It's by far the most popular one and has nothing to do with higher math.

The foundations of higher math are Linear Algebra(again, different from what's on Khan Academy), Abstract Algebra, Real Analysis etc.

You could, probably, skip all the micky mouse classes and start immediately with rigorous(proof-based) Linear Algebra.

But it's probably best to get a good foundation before embarking on Real Analysis and the like:

Discrete Mathematics with Applications by Susanna Epp

How to Prove It: A Structured Approach Daniel Velleman

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Book of Proof by Richard Hammock

That way you get to skip all the plug-and-chug courses and start from the very beginning in a rigorous way.

1. The man who loved only numbers (just generally a good read):
   
   http://www.amazon.com/MAN-WHO-LOVED-ONLY-NUMBERS/dp/0786884061
   
   
2. Four colors suffice (really good if you like graph theory):
   
   http://www.amazon.com/Four-Colors-Suffice-Problem-Solved/dp/0691115338
   
   

There are a ton here just to get you started. Honestly, googling "proofs without words" or "visual proofs" should get you a lot that you'll like :)

If you're hungry for more, there are entire books that collect them(2nd and 3rd volumes, too)

:)

Calculus Made Easy by Silvanus P. Thompson. I read this book 10 years after college and I felt like I finally really understood what calculus is all about. Kind of rekindled my interest in math.

You're very welcome - if you haven't had enough Erdos then I strongly recommend The Man Who Loved Only Numbers. It was actually thinking about this book (which I read a few years ago now) that prompted me to search for and watch this documentary yesterday.

Hi, here I will post some great books, some free (by Santos), some not (others).

Junior problem seminar: Santos

Number Theory for Mathematical contests: Santos

The Art and Craft of Problem solving: Zeitz

Problem-Solving Strategies: Engel

Mathematical Olympiad Treasures: Andreescu, Enescu

Mathematical Olympiad challenges: Andreescu, Gelca

Problems from the book: Andreescu

Those are more or less the "general" books, they always contain the main topics of mathematical olympiads, they usually aren't focused on just one topic, for one-topic books see here: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=319&t=405377

Division by zero is just one particular topic. You may as well ask for a book about cross-multiplication.

There is a book about zero. I really enjoyed this book, but I haven't read it in a long time, so I don't remember much about it. But I'm pretty sure it discusses division by 0.

it most certainly is! There's a whole approach to statistics based around this idea of updating priors. If you're feeling ambitious, the book Probability theory by Jaynes is pretty accessible.

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.

These are my personal favourites for introductory books on ODEs - [Simmons & Krantz's Differential equations: theory, techniques and practice](https://www.amazon.com/Differential-Equations-Steven-Krantz-Simmons/dp/0070616094) is a great book with examples from physics and engineering along with lots of historic notes.

[Braun's differential equations and their applications](https://www.amazon.com/Differential-Equations-Their-Applications-Introduction/dp/0387978941/ref=sr\_1\_1?crid=35EOUTZZ32HDA&keywords=braun+differential+equations&qid=1556968795&s=books&sprefix=braun+Differenz%2Cstripbooks-intl-ship%2C215&sr=1-1) is another applications oriented differential equations book that is a bit more involved than Simmon's but has a much broader perspective with introductions to bifurcation theory and applications in mathematical biology.

​

If you're not planning to do research in ODE theory, but want to learn the basic theory more rigorously, then [Hurewicz's Lectures](https://www.amazon.com/Lectures-Ordinary-Differential-Equations-Hurewicz/dp/1258814889/ref=sr\_1\_1?crid=341Z3D48AUTBU&keywords=hurewicz+differential&qid=1556969136&s=gateway&sprefix=hurewicz+%2Cstripbooks-intl-ship%2C216&sr=8-1) is a perfect short book that covers the basic theorems for existence and uniqueness of solutions of ODEs.

Number theory is pretty cool. I enjoyed Dudley's book for a number of reasons.

