Math at its core is about establishing truths separate from sensual qualities, seeking patterns based upon these truths, systematically removing contradictions/inconsistencies from the patterns, and formulating conjectures with all of the above in mind. It is the one true language apart from reality which makes it ironic that it is so useful.
Here we wish to provide resources which will help you to develop mathematical skills for where ever you choose to apply them.
Precalculus
Note that any autodidactic education requires a minimum amount of fundamentals, and to grasp the higher levels of math you absolutely need to understand the basic concepts known as precalculus, which is generally the math you will see up to high school. If you lack any of these fundamentals, you should refresh your knowledge at pages like Khan Academy or PatrickJMT. If you think you are fit, you can also directly start with calculus, although I would advise to skim Stewart's Precalculus before you do so.
If you are a total beginner, it is recommended to do the interactive exercises on Khan Academy, because they are a really helpful tool to quickly refresh your school knowledge up until calculus. You should do all the chapters up to Precalculus, that is: Early Math, Arithmetic, Pre-Algebra, Basic Geometry, Algebra I, Geometry, Algebra II, Trigonometry, Probability and Statistics. You don't need to listen to every video, but you should cover each exercise once to check if you understand it. Once you finish the Precalculus module, you can continue with your first book.
If you are still fit regarding math, you should at least do the Precalculus module on Khan Academy to be sure you have grasped everything necessary. For a general overview on the topics to come, choose any book on Precalculus, though I recommend the following:
Stewart's Precalculus
You can finish this book in a few weeks. It is already structured in a way that you can do 1-2 chapters per day for 6 days and do a review day on the 7th. You will be familiar with most concepts in this book, but especially if you just come out of High School or have just finished Khan Academy from zero, it will be a good exercise for you.
Problem books
These elementary problem books are meant for additional non-routine practice, challenges & puzzles, killing time, preparing for school competitions and exams, or to steal interesting questions from when you're teaching or tutoring students >:3
- Challenging Problems in Algebra by Posamentier
- Challenging Problems in Geometry by Posamentier
- The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics by Shklarsky, Chentzov, and Yaglom
- 103 Trigonometry Problems: From the Training of the USA IMO Team by Andreescu and Feng
- 104 Number Theory Problems: From the Training of the USA IMO Team by Andreescu
- 105 Algebra Problems from the AwesomeMath Summer Program by Andreescu
- 106 Geometry Problems from the AwesomeMath Summer Program by Andreescu
- 107 Geometry Problems from the Awesomemath Year-Round Program by Andreescu
- 108 Algebra Problems from the Awesomemath Year-Round Program by Andreescu
- "Problems From the Book" and "Straight From the Book" by Andreescu
- The Stanford Mathematics Problem Book by Polya and Kilpatrick
- Sequences, Combinations, Limits by Gelfand, Gerver, Kirillov, Konstantinov, and Kushnirenko
- Challenging Mathematical Problems With Elementary Solutions by A. M. Yaglom and I. M. Yaglom
- Hungarian Problem Book I-IV containing the Eötvös Mathematical Competitions from 1894–1963
The following have more advanced problems at the university level up to preparing for qualifying exams during graduate school
- The Green Book of Mathematical Problems by Hardy and Williams
- The Red Book of Mathematical Problems by Williams and Hardy
- William Lowell Putnam Mathematical Competition: Problems & Solutions: 1938-1964
- The William Lowell Putnam Mathematical Competition: Problems and Solutions 1965–1984
- The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions and Commentary
- Contests in Higher Mathematics: Miklos Schweitzer Competitions, 1962-1991 by Szekely
- Problems in Mathematical Analysis I: Real Numbers, Sequences and Series by Kaczor and Nowak
- Problems in Mathematical Analysis II: Continuity and Differentiation by Kaczor and Nowak
- Problems in Mathematical Analysis III Integration by Kaczor and Nowak
- A Collection of Problems on Complex Analysis by Volkovyskii, Lunts, and Aramanovich
- Problems in Group Theory by Dixon
