Math books that made you significantly better at math?

* The Art of Probability by Hamming. An opinionated, slightly quirky text on probability. Unlike the text used in my university course its explanations were clear and rigourous without being pedantic. The exercises were both interesting and enlightening. The only book in this list that taught skills I've actually used in the real world.

* Calculus by Spivak. This was used in my intro calculus course in university. It's very much a bottom-up, first-principles construction of calculus. Very proof-based, so you have to be into that. Tons of exercises, including some that sneakily introduce pretty advanced concepts not explicitly covered in the main text. This book, along with the course, rearranged by brain. Not sure how useful it would be for self-study though.

* Measurement by Lockhart. I haven't read the whole thing, but have enjoyed working through some of the exercises. A good book for really grokking geometric proofs and understanding "mathematical beauty", rather than just cranking through algebraic proofs step by step.

* Naive Set Theory by Halmos. Somewhat spare, but a nice, concise introduction to axiomatic set theory. Brings you from nothing up to the Continuum Hypothesis. I read this somewhere around my first year in university and it was another brain-rearranger.

The classical stuff is great:

* Geometry and the imagination by Hilbert and Cohn-Vossen

* Methods of mathematical physics by Courant and Hilbert

* A comprehensive introduction to differential geometry by Spivak (and its little brothers Calculus and Calculus on manifolds)

* Fourier Analysis by Körner

* Arnold's books on ODE, PDE and mathematical physics are breathtakingly beautiful.

* The shape of space by Weeks

* Solid Shape by Koenderink

* Analyse fonctionnelle by Brézis

* Tristan Needhams "visual" books about complex analysis and differential forms

* Information theory, inference, and learning algorithms by MacKay (great book about probability, plus you can download the .tex source and read the funny comments of the author)

And finally, a very old website which is full of mathematical jewels with an incredibly fresh and clear treatment: https://mathpages.com/ ...I'm in love with the tone of these articles, serious and playful at the same time.

Some that stand out

"Concrete Mathematics: A Foundation for Computer Science" by Knuth, Graham, and Patashnik - solid foundation in mathematical concepts and techniques, and it helped me develop a deeper understanding of mathematical notation and problem-solving.

"Introduction to the Theory of Computation" by Michael Sipser - introduced me to the theoretical foundations of computer science, and it helped me develop a strong understanding of formal languages, automata, and complexity theory.

"A Course in Combinatorics" by J.H. van Lint and Wilson - provided a comprehensive introduction to combinatorics, and it helped me develop a strong understanding of combinatorial techniques and their applications.

"The Art of Problem Solving" by Richard Rusczyk - This book is a comprehensive guide to problem-solving, with a focus on mathematical problem-solving strategies. It helped me develop my problem-solving skills and learn how to think critically about mathematical problems.

To add one more thing: the "thing" that has helped me most lately isn't a specific book or video or anything, but rather simply committing to spending 1 hour every day on math. I even set up a Google Calendar task to remind me of this every. single. day.

And so far this year I haven't missed a day yet. Now what constitutes that hour can vary. It can be watching math videos, it can be solving problems on paper, and I might even let myself count futzing around with numerical computing stuff or something at some point. In practice so far it's basically always either watching videos, reading books, or doing exercises (from books).

I won't claim that everybody must do this, or that you need to commit 1 hour every day. Maybe 30 minutes would be fine. Or maybe some people who can spare the time would be well served to commit 2 hours a day. Who knows? But having some kind of routine strikes me as something that most people would probably find valuable.

This is different from the other answers, but it does answer your question: When I was a kid I had tons of math and logic puzzle books. Two I remember specifically are "Aha! Insight" and "Aha! Gotcha" by Martin Gardner. Decades later, when a math problem comes up in my work, I have an apparently unusual ability to cut to the heart of it ("by symmetry, we must have X" or "looking at this extreme case, we must have Y" or "this looks like a special case of Z" sort of things) instead of starting by soldiering through equations, and I credit a lot of that to all the puzzle-solving I did as a kid.

“Mathematical Notation: A Guide for Engineers and Scientists”[0] really changed my abilities with being able to read papers and decipher what was going on. I had university math experience but it was a long time ago. When I started reading papers for algorithms later in my career I couldn’t get past the notation. Once the symbols are explained, as a programmer, I was able to grok so much more. This should be on everyone’s shelf.

[0] https://a.co/d/gQmDIo7

I would not call myself great at math – I struggled with it in school, in fact – but in recent years I’ve begun “correcting” my lack of mathematical knowledge. The single best decision I’ve made is to first start with the philosophy of mathematics. Maybe it’s because my background is in philosophy, but I also think that for certain people like myself, understanding what math is makes me far more interested in understanding how it works, rather than just doing context-less calculations using formulas I don’t know the history or deeper purpose of. When I learned math in school, it was entirely cut off from any of these deeper questions.