First, to get a sense as to the world of math and what it encompasses, and what different sub-subjects are about, watch this: https://www.youtube.com/watch?v=OmJ-4B-mS-Y

Ok, now that's out of the way -- I'd recommend doing some grunt work, and have a basic working knowledge of algebra + calculus. My wife found this book useful to do just that after having been out of university for a while: https://www.amazon.com/No-bullshit-guide-math-physics/dp/0992001005

At this point, you can tackle most subjects brought up from first video without issue -- just find a good introductory book! One that I recommend that is more on computer science end of things is a discrete math
book.

https://www.amazon.com/Concrete-Mathematics-Foundation-Computer-Science/dp/0201558025

And understanding proofs is important: https://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472108

I would highly recommend spending some time learning number theory first. Much of crypto relies on understanding a fair amount of number theory in order to understand what and why stuff works.

The book antiantiall linked is fantastic (I have a copy), however if you don't have a strong foundation in number theory will likely be a bit over your head.

Here is the textbook that was used in my number theory course. It isn't necessarily the best out there, but is cheap and does a good job covering the basics.

Hey I'm a physics BSc turned mathematician.

I would suggest starting with topology and functional analysis. Functional analysis is the foundation of quantum mechanics, and topology is necessary to properly understand manifolds, which are the foundation of relativity.

I would suggest Kreyszig for functional analysis. It's probably the most gentle functional analysis book out there.

For topology, I would suggest John Lee. This topology text is unique because it teaches general topology with a view towards manifolds. This makes it ideal for a physicist. If you want to know about Lie algebras and Lie groups, the sequel to this text discusses them.

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.

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.

Your professors really aren't expecting you to reinvent groundbreaking proofs from scratch, given some basic axioms. It's much more likely that you're missing "hints" - exercises often build off previous proofs done in class, for example.

I appreciated Laura Alcock's writings on this, in helping me overcome my fear of studying math in general:
https://www.amazon.com/How-Study-as-Mathematics-Major/dp/0199661316/

https://www.amazon.com/dp/0198723539/ <-- even though you aren't in analysis, the way she writes about approaching math classes in general is helpful

If you really do struggle with the mechanics of proof, you should take some time to harden that skill on its own. I found this to be filled with helpful and gentle exercises, with answers: https://www.amazon.com/dp/0989472108/ref=rdr_ext_sb_ti_sims_2

And one more idea is that it can't hurt for you to supplement what you're learning in class with a more intuitive, chatty text. This book is filled with colorful examples that may help your leap into more abstract territory: https://www.amazon.com/Visual-Group-Theory-Problem-Book/dp/088385757X

Kreyszig is the best first book on functional analysis IMO. For measure theory I liked Royden, specifically the 3rd edition.

The Man Who Loved Numbers, biography of Paul Erdos, one of the most prolific and bizarre mathematicians of the 20th century. It is pretty light on the actual math but is a very entertaining read regardless. Also he was from Poland, and the book has quite a few stories about being a Mathematician in Eastern Europe.

Here's how I was taught, but I was taught in physics not math.
Fourier transforms are more intuitive, so think about how you take a derivative of a FT. You carry the derivative operator into the integral and you just get a factor of 2(pi)ix under the integrand. Logically, if you want a second derivative, just take the FT of the functions transform times x^2 etc. If you want a 1.3^th derivative (yes fractional derivatives exist) then FT the function times x^1.3 etc. This means taking a n^th derivative in real space is the same as multiplying by x^n in transform space. Sounds alot like what logarithms did for multiplication back in the day doesn't it? So now you can turn a differential equation into a polynomial equation if you just take the Fourier transform of it. However, if the diff eq is more complex than just n^th order with constant coefficients, maybe the FT isn't the best transform available for simplifying it? Then use a transform that's tailored for the particular function you have.

If I remember correctly this book has a nice description. I consider this book to be the "readable" version of this one

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.

There are a couple of easy-ish sources on category theory that are good to have under your belt.

Category Theory for Programmers is available for free: https://github.com/hmemcpy/milewski-ctfp-pdf
It's not amazing, but it's good for programmers who want to start having basic intuitions about category theory.

Lawvere's Conceptual Mathematics is enjoyable and accessible
https://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X/ref=mp_s_a_1_1?keywords=conceptual+mathematics&qid=1568389352&s=gateway&sr=8-1

To answer your general question: in my experience, your question is less about math and maybe more about chasing something you think has the answers. You'll meander as long as you feel like something is lacking.