- Berkeley Problems in Mathematics by Paulo Ney de Souza and Jorge-Nuno Silva
- Problems and Solutions in Mathematics (Major American Universities PH.D. Qualifying Questions and Solutions)
Problem Solving and Heuristics
Some strategies on how to approach difficult problems to solve them exactly or heuristically and dealing with Fermi problems :
- How to Solve It: A New Aspect of Mathematical Method by Polya
- How to Solve It: Modern Heuristics by Michalewicz and Fogel
- Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin by Weinstein and Adam
- Guesstimation 2.0: Solving Today's Problems on the Back of a Napkin by Weinstein and Edwards
- Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving by Sanjoy Mahajan
- Problem-Solving Through Problems by Larson
- Putnam and Beyond by Gelca and Andreescu
Calculus
Calculus is the study of change (derivatives) and accumulation (integrals) of functions. These topics are linked by the Fundamental Theorem of Calculus which states informally that "the accumulation of the changes of a function" and "the change of the accumulation of a function" result in the original function. The rest of calculus is just spamming these idea on various applications and situations.
Some students struggle with calculus but honestly it is a really straight forward subject, especially compared to other advanced subjects in math. As long as you pay attention and go in trying to learn, you should quickly end up joining the rest of the math students in calling it 'trivial' in retrospect. Don't set yourself up to failure by thinking you're not capable of learning it because it will all click once you look at it right.
Single Variable Calculus
The standards texts (Stewart, Rogawski, et al.) you see required for college classes are, in all honesty, quite terrible since they are not written with self-study in mind but just as a collection of exercises and a review of the basic methods. Do use them to practice calculus by not as a means to learn it.
For a well done intuitive approach using infinitesimals, which is the way everyone ends up thinking about calculus which is also technically nonstandard but by no means mathematically incorrect, "Elementary Calculus: An Infinitesimal Approach" by Jerome Keisler is a fantastic and free public domain book (also available in an inexpensive Dover paperback edition). If the infinitesimal approach intrigues you but you've already done a course in calculus or currently reading through another 1000 page book on calculus, Infinitesimal Calculus (Dover Book) by Henle and Kleinberg is a nice short 144 page book that develops the theory of infinitesimals in calculus in an accessible and clear manner. As you can probably infer from the page count, Henle's book doesn't have any material on the applications of calculus so don't use it as a standalone book to learn all of calculus from but as a supplement to see a different approach in understanding the subject the way it was originally invented.
For a rigorous standard (δ-ε) approach to the subject, your options are "Calculus" by Spivak or the classic "Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra" by Apostol. Spivak's writing certainly has its fans but it sadly lacks much of the applications and motivation (related rates, optimization) that are standard in calculus making it hard to use on it's own. Apostol's Calculus doesn't have this oversight and it's probably the best one to learn the material from on your own. Another rigorous option that also has copious amounts of physics applications, motivation, and intuition presented at the same time is "Introduction to Calculus and Analysis, Volume I" by Richard Courant and Fritz John. The book is a modern rewrite of the classic "Differential and Integral Calculus" by Richard Courant and includes the most material of the three and it's exercise are the most difficult (perhaps a bit too difficult in places).
To just learn the methods of calculus, there are plenty of lectures on calculus for you to choose from on Youtube.
Multivariable and Vector Calculus
Again, the usual suspects you'll find assigned in college courses tend to make good exercise books but terrible introductions to the subject. You're options are the latter part of Keisler's book above, "Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability" by Apostol, or the two split parts of "Introduction to Calculus and Analysis, Vol. II/1 and Vol. II/2" by Richard Courant and Fritz John.