Here’s a good starting point for philosophy of mathematics :

https://plato.stanford.edu/entries/philosophy-mathematics/

The best resource I've found is this random, somewhat obscure website (though I've learned that it has grown in popularity) called Paul's Online Notes. The professor has a real knack of pedagogy, and the problems are perfectly structured in terms of their difficulty. His explanations are clear and without jargon, and it goes from algebra to diff eq.

A note: this isn't a resource for higher-level, proof based maths. It will give you a solid foundation and a pragmatic understanding to build upon. Very useful for STEM.

Link: https://tutorial.math.lamar.edu

In my experience, the best way to get better at math is to do a lot of it. Find some book that's "good enough" for some topic you're interested, and work many many problems from the book. You'll learn about the topic, but more importantly you'll learn problem solving skills. I recommend working the problem until you're sure the answer is right -- in grad school problem sets didn't have answers you could check, and full understanding was necessary to get the problem sets correct.

For me, a watershed book was Introduction to Analysis by Rosenlicht [1]. Proof-based, very "mathy", small and compact (so to speak) but with a massive scope. A great introduction to a really important topic, and it'll put your brain through its paces.

Again, I recommend working nearly every problem.

[1] https://www.amazon.com/Introduction-Analysis-Dover-Books-Mat...

This is a bit of an odd suggestion, but I learned the basics of category theory from the appendix to Weibel’s “An Introduction to Homological Algebra”.

I’m not sure why, but I think the fact that it’s an appendix meant the author had no motivation to inflate the content unnecessarily. So it’s more like a pamphlet; only about 30 pages IIRC, and it’s really just the bare-bones definitions and facts. The full-on textbooks dedicated to category theory have way too much superfluous content IMO, unless your aim is to be a researcher in that field specifically.

Lot's of good books already here! In the same spirit as Polya's book "Thinking mathematically" by J. Mason, L. Burton and K. Stacey I learned a lot in the early days with Demidovich book on 5000 problems on mathematical analysis. Tom Apostol books on calculus, but for me his book on analytical number theory. Alongside alan baker's thin book on number theory. Gilbert Strang book(s) on linear algebra. Rudin book on functional analysis. Oh Hardy's book on divergent series! Ian Steward books on transcendental numbers and Galois theory. Elements of algebraic topology by Munkres is a fantastic book. So many books are invaluable to me in teaching not only math but mathematical thinking.

I guess if you want to learn thinking but not necessarily math "thinking mathematically" above mentioned is your friend.

Linear Algebra Done Right by Sheldon Axler for the following reasons:

- I was revisiting a topic in greater depth, which is a common theme in university-level math courses.

- It is a rigorous book, written in the style of definition, proposition, theorem, etc.

- It was the first math book where the exercises don't just reinforce what you learned in the chapter, but teach you new material (another common theme in advanced math textbooks).

- Linear Algebra is arguably the most important math subject these days.

I wonder how good you can get at maths just by casually reading books. You need to work on problems for hours and hours to get a grasp on the theories. Programming is different in the sense that it's something people routinely do as a hobby because it's quite fun and addictive. But maths? maybe if you have already strong foundations you can pick up a new topic and develop your culture. But I doubt one can get these foundations without actually graduating in maths as it's an extremely strong commitment.

Not a direct answer, but I once read that the best book about a technical topic is the third book you read on it. Often you'll see things in comments sections like: "I have heard this explained so many times by others, but this explanation finally clicked!". The assumption is that that's the case because the explanation is better, rather than assuming it's the case because you've struggled with the material before and you're still going at it.

Statistics by Freedman, Pisani and Purves. Don't know if I got better but loved the real world examples and cartoons. Does not have too many pre-requisites. Each section presents a tiny concept which is followed by plenty of exercises that have answers at the end. The furthest I got in a book in recent days, Math or not.

* The Language of Mathematics: Utilizing Math in Practice by Baber https://www.amazon.com/Language-Mathematics-Utilizing-Math-P... really helped me "get it", as I always found programming natural but math hard. This one is written by a CS professor and it really makes all the difference.

* How to Solve it by Polya https://www.amazon.com/How-Solve-Mathematical-Princeton-Scie... and How to Prove it by Velleman https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/... helped strengthen that understanding.

* This year I am trying to master https://www.amazon.com/Methods-Mathematics-Calculus-Probabil... which focuses on how to "connect the dots".

* I am using Geometry and the Imagination by Hilbert https://www.amazon.com/Geometry-Imagination-AMS-Chelsea-Publ... as an attempt to "immerse" myself in Geometry. I just love this book.