I've seen this a lot with people who have massive textbook collections. A massive collection of textbooks is debt, and it provokes anxiety. You may have to figure out some squishy human stuff in addition to the technical math stuff.

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

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!

One of my favorites on the history of zero by Charles Seife. Short, interesting, and well-explained. Has some challenging math concepts later in the book.

http://www.amazon.com/Zero-The-Biography-Dangerous-Idea/dp/0140296476/ref=sr_1_1?ie=UTF8&qid=1346799685&sr=8-1&keywords=zero+the+biography+of+a+dangerous+idea

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

Hmm I'm surprised you've had point-set topology, linear algebra, and basic functional analysis but have yet to encounter locally convex topological vector spaces! No worries, you have most likely developed all oft the machinery to understand them. I agree with G-Brain, Rudin's function analysis will do. Most functional analysis books should cover this at some point. The only I use is Kreyszig. Hope that helps!

Since you like the visual proofs, I was wondering if you had seen the book Proofs Without Words. It is really cool. Although I probably need someone like you to talk me through some of the more complicated ones, because the author (editor?) has taken the 'without words' part very seriously.

*formatting link

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

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".

Still the best math book cover ever

This is the best one I have used.

I love this book, personally.

Concepts of Modern Mathematics by Ian Stewart is an excellent book about modern math. As is Foundations and Fundamental Concepts of Mathematics by Howard Eves I would recommend these two along with the far more expensive Naive Set Theory by Halmos

Here's a later edition, reworked by the late, great Martin Gardner: https://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/dp/0312185480/ref=as_li_ss_tl?ie=UTF8&qid=1492709553&sr=8-1&keywords=calculus+made+easy&linkCode=sl1&tag=pausnisblo-20&linkId=1818a78067c98161b0aea3561b88ac0b

The Book of Proof was such a great book, I bought a copy that I regularly refer back to. It's full of worked examples, exercises, and explanations. This should be on the bookshelf of every undergrad.

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.

Here it is on Amazon: https://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/dp/0312185480/ref=sr_1_1?s=books&ie=UTF8&qid=1492741701&sr=1-1&keywords=calculus+made+easy

It is noted in the reviews that the hard back copy is superior

IMHO if you don't understand AC and its equivalents, then Jech is not the book you should be reading. That book is pretty heavy and is used (as far as I know - I have a handful of friends who work in set theory) as a research reference. Maybe read Naive Set Theory first. Despite its name, it's reasonably advanced but way more readable.

Don't think of your abilities as fixed. The number of proofs you encounter grows from where you are now. You did not know algebra when you started. You will be increasingly exposed to proofs as you go along. Spend time on them. I recommend you get a tutor, or at least read some extra material. https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321390539/ref=sr_1_69?ie=UTF8&qid=1494805054&sr=8-69&keywords=proofs+math

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.

If you enjoyed this I highly recommend Paul Hoffman's biography of Erdos

https://www.amazon.com/Man-Who-Loved-Only-Numbers/dp/0786884061/ref=sr_1_1?ie=UTF8&qid=1484066739&sr=8-1&keywords=the+man+who+loved+only+numbers

Well the first book details much of what you would need to know as a first year grad student in math and has recommendations for other books as well.

This book might also be something to try or perhaps Knuth's Concrete Math.

You could also try what this guy did, only with the MIT math curriculum on MIT OCW and without the time constraints.

You may also enjoy Probability Theory: The Logic of Science by E.T. Jaynes, and Information Theory, Inference and Learning Algorithms by David McKay.

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

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

Naive Set Theory if you want a more textbook approach, the book mentioned in my other response if you're looking for something more like a story with proofs.

If you're coming from a more applied background (or physics / engineering) https://www.amazon.com/Introductory-Functional-Analysis-Applications-Kreyszig/dp/0471504599 is pretty easy to follow. Obviously it goes into the infinite dim too but it covers all the finite stuff first.

> then maybe something titled along the lines of “Math Methods for Physics”.

Boas is good.

There is this online Category Theory book (PDF). Also, the book Conceptual Mathematics has been well recommended as an introduction to CT starting from the basics.