The following texts take a slightly more rigorous approach and go a bit deeper into the subject by covering differential forms and manifolds. Most single semester courses on vector calculus do not have time to reasonable cover this material, and consequently is usually skipped until later, but this advanced perspective can greatly aid in the understanding of the subject. You could study this material either when you first learn multivariable calculus or when you want a second pass on the subject, after just learning the basic methods, to improve you understanding while deepening your knowledge by generalizing what you've seen before. They can also be used as supplements to an advanced multivariable analysis course.
- C. H. Edwards Jr.'s Advanced Calculus of Several Variables (Dover Book)
- Hubbard and Hubbard's Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach
- Harold M. Edwards' Advanced Calculus: A Differential Forms Approach
- Sternberg and Loomis' Advanced Calculus (for the utterly fearless)
Curves and Surfaces in ℝ² and ℝ³
This subject is the study of Geometry using the tools that you learned in Vector Calculus and serves as a preparation to more abstract approaches to Differential Geometry you'll see in the future. Most schools only quickly pass through the subject during multivariable calculus but it will help in the long run if you study the material early on.
- Pressley's Elementary Differential Geometry
- do Carmo's Differential Geometry of Curves and Surfaces
Ordinary Differential Equations
The standard text used in college courses is "Elementary Differential Equations" by Boyce and DiPrima, which many people do seem to like (not me however). A cheap and very good alternative is "Ordinary Differential Equations" by Tenenbaum & Pollard (published by Dover) which is perfect for self study. Another great book is "Differential Equations with Applications and Historical Notes" by Simmons.
Linear Algebra
When speaking of Linear Algebra, people refer to one of 2 complementary but different subjects: Matrix Algebra/Computational Linear Algebra and Theoretical Linear Algebra/Finite Vector Space Theory. Oddly enough, you could study them in any order but canonically you're typically expected to learn some matrix algebra first and then transition to vector spaces and/or more applied/numerical topics second. The necessary prerequisite knowledge is just precalculus but some calculus knowledge is useful and may appear in a few examples.
Matrix Algebra
For a first exposure to the subject there really isn't that much to learn. You typically cover systems of equations, matrix operations, Gaussian elimination (also known as row reduction), LU decomposition, determinants, eigenvectors and eigenvalues, and diagonalization possibly with a few additional fluff subjects to round out a whole course. Many times you can pick up this material while studying calculus or ODEs (like with Apostol or Hubbard2's book) so you can just skip to more advanced material. Also, the introductory material in first few chapters of advanced textbooks are often good enough to learn matrix algebra from if you're truly in a rush.
But while some students seem to inhale these topics and quickly move on, others will need to take their time before operating with matrices becomes natural to them. A gentle introduction for learners with weaker math skills can be found in "Matrices and Linear Algebra" by Schneider and Barker. Learners with slightly better math abilities can benefit more from "Matrices and Linear Transformations" by Cullen which is aimed at STEM students. A free book for students seeking a honors introduction to linear algebra (and basic proofs) is "Linear Algebra Done Wrong" by Treil (Don't worry, the title is a pun on Axler's "done right" book below). There's also a whole host of vulgarly over expensive textbooks used by college courses at this level (like Strang's Introduction to Linear Algebra, Lay's Linear Algebra and Its Applications, Friedberg's Elementary Linear Algebra, etc) but most of them aren't very good and even if they were, the 2 aforementioned books above are far cheaper thanks to them being published by Dover and the last one is free.
Applied Linear Algebra
For a first book in applied linear algebra, "Linear Algebra and Its Applications" by Strang is the standard text used but it is one of those love it or hate it texts. If you fall into the hate it camp, then Meyer's "Matrix Analysis and Applied Linear Algebra" is a good alternative. After reading one of them, you'll be more than ready to move onto advanced Numerical Linear Algebra and Matrix Analysis textbooks.