Measure and Category by John Oxtoby. This book studies duality results between different notions of "small" sets in measure theory and topology. It's the first (and to some extent the only) math book where things just clicked and I didn't feel like I was drowning in a sea of notation and ideas. Here are some more thoughts on it: https://bcmullins.github.io/Top-Books-2019.

Honestly, unless you're very gifted, a book on its own is not going to be enough to really develop your skills. You need to work interactively with a teacher. Getting a degree in math is a good start, but even then it will be limited if you don't work with other people, go to office hours, form relationships with professors---embed yourself in the culture, so to speak. I think getting engaged with an online community like MathOverflow or similar could be a substitute for this.

IMO, programming is "easier" to learn on your own for a few major reasons:

1) The sorts of things most people are interested in building just aren't unforgiving intellectually in the same way that math is.

2) You have a compiler to check if you're right, and your code will often still work even if it's "wrong" (not as efficient as it could be, has unwanted side effects, etc.). In some sense the compiler is a bit like a teacher this way.

3) With programming, you can upload and make it available for free, and whether it's legit or not is largely disconnected from your pedigree or how "correct" it is. This makes programming far more accessible. This makes sense considering that programming is primarily a practical tool. On the other hand, mathematics is primarily a field of scientific inquiry and is judged by different standards. If you learn a bit of math, try to write a paper, submit it to the arXiv... well, people will probably think you're a crank.

On the other hand, if you're just interested in math for the love of the game... you can certainly pick up a book and read it, maybe work some problems, but I think at this point it's quite easy to fool yourself into thinking you understand more than you actually do. I guess there's no real harm in being a charlatan, but probably the average person is interested in having some kind of real relationship with mathematics that they can be confident has a firm foundation. I'm very skeptical most people can truly pull this off by just reading books and not actually going to school.

---

As an aside, I think the fetishization of math in programming communities is very interesting...

Calculus Made Easy and Probability Through Problems. I'm not sure that I'd have gotten through either my university Calculus courses or Probability and Statistics without these two books. I used them as supplementary material to the course textbooks and homework. They both have a style that is approachable and helped me build an intuition for the material unlike anything else I found.

If you have a Love of Books and Math Books in particular, you can't miss this playlist by The Math Sorcerer: https://www.youtube.com/playlist?list=PLO1y6V1SXjjM-1azbCNYq...

Velleman's How to Prove It greatly helped my ability to construct set theoretic proofs, which better prepared me for Spivak's calculus and Baby Rudin. Hamkins' Proof and the Art of Mathematics is designed as a a good, less set-theory heavy, introduction to proof writing that leads more naturally to analysis. OpenStax books are FREE.

No specific book, but generic advise about how to use a math book. Homework, homework, homework. Read the whole thing, but focus on the exercises. Do every exercise as soon as you can manage: don't wait until you've read the whole chapter -- once you get confused and stumped, the lesson of the chapter becomes urgent and I find that sharpens my attention.

Rudin's Principles of Mathematical Analysis has a really special place in my heart. Chapter 3 is great- it's a great reference for derivations of a lot of fundamental identities about limits used in undergrad calculus.

Chapter 4 is a great place to learn about topology for the first time.

In general, it kicks up the mathematical rigor you're used to a notch. Seeing ">" defined as "not <" really blew my mind when I first read it! "<" is just something that satisfies some axioms, like anything else in math.

I don't think there is any book I've read as an adult that was particularly special. If I wasn't already good at math, I wouldn't be reading these books in the first place. Not that there weren't good and helpful books, but I wouldn't say any of them were revolutionary to me.

But I'd like to mention two books I read as a child which had a life-altering effect. They probably wouldn't do any good for an adult, but might really help your kids... Unfortunately, I don't remember the specific titles or authors (I was probably around 10 yrs old). The first was similar to this book:

"Speed Math for Kids: The Fast, Fun Way To Do Basic Calculations." This gave all sorts of advice and tips to quickly do math in your head... simple things, mostly. For example, to multiply by 18 just double, multiply by 10, and subtract 10%; or how it's frequently faster to multiply numbers by moving from most significant digits to least, which is opposite of how we're taught; or how to quickly estimate square roots. This really didn't teach new concepts, but by making routine and tiresome math operations faster and easier, it made the entire field more enjoyable to engage with.

The second book was a guide to slide rulers, and I couldn't even find a similar book on Amazon. But learning advanced slide ruler techniques can trigger an epiphany; you learn mathematical relationships, how you can transform how numbers are represented. It was the first time I really saw an elegant structure behind the math.

I can recommend Teschls book on ODEs, and it's completely free: https://www.mat.univie.ac.at/~gerald/ftp/book-ode/index.html

And if you like something very applied: Modern Statistics for Modern Biology https://www.huber.embl.de/msmb/

How To Solve It by G. Polya https://press.princeton.edu/books/paperback/9780691164076/ho...