So, you are gonna be an engineer/scientist, rather than a pure math major which, probably, means techniques will take precedence over ideas and rigor. To that end, you might like:

Engineering Mathematics

Advanced Engineering Mathematics

Numerical Methods for Scientists and Engineers

Mathematical Methods in the Physical Sciences

Basically, you need to put yourself through technical boot-camp that involves Calculus, Applied Linear Algebra, some Stats, Diff. Equations.

I hate to disappoint you OP, but here are some books just in my wish list on Amazon that outdo that:

This has 14, 5 star reviews.

This has 20, 5 star reviews, and 1, 4 star review.

I'm in engineering, and back when I took DE I mainly used Braun for applications I think. I remember liking it quite a bit. http://www.amazon.com/Differential-Equations-Their-Applications-Introduction/dp/0387978941/

Edit: also IIRC, Churchill (Complex Variables and Applications) had a section about applications of DE in Laplace transforms.

Doing math is all about problem solving and this is a really terrific book for building your intuition and confidence in problem solving.

I like Underwood Dudley's book, and it's now available cheap from Dover: https://www.amazon.com/Elementary-Number-Theory-Second-Mathematics/dp/048646931X/ref=sr_1_1?s=books&ie=UTF8&qid=1468275037&sr=1-1&keywords=dudley+number+theory

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

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

1. Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
   
   
2. [A Book of Abstract Algebra by Charles Pinter](
   http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178/ref=sr_1_1?ie=UTF8&qid=1394386195&sr=8-1&keywords=pinter+abstract+algebra)
   
   
3. Elementary Real Analysis by by Brian S. Thomson, Judith B. Bruckner, Andrew M. Bruckner
   
   
4. Topology Without Tears y Sidney Morris
   
   
5. Conceptual Math: Categories by F. William Lawvere, Stephen H. Schanuel

I haven't read it myself, but I have heard Naive Set Theory recommended here several times before.

• Proofs without Words
  
• Proofs without Words II
  
• Math Made Visual: Creating Images for Understanding Mathematics
  

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...

If you can find this at your library, I suggest you pour over it in the weekend. You will not regret it.

Others have already answered this question, but I thought I might direct you toward a book on the the subject that I enjoyed reading: Zero: The Biography of a Dangerous Idea

Kepler's Conjecture: How Some of the Greatest Minds in History Helped Solve One of the Oldest Math Problems in the World and Zero: The Biography of a Dangerous Idea are riveting histories.

Having a proof explained to you isn't even close to the thrill of proving something yourself, IMO. My advice would be to get your hands on an intro to proofs text and work through some of it. If you don't like writing proofs or think it's boring, your time at university is probably going to bore you to tears.

If you want an intro proofs book, you might start here. The text is very clearly written and chapters 9-13 will give you a very basic notion of what ideas will be at the core of some of your upper-level math classes (abstract algebra, real analysis, etc).

It's difficult to make recommendations without being certain of what you actually know and what you imagine mathematics to be like. A lot of university-level mathematics is technical and requires familiarity of high-level concepts. This is in contrast to softer popular mathematics, which is more related to solving problems and contest questions. One of the things I've noted about pre-university students passionate about mathematics is that they assume that the subject is only about problem-solving and fail to take into mind the level of technical knowledge that must be learnt and memorized to be a mathematician.

If you're simply looking for problems to solve, try The Art and Craft of Problem Solving by Zeitz or Problem Solving Strategies by Engel. Generally any book geared to the Olympiad or regional competitions will be alright. Here, you're not looking for a specific body of knowledge, but rather an approach to thinking and persevering when handling tough problems.

But if you're looking to learn more about 'technical' mathematics, you'll need to know the basics of numbers and sets. Numbers & Proofs by Allenby is a good introduction, using an approach that gets you to actively solve problems. Once you get past that, then you can try your hand on analysis or group theory or linear algebra or even basic graph theory. But keep in mind that with 'technical' mathematics, all knowledge is built on understanding of previous fields, so don't rush through it or you'll get discouraged by any difficulty or unfamiliarity you'll encounter.