Finite Vector Spaces
To get started on the theoretical side of linear algebra you obviously should be familiar with the basics of proofs. Once you are, theory side has a lot of classic and well loved textbook to choose from:
- Linear Algebra by Shilov (Dover Book)
- Finite Dimensional Vector Spaces by Halmos
- Linear Algebra by Friedberg, Insel, and Spence
- Linear Algebra by Hoffman and Kunze
Of course there's also "Linear Algebra done Right" by Axler and on the one hand, the stuff he does is great... but on the other hand, he fucking HATES determinants and goes crazy avoiding them. Because of that you shouldn't use his book alone to learn from and you really should read Shilov alongside of it. But Axler certainly gives an unique development of the subject.
Refreshers and Advanced Books
Now if you want a challenge, start off with "Linear Algebra and Its Applications" by Lax. It is good for learning linear algebra for the first time if you're a hot shot freshman, using it as a second book on linear algebra, or as a 3rd refresher book for those who are entering graduate school. Another good 3rd book for deeper linear algebra study, and if you have the abstract algebra background for it, is Roman's "Advanced Linear Algebra".
Partial Differential Equations
Historically, the study of PDEs was a major impetus for the development of many results of analysis. Without this advanced math knowledge, the study of PDE is destined to be somewhat more trickier than what you've seen before in your studies. Be prepared to do some real work.
For a quick primer on PDEs, "Partial Differential Equations for Scientists and Engineers" by Farlow is pretty good albeit somewhat shallow. Fuller undergraduate treatments can be found with:
- Applied Partial Differential Equations: With Fourier Series and Boundary Value Problems by Haberman (The best applied text at this level)
- Partial Differential Equation: An Introduction by Strauss
- Partial Differential Equations by Fritz John
Graduate Partial Differential Equations
Now if you have the required background in analysis, you can really study the meat of PDEs in detail with the following:
- Partial Differential Equations by Jost (Strong bias for elliptic equations)
- Partial Differential Equations by Evans (The standard introduction text for graduate PDEs)
- Introduction to Partial Differential Equations by Folland (An more intermediate graduate level PDEs book than Evans)
- Partial Differential Equations I: Basic Theory; II: Qualitative Studies of Linear Equations; III: Nonlinear Equations by Taylor
Proofs and Advanced Mathematics
True mathematics involves proofs, lots and lots of proofs (cry more physicists). You really need to master the art of writing valid proofs that do not make careless unstated assumptions or unproven assertions because you view they are obvious, which in fact are often dead wrong or at the very least hold most of the meat of the proof in them. Another part is following along when reading proof in mathematical texts which requires diligently filling in all the skipped steps and checking which assumptions could be removed/weakened or what fails when they are removed/weakened. At their core, basic proofs are really easy and frequently just a matter of unwrapping the definition and following your nose, but getting into the right mindset for them might take the neophyte some practice in order to see it that way. Therefore you should work through a book or two on proofs before moving onto advanced mathematics and then blaming those books for being written badly because you lacked the prerequisite mathematical maturity from skipping this step. Some good books to learn proofs are:
- A Transition to Advanced Mathematics by Smith, Eggen, and Andre
- A Primer of Abstract Mathematics by Ash
- Conjecture and Proof by Laczkovich (An excellent supplement to either of the above books that shows a larger variety of proofs in mathematics)
- Proofs from THE BOOK by Aigner and Ziegler (Not a textbook on proofs but it is an excellent collection of well done and elegant proofs to appreciate and draw inspiration from)
After this, books on set theory and mathematical logic make excellent next steps in getting better at proofs and abstract mathematics in general before moving on to the much more difficult subjects like algebra and analysis. Combinatorics, graph theory, linear algebra involving vector spaces, and number theory textbooks would then be the next level to practice on and are fairly easy to read at this stage of mathematical maturity.
Since you will like be revising your proofs quite often, now would be an ideal time to finally learn LaTeX (pronounced "lay-tech") to typeset your proofs and future papers in.