A Logical Approach to Discrete Math by David Greis and Fred Schneider, https://link.springer.com/book/10.1007/978-1-4757-3837-7

I'm self-taught so for me it was learning how to write proofs that gave me a big boost in being able to branch out into different area of interest and not give up. :)

I have not yet become significantly better, and not a math book, but I recently read A mathematicians Lament by Paul Lockhart and it resonated so much with me that I plan to take another stab at math different from how it is taught in school.

Waiting to get my hands on his book 'Measurement' and approach it more like art.

If what he says is true, perhaps many who would have turned out great at math are locked out by how it's taught in school.

For now, I have a test subject of one :)

A lot of people have mentioned Keith Devlin's book "Introduction to Mathematical Thinking" by Keith Devlin but for me his other book "Mathematics: The Science of Patterns" was something that really had a huge impact on me just to put mathematics in perspective. Probably has something to do with my own personal character and education but I needed that perspective before I could take the next step. Then the "Introduction to Mathematical Thinking" is a great following read. So it depends where you are.

Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach

It's a rigorous but chatty textbook in the style of Spivak but written by someone who is sensitive to applied maths. I would not have survived my astrophysics classes without it.

(Not to mention it's where I first saw this really intuitive way of doing matrix multiplication: https://blogs.ams.org/mathgradblog/2015/10/19/matrix-multipl...)

For sure I got significantly(?) better with classics like Spivak, Apostol, Rudin.

"Real and Complex Analysis" by Rudin, and the two books both named "Calculus" from Spivak and Apostol. But also from Apostol his more concise and far-reaching "Mathematical Analysis". And from Spivak his small gem "Calculus On Manifolds" made quite a dent on me.

Other than more "classic math" books, I also wanted to mention two outliers that I found eye-opening and generally awesome:

* Street-Fighting Mathematics, by Mahajan (http://streetfightingmath.com/). Intuitive, useful and fun.

* Geometric Algebra for Physicists, by Doran and Lasenby. I found the power and elegance of geometric algebra mesmerizing, and even if this book is also about physics and there may be more appropriate math-only books about geometric algebra, this is the one that made it for me.

I think you guys might find this list I found long ago very useful when deciding on a mathematics book you want to read.

This is an introduction written by the original author of the list:

"Somehow I became the canonical undergraduate source for bibliographical references, so I thought I would leave a list behind before I graduated. I list the books I have found useful in my wanderings through mathematics (in a few cases, those I found especially unuseful), and give short descriptions and comparisons within each category. I hope that this list may serve as a useful “road map” to other undergraduates picking their way through Eckhart Library. In the end, of course, you must explore on your own; but the list may save you a few days wasted reading books at the wrong level or with the wrong emphasis.

The list is biased in two senses. One, it is light on foundations and applied areas, and heavy (especially in the advanced section) on geometry and topology; this is a consequence of my interests. I welcome additions from people interested in other fields. Two, and more seriously, I am an honors-track student and the list reflects that. I don't list any “regular” analysis or algebra texts, for instance, because I really dislike the ones I've seen. If you are a 203 student looking for an alternative to the awful pink book (Marsden/Hoffman), you will find a few here; they are all much clearer, better books, but none are nearly as gentle. I know that banging one's head against a more difficult text is not a realistic option for most students in this position. On the other hand, reading mathematics can't be taught, and it has to be learned sometime. Maybe it's better to get used to frustration as a way of life sooner, rather than later. I don't know." - by original author.

[List] https://www.ocf.berkeley.edu/~abhishek/chicmath.htm

Any good book about the history of mathematics that will teach you a natural historical development of concepts to reach more generalizations.

History of mathematics will give you a very subtle entry into the minds of mathematicians and the motivation behind their theorems.

This will surely make you more appreciative of subjects and concepts you are learning.

During my undergraduate studies, I loved "Discrete Mathematics and Applications" by Kenneth Rosen. I really enjoyed reading through the various examples and biographies of famous mathematicians included in each chapter.

For those looking to delve into discrete mathematics, I highly recommend the lecture notes from L. Lovasz and K. Vesztergombi (Yale University, Spring 1999) and from Eric Lehman, Tom Leighton, and Albert Meyer (MIT, 2010).

Div Grad Curl and All That

https://www.amazon.com/Div-Grad-Curl-All-That/dp/0393925161

Not nearly as rarefied as many of the books cited here, for me it was John Saxon's excellent Algebra 1/2, Algebra 1, and Algebra 2. I didn't get a good enough grasp on basic Algebra in high-school. When I was in my early–mid 20s, a friend gave me these three algebra textbooks. In a marathon session lasting about two weeks, I went through all three books from end-to-end and really learned algebra. For whatever reasons, the Saxon books worked really well for me — better than any other learning I have ever gotten from a textbook. Although I was very motivated, I attribute a lot of my success learning algebra well to those three books. I still own them.