Calculus Made Easy by Silvanus Thompson and Martin Gardner

http://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/dp/0312185480/ref=pd_sim_b_2/176-3762013-5071850

UK Amazon

US Amazon

Halmos's Naive Set Theory is a classic.

http://www.amazon.com/Naive-Theory-Undergraduate-Texts-Mathematics/dp/0387900926/ref=sr_1_1?ie=UTF8&qid=1320969231&sr=8-1

This is the math bible for undergrad physics.

https://www.amazon.ca/Mathematical-Methods-Physical-Sciences-Mary/dp/0471198269

You could look thru the math for physicists compendiums by Arfken, Webber, Harris and Boas and many others, google around

http://webusers.physics.illinois.edu/~m-stone5/mma/notes/amaster.pdf

http://math.ucr.edu/home/baez/books.html

http://arxiv.org/abs/1209.5665

--------


http://www.amazon.com/Mathematical-Methods-Physical-Sciences-Mary/dp/0471198269/

http://www.amazon.com/Mathematical-Physics-Modern-Introduction-Foundations/dp/0387985794


also http://www.physics.miami.edu/~nearing/mathmethods/

http://www.amazon.com/Art-Craft-Problem-Solving/dp/0471789011

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

http://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753 click to look inside.

learn Haskell, learn how to make money, or learn some probability theory

Probability Theory: The Logic of Science

Jaynes's Probability Theory: The Logic of Science

https://www.amazon.com/Man-Who-Loved-Only-Numbers/dp/0786884061

From Proofs Without Words --
a trisection device
I don't know GSP well enough to know if it's up to these geometric constraints.

I don't have a great book yet, but the book that got me back into mathematics was actually a biography of Erdős.

Zero: The Biography of a Dangerous Idea

​

and

​

The Golden Ratio

​

are two of my favorites

I'm not so sure this is a fundamental difference, so much as a distinction in who is looking at each field. For the most part, category theory is studied by those who are looking to make advances in knowledge. Sure, the things researchers are looking at can be complex. But if you look at current research in abstract algebra, it's equally difficult to get up to speed and comprehend. The reason abstract algebra can be seen as simpler is that there is also introductory material, aimed at undergraduates, and even the general population.

Is it fundamentally impossible to produce such introductory material in category theory? Of course not! Several people have made serious and credible attempts. For example, here and here

Charles Seife's Zero: The Biography of a Dangerous Idea

You may want to study regression as a machine learning problem. I don't know your background, but if you are a mathematician, this approach probably isn't the best for you.

If you are only thinking of fitting polynomials, you only need a little trick to adapt linear regression.

I've always wanted to recommend this book to someone who knows no math. I find the writing infuriating. It is a dialog but this approach to dialog totally sucks. On the other hand, this is a stunning introduction to categorical logic. It will not help you solve problems etc. but I can guarantee that this book will change your entire outlook on the world.

http://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X/ref=sr_1_1?ie=UTF8&qid=1320560710&sr=8-1

I second the recommendation to find someone more experienced to help you one-on-one. Is there any way you could hire a private tutor? A big benefit of a tutor is that they'll be able to point out the gaps in your knowledge and point you to relevant resources. This can be tough to do on your own or through web discussions. For example, let's say one thing that's holding you back is that you haven't memorized your times table. This would be a major problem and a blind spot for you that would be immediately obvious to me if we were working face to face, but it would be impossible to see from reading your reddit comments.

Let me make a few more concrete suggestions. First, experiment with different study techniques. Take a look at this comment and the linked video. Try the "Feynman Technique" (video) -- this is not easy but it's the only way to really get a solid understanding. Don't expect to be spoon-fed knowledge when you're watching videos: you need to be spending most of your study time with a pen and paper, puzzling out for yourself why things work.

Second, for algebra, I can recommend two textbooks:

• Rusczyk's Introduction to Algebra. Probably right around your level. Lots of interesting problems that will make you think.
  
• Gelfand and Shen's Algebra. It has excellent problems, although it is quite terse and probably a little too advanced for you: you'll need to be willing to do a lot of extra thinking to fill in the gaps.
  