Analysis
Mathematical Analysis' origins are found in the age old struggle of mathematicians to deal with the infinite and infinitesimal dating all the way back to Eudoxus of Cnidus and Archimedes of antiquity. With the piecemeal development of calculus by Cavalieri, Pascal, Fermat, Desecrate, Leibniz, Euler, Lagrange, Fourier and many others, calculus gradually showed itself to be a powerful yet deeply troubled tool. As much as mathematicians tried, they struggled with clearly defining key stumbling points: the concept of an infinitesimal number smaller than 1/n for all integers n yet nonzero in a logically consistent manner, the concept of infinite approach, division by 0, and the rules in which an infinite series may be manipulated and examined. These were not just pointless trifles that could be brushed off as more philosophy than math but of increasing practical importance. As time went on, many counterexamples (and not just pathological ones) where the naive application of the methods of calculus would produce erroneous results cast a shadow on the validity of all other results of calculus and many critics wanted to end its study altogether and relentlessly mocked the concept of infinitesimals as "the ghosts of departed quantities". Since the triumphs for calculus were both numerous and far reaching, mathematicians strongly sought to make the results of calculus proven rather than discarding the subject all together. This situation was finally resolved only in the early 19th century with the work of Cauchy and Weierstrass and the ε-δ definition of a limit (which would ironically kill off infinitesimals until the 1960s and the advent nonstandard analysis) that birthed the new field of analysis. This sparked off a massive revolution in mathematics and the field of analysis quickly exploded into various distinct but interconnected directions.
Students just finishing the study of calculus and basic proofs often fail to realize the sheer importance of careful work in analysis and scoff the whole subject off as merely "intellectual or autistic masturbation". This mentality comes from being coddled with the toy-problems you see in calculus that are selected to hide any possible nastiness that comes from complicated situations that frequently arise in science and engineering. Even if the student is aware of importance of being careful, they are often insulted when forced to work through "obvious" theorems. The problem here is that many results in analysis that seem obvious are frankly very difficult to prove (for example see the Jordan curve theorem) or even dead wrong. In order to gain the ability to prove important and powerful theorem hidden away in analysis, students need plenty of practice working through basic problems to gain familiarity and mathematical maturity to move on to difficult work even if this means you need to spend time proving that "every open ball is open".
A good reference to keep with you and refer to often is "Counterexamples in Analysis" by Gelbaum and Olmsted.
Inequalities
A lot of the exercises in analysis often boil down to spamming the triangle inequality until you get the result you want. If you haven't done much work with inequalities since grade school, practicing them can make the subject seem vastly easier.
- The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities by Steele
- Inequalities by Hardy, Littlewood, and Polya
Real Analysis (Metric Space based)
This is where the results of single variable calculus are finally made both rigorous and generalized.
The gold standard for the subject is the first 8 chapters of Rudin's "Principles of Mathematical Analysis" who's slick proofs and challenging exercises can't be beat <\shameless shilling>. Chapters 9 and 10 of Rudin on multivariable analysis are bit sparse to learn from so you're better off moving fuller treatment on analysis on manifolds (see below) to learn from. The final chapter 11 is completely skippable as other books are far better in their treatment of Lebesgue integrals/measure theory than Rudin's brief survey. Apostol's "Mathematical Analysis" goes through a bit more material than Rudin, gives more worked out proofs, and has relatively easier problems. If you're struggling with Rudin, give Apostol a try. Zorich's "Mathematical Analysis I && II" starts lower level than Rudin but ends on much higher level covering many additional topics including manifolds. The price paid is that his books is quite longer than Rudin and larger time investment.
Analysis on Manifolds
This is the study of analysis on multidimensional spaces making multivariate and vector calculus rigorous and pushing the subject further. A good grounding in linear algebra is required.
- Munkres' Analysis on Manifolds
- Spivak's Calculus on Manifolds
- do Carmo's Differential Forms and Applications
Books
Chicago undergraduate mathematics bibliography
Amazon's "So you'd like to... Learn Advanced Mathematics on Your Own"