A lot of comments about textbooks that helped in specific topics but I don't think that answers the spirit of OP's question. Sure working through ANY Linear Algebra textbook is going to improve your Linear Algebra skills.

In the spirit of OP's question:

How to Solve it by G. Polya

Solving Mathematical Problems by Terrence Tao

Introduction to Mathematical Thinking by Keith Devlin

Are all amazing, How to Solve it in particular is an all time classic.

Also, Geometry for Programmers but only because I wrote it. I had to update my skills significantly while gathering material and doing all the experiments. Not sure if reading the book would have the same effect :-)

A Programmer's Introduction to Mathematics https://pimbook.org/

It introduces math from a mathematician's point of view (complete with proofs, etc.) rather than rote memorization and exercises, but it does so from the perspective of a programmer.

Since I haven't seen many discrete maths books, he's my list:

Beginner: NL Biggs, Discrete Mathematics, Oxford University Press

Intermediate: PJ Cameron, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press

Advanced: JH van Lint & RM Wilson, A Course in Combinatorics, Cambridge University Press

Not a book. But a course by Prof Keith Devlin on Coursera called Introduction to Mathematical Thinking.

K A Stroud Engineering Mathematics is probably the book that helped me most. 31 chapters each composed of about 60 problems. The problems are progressive and contain explanations of the new concepts that they contain. All the answers are at the back.

Several people who have mentioned "How to Solve It" by George Polya. It's been decades since I've looked at it, but a favorite Polya quote of mine is " "The open secret of real success is to throw your whole personality into your problem."

I can't remember if the book addresses this, but for myself my inability to tolerate frustration really impeded my ability to work on any mathematical challenge for decades.

Math always seemed a bit arbitrary to me. Why and how did all those fields and branches develop, why are some so much more intuitive than others etc. What helped me cope with that challenge (and ultimately be a better learner / user of mathematics) was digging into the history of mathematics. Many great books in that genre but a very influential one for me was the Concise History of Mathematics by Dirk Jan Struik

A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekeres

My intro to abstract math... Wide range of topics, very clearly written and very well structured. Sets, groups, vector spaces, tensors, topology, differential geometry, lie groups and more.

An Introduction to Category Theory by Harold Simmons

Very enjoyable read. You cannot go wrong with this as your first book on the subject.

The classic, How to Solve It by Polya.

A lot of the advice seems obvious in retrospect but being systematic about a problem solving framework is enormously helpful.

Probabilistic Graphical Models by Koller & Friedman. In anything statistics and ML related, being able to deal with complicated probabilitistic things that are all related is really useful. This book gives you that toolkit. It's a "strong foundations" kind of book, rather than a bunch of methods you'll use directly.

Meta comment: might be good to add the level of mathematical maturity needed to enjoy the book.

1. Principles of Mathematical Analysis by Walter Rudin (baby Rudin) - I'd studied real analysis in the past, but this book is direct and rigorous and provided a good framework to move forward into things like functional analysis in a way that I was not prepared for with other books.

2. Differential Equations and Dynamical Systems by Lawrence Perko - Solidified for me how dynamic systems behaved and were solved. Very much helped my understanding of control theory as well.

3. A Concise Introduction to the Theory of Integration by Daniel Stroock - Helped solidify concepts related to Lebesgue integration and a rigorous formulation of the divergence theorem in high dimensions.

4. Convex Functional Analysis by Kurdilla and Zabarankin - Filled in a lot of random holes missing in my functional analysis knowledge. Provides a rigorous formulation of when an optimization formulation contains an infimum and whether it can be attained. Prior to this point, I often conflated the two.

I read the first chapter of “Mathematics; Its Content, Form, and Meaning” (or something close to that) and it explained the entirety of my high school mathematics. If I had been given that to read back then I might have gone into mathematics in college. Instead I got burned out and quit doing any math for a decade or more. Sigh.

A book of abstract algebra - Charles C. Pinter. Each chapter is a few pages of explanation, and the rest you solve yourself by doing exercises that introduce aspects of the theory step by step.

I'm coming from an applied math perspective. A few of my favorites (and ones I find myself regularly referring to) are:

Matrix Analysis by Horn and Johnson (perhaps the best end-of-chapter problem sets of any math book I've encountered!)

Matrix Computations by Golub and Van Loan

Elements of Statistical Learning by Hastie, Friedman, Tibshirani

Functional Analysis by Reed and Simon

Differential and Integral Calculus (Volumes 1 & 2) – Nikolai Piskunov

Not as colorful and attractive but the adage "do not judge a book by its cover" applies so well to this masterpiece. With brief and precise explanations and high quality exercises with solutions, I went from struggling to getting A+

> Discrete Mathematics with Applications by Susanna S. Epp

Fantastic book for Discrete Mathematics with lucid explanation and good exercises. The other one would be concrete mathematics.