  Khan Academy is a good supplement, but in my opinion it's too passive to be used as your main resource. It doesn't encourage independent thinking and it has no problems (easy drill exercises don't count as problems.) You need to do lots of problems. In particular, you need to struggle through problems that you're not explicitly told how to solve ahead of time.
  
  Finally, mechanical knowledge is incredibly important, but of course it does need to be built upon a conceptual foundation. For every technique you learn (like solving 2/3 = 3R) you should first be able to explain why the technique works in simple, obvious terms, and then practice it (invent your own problems!) and add it to your collection of techniques. Math is (arguably) simply a grab bag of such techniques together with explanations of why they work. It's often not obvious which technique to apply in a specific case: this can only be learned through experience. Avoid problem sets with ten variants of one specific problem -- they don't teach this skill! Instead look for varied problems which require creativity (Rusczyk's book is a good start.)
  
  You might also want to check out /r/learnmath and #math on freenode if you have more specific questions.

Zero had a long road before it was accepted as a rational concept. See the book Zero: The Biography of a Dangerous Idea. It's not preposterous to assume someone was having troubles with 0 conceptually.

Calculus Made Easy -- Can't get much better as far as bang for the buck. Follow it up something more rigorous. Maybe, Calculus, Vol 1 by Apostol. The problem with Apostol, as most calculus texts, is price.

>If I can barely do long division and I’m horrible at math, how would I bring my skills up to be able to pass a statistic class.

Go to the library, find some books appropriate for your level. If you want me to pick a book for you, start with Algebra by Gelfrand. Khan Academy might also help.

>Is math just memorization

No, this is why you're bad at it. You need to memorize as little as possible.

>and practice or is it something that I need to read books about?

You should practice a lot and spend time reading. When reading,
go slowly, take notes, and work out details for yourself. Your primary goal needs to be understanding. Everything in math is supposed to make sense, you learn when it makes sense to you.

>I absolutely hate math but desperately need help in order to graduate.

You need to kill this mentality, if you continue to feel like this it's just not going to be possible to learn. It's abundantly clear from your post that you've approached math wrong your whole life. This isn't really your fault. Math is highly cumulative, and if you lose track at some point it'll be impossible to follow meaningfully after. Most teachers are bad, and the system doesn't incentivize learning math well. This is your chance to change all that. You can start from zero, and learn things right this time. If you approach it with an open mind and put in the necessary work (it won't be easy!), you'll come to see it's a beautiful subject.

> Is there any good book with problems/examples that I could work through in order to thoroughly prepare myself to be able to write proofs for a Real Analysis I course?

Besides Velleman's "How to Prove it," try Mathematical Proofs: A Transition to Advanced Mathematics or maybe How to Read and Do Proofs: An Introduction to Mathematical Thought Processes.

The book I used in my "Intro to Proofs" course was A Transition to Advanced Mathematics. It was pretty good, but the edition that I used had several mistakes in it. Also, it's waaaay too expensive.

Now for the unpleasantries —

Suggestions aside, the main problem here is your "thoroughly prepare myself to be able to write proofs for Real Analysis" goal. Working through a proofs book on your own will be seriously challenging, but the thought of taking Real Analysis without at least two other proofs courses under your belt is terrifying to me. I had to take "An intro to mathematical proofs" followed almost immediately by a proof-based Linear Algebra course before I was even allowed to contemplate a Real Analysis course.

Come to think of it, how in the hell are you even allowed to do this if you haven't taken a proofs course before? Are you sure this is even possible? Are prerequisites not enforced at your school? No one, and I mean no one was permitted to take Abstract Algebra or Real Analysis without the required prerequisites at my university. The only way you could get around it was by being the next Andrew Wiles.

Just to drive all this home, I was a straight-A Physics/Math major with the exception of two courses: Thermodynamics and my first proofs course. I've never worked so damn hard for a B in my life. Come to think of it, I actually recall quantum mechanics being easier than my proofs course.

I'm being sincere when I ask you to reconsider this plan. You are asking for a world of pain followed by the very real possibility of failure if you do this.

TL;DR: Unless you are remarkably sharp and have loads of time on your hands, this is probably a mistake. You should take a more elementary proofs course before tackling Real Analysis. Good luck, whatever you choose to do.