The Road to Reality: A Complete Guide to the Laws of the Universe

In this book, Roger Penrose a Nobel Prize winner in Physics for his contributions in mathematical physics of general relativity and cosmology, provides background math to understand the book's contents in the first half of the book.

As a child, I re-read The Number Devil constantly. It introduced me to some really cool mathematical ideas, couched in a cute story. The very last chapter includes a picture of the Principia Mathematica's proof of 1 + 1 = 2 -- mostly for shock value, I think, but also "even this is within your reach". I recently got to see a friend's copy of the Principia, and I realized just how much of it I actually did understand, which was a really nice closing of the loop.

As an adult, Imre Lakatos' Proofs and Refutations gave me a much richer understanding of definitions in mathematics -- what job they're meant to do, and when it makes sense to change your definitions instead of adding premises to your theorems.

Knuth's "Concrete Mathematics" is fantastic, precisely because it's very applicable.

A pair which I am most of the way through:

_Make: Geometry: Learn by coding, 3D printing and building_ https://www.goodreads.com/en/book/show/58059196

and

_Make: Calculus: Build models to learn, visualize, and explore_ https://www.goodreads.com/book/show/61739368-make

I'd really like to find a similar book on conic sections --- my next major project seems to need them, and when I tried to solve it using trigonometry alone, I wound up 7 or 8 levels deep in triangles and wasn't much more than half-way to where I needed to be.

Going to echo the suggestions for the Art of Problem Solving books, particularly I recommend the contest books (vol 1 or 2). Several very talented people have said to me that these books taught them how to think. Maybe a bit exaggerated, but they’re very good.

In early high school in the 90s, I got my parents to buy me an (expensive) copy of Chaos and Fractals: New Frontiers of Science by Peitgen, Jurgens, Saupe. The end of each chapter was a Basic program for calculating and displaying various fractal/chaos theory images, and that's what got me started programming.

It also included a bunch of mathematics involving "neighborhoods", meaning the set of all points within a distance of an arbitrarily small epsilon from some point X. Although I never did any of the math problems from the book, that early exposure to epsilon made calculus vastly easier to understand, and for that, it's close to my heart.

Counter-intuitively, reading Tim Ferris and his DSSS approach made me much better at math

Deconstruct: Break down the math you want to know into big problems and concepts. Pick a math-related goal that is Measurable and Time-Bound

Selection: What the 20% of math concepts, that if made really strong, would solve 80% of math problems

Sequencing: What order of material should you study for maximal progress

Stakes: Find some incentive to complete the problem. Some nice view of mathematical terrain, as part of a masters program, applications to another field, a prize, a cookie. Anything that motivates you to actually make progress towards the goal

This approach helped me learn a bunch of high level math like abstract algebra, analysis, linear algebra, etc.

Calculus on Manifolds by Spivak. Brilliant. And relatively thin too.

”Road to reality” by Roger Penrose is an interesting book as a refresher and review if the content is otherwise within familiar territory.

Nonlinear Dynamics and Chaos by Strogatz

Chaos theory and deterministic systems are a fascinating vantage point for thinking about the dynamics of large computer systems. Thinking of them as stochastic systems is sometimes useful, but most of the systems are actually just operating in unstable periodic processes which are much closer to being a chaotic system rather than a stochastic system. This influences how I think about testing and debugging large distributed systems.

I will say, I'm not sure I could have learned it well without a class and a good professor. The author has a number of books though and is a professor at Cornell.

The Art of Problem Solving: https://artofproblemsolving.com/store/book/aops-vol1

Yes, it is targeted towards middle and high school students. Yes, I read it and (more importantly) worked through most of the problems in my mid-30's. It is great if, like me, you coasted/crammed through your early mathematics education and never felt like you dialed in the fundamentals. It is also great if, like me, you needed some pen-on-paper practice and did not know where to start.

I was reading this book, when the ideas of function spaces, functions as vectors, functions as elements of vector spaces, functional analysis clicked on me.

I am not sure if this book is particularly good or better than other books. (Well, it still looks like a very gentle introduction to the topic.) But as per your question, this was the book at the right time for me.

"Fourier Series and Orthogonal Functions" by Harry S. Davis. https://www.amazon.com/dp/0486659739/

A lot of recommendations depend on what you're trying to learn.

But I've enjoyed the following texts to a larger extent than others:

- Algebra: Chapter 0 (Aluffi)

- Real Mathematical Analysis (Pugh)

- Mathematics and its History (Stillwell)

- An Introduction to Manifolds (Tu)

- Gauge Fields, Knots and Gravity (Baez)

- A First Look at Rigorous Probability Theory (Rosenthal)

- All of Statistics (Wasserman)

There are some authors I trust and am happy to buy so long as the topic vaguely interests me: VI Arnold, Tristan Needham and John Stillwell.

I really like the list put out by @enriquto in a separate comment, but I've avoided duplicating those recommendations in the list above.

Donald Sarason's "Complex Function Theory". There are bigger more complete books on complex analysis. There are even ones that are more appealing, like "Visual Complex Functions" by Elias Wegert.

Sarason's book is only 160 pages long, with legible text and clear examples. It covers the length of an undergraduate university class, and explains Holomorphic functions perfectly. The proofs are crystal clear, and so are the motivations. I haven't seen a better introduction.

Mathematik für Ingenieure und Wissenschaftler I, II and III from Lothar Papula (in German). The solutions are detailed, making it perfect for self-studying.

Book of Proof by Richard Hammack. A great introduction to proofs in mathematics. The book is available free online [0], but also I bought the physical version because I really enjoyed it.

[0]https://jdhsmith.math.iastate.edu/class/BookOfProof.pdf

For what level and in which area? Books like _Methods of mathematical physics_ can be both too hard and irrelevant to your needs. For starters,

To become a better problem solver with high-school level maths:

  - Polya's How to Solve It.
  - Books of your choice about math contests.
  - Concrete Maths. I understand that this book is taught in college, but it requires very little advanced maths, and its techniques are hugely useful for high school students too.

To hone my intuitions. I learned it the hard way that college maths were different from high school math: in high school, my teachers painstakingly drilled intuitions into us with very targeted explanations and tons of well designed exercises. In college, we won't get such luxury. So, it's really up to us to understand mathematical concepts intuitively before diving into technical details. For that matter, the following books helped me a lot:

  - The visual series. Visual Complex Analysis and Visual Group Theory, for instance
  - Pinter's A book of Abstract Algebra
  - Strichartz's The Way of Analysis
  - Linear Algebra Through Geometry by Wermer. The book offers a comprehensive geometric interpretation to linear algebra concepts. It's especially helpful for me to understand quadratic forms.

To understand Analysis better. This area is vast, so I'll skip recommendations of excellent text books:

   - _Counterexamples in Analysis_. Those counterexamples in Analysis play a huge role in helping me truly appreciate the intricacies of Analysis. Similarly, books like _Counterexamples in Probability and Real Analysis_ are of great help too.
   - The Way of Analysis by Robert S. Strichartz. This books is AMAZING for laymen like me. You'd want someone to *explain* how concepts emerge, and how intuitions evolve.

To become good at maths by doing maths, so the following books used to help me a lot:

  - Problems and Proofs in Real Analysis
  - Putnam and Beyond. I still suck at maths, but those well designed problems in Putnam really taught me how to seek insights in higher maths.
  - Piotr's Problems in Mathematical Analysis. But really, any problem books that challenge you will do. I'd recommend you find problem sets from the website of university courses. They cover essential techniques, and will not be as overwhelming as the books.

I highly recommend working through Claude Shannon's Mathematical Theory of Communications [0]. It's originally a paper but was later restructured as a book, in either form it works quite well.

The reason I recommend it is because it shows mathematical reasoning that is easy to follow and relevant to your daily life. It's real math, but very easy to read through and understand. If your unfamiliar this paper is where the very idea of "bits" comes from.

One of the most important things in the paper for non-mathematicians to see is that the definition Information Entropy is derived simply from the mathematical properties Shannon desires it to have.

This is important because I find that one of the biggest questions people ask about mathematical formula and idea is "What does this mean? Why is it this way?" without realizing that math is really not engineering nor physics. When deriving his definition of Information, Shannon simply states that information should have the following x,y... properties and then goes on to show that the now standard definition of information meets all these criteria.

In mathematics it is very often the case that only after an idea is created to we start realizing the applications. This is quite different than science where a model is only adopted if it correctly describes a physical process.

Work through the paper and you will have worked through the mathematical underpinnings of the information age and will likely have understood most of it pretty well.

0. https://people.math.harvard.edu/~ctm/home/text/others/shanno...

Steele's The Cauchy-Schwarz Masterclass is actually quite good and seemingly designed for self study. (A lingering result of this book is I heavily use inequalities even outside of analysis.)

Artin's Algebra probably has had the most impact on my math thinking. The development of groups and rings while tightly linking them to linear algebra was rather brilliant.

Khan Academy. Not a book per-se, but have worked through the courses over the past few years, I'm confident that my college and university results would be about 2 or 3 grades higher we I to retake them.

https://www.khanacademy.org/

It's unclear what kind of mathematical thinking and skills you want. "How to solve it," by Polya perhaps?

If I could pick one, that would be How to Solve it by George Polya.

None. As far as I'm concerned, they all suck because academia goes about teaching math in all the wrong ways.

I finally learned the point behind math thanks to dabbling in programming. All the math classes and teachers and textbooks in the world will never teach me what the importance of 1+1 is.

Books do not make you better at math. Working math problems makes you better at math. Go ahead, down vote all you want.

Reading about running does not make you a better runner. You can watch 1000 marathons, sprinters, Olympians. You may get _ideas_ for running, but it will never make you a better runner. To be a better runner you have to do it. To be a better programmer/mathematician/physicist/whatever, you need to go work at it.

I suppose I am just taking action against how the question is written, but I see a lot of people seemingly hoping that "if they just found the correct book, tutorial, or video, they would be better". A lof of those people are my students. When I ask how many problems they have worked, I typically always get the same response. Zero, or the bare minimum.

The classic "Advanced Engineering Mathematics" by Erwin Kreyszig was absolutely 'good enough'. Maybe even more important - for me: it was one of the easiest books to follow and digest during my undergrad. See it as a solid base for other heavier/purer titles.

I love math books with great exercises, I just wish I could code them up in a theorem prover and solve and store my proofs that way. I've tried a bunch of tools but haven't found a language or workflow that really meets the needs for computer assisted study of mathematics.

If the HN crowd is up for it, I'd love similar book threads for physics, chemistry and biology.

While I have a ton of favorite math books, the two books that I felt really helped me the relevant subject are:

* An Introduction to Manifolds by Loring Tu

* The Elements of Integration and Lebesgue Measure by Robert G. Bartle.

These two books were instrumental to my studying for my qualifying exams.

Not strictly math books but these three books turned me to live mathematics and appreciate for what it is:

[1]: 3D Math Prime for Games for Graphics and Game Development; https://www.gamemath.com/ (free to read online) [2]: Essential Mathematics for Games and Interactive Applications; https://www.essentialmath.com/book.htm [3]: Mathematics for 3D Game Programming and Computer Graphics; https://www.mathfor3dgameprogramming.com/

Not a book, but I loved this:

* The Natural Number Game https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_g...

Calculus Made Easy (1910) simply for the quote at the beginning: "What one fool can do, another can."

I did horribly in math because I figured it was hard and just accepted I'd never be good at it. That quote somehow managed to dissolve my mental block.

* The Man Who Loved Only Numbers, by Paul Hoffman

https://www.goodreads.com/book/show/714583.The_Man_Who_Loved...

* Algorithms to Live By, by Brian Christian & Tom Griffiths (not really a Math book, mostly computer science, but still has some math algorithms and their implementations to real life)

https://www.goodreads.com/book/show/25666050-algorithms-to-l...

V.I. Arnold "Problems for children from 5 to 15" [0]. The book was discussed on HN in 2021 [1] (325 comments)

If you have kinds and teach them math this book has mind-opening problems that even curious adults would enjoy.

[0] https://www.imaginary.org/sites/default/files/taskbook_arnol...

[1] https://news.ycombinator.com/item?id=27884973

Analysis Now, Gert Pedersen https://link.springer.com/book/10.1007/978-1-4612-1007-8

Probably the most elegant math book I have ever seen is Probabilty theory a graduate course by Achim Klenke. A very nice exposition into the abstract, measure theoretic prob. thoery (but it assumes some prior knowledge).

Elementary Differential Equations by Boyce and DiPrima

I'm not sure I'd say that it made me significantly better at math, but I keep coming back to it time and again, and usually via very different paths.

For the German speakers:

Something that really helped me with my mathematic modules at university: Lothar Papula's "Mathematik für Ingenieure und Naturwissenschaftler" [1]. If you get the stuff in this book right, you're set for life.

[1] https://www.amazon.com/Mathematik-f%C3%BCr-Ingenieure-Naturw...

You know how Feynman has these lectures where he digs into physics but you don’t need to know any to get lots out of it?

Is there that for math? Books or lectures that talk about math without doing the math?

Significantly better I don't know but when I was a child I was given Der Zahlenteufel. Ein Kopfkissenbuch für alle, die Angst vor der Mathematik haben (The Number Devil) and I liked it very much

The three for me were:

- Principles of Mathematical Analysis by Walter Rudin (aka “Little Rudin”)

- Linear Algebra and its Applications by David Strang

- Elementary Differential Equations and Boundary Value Problems by Boyce and Diprima

There exist only two kinds of modern mathematics books: ones which you cannot read beyond the first page and ones which you cannot read beyond the first sentence. -- Chen Ning Yang

Spivak, Abbott, Hubbard & Hubbard, Linear Algebra Done Right, Gallian’s Abstract Algebra, and believe it or not MTW taught me more good math than many math books.

If you are into numerical optimization, a nice source of intersting problems and examples (that e.g. contradict the intuition) can be found in

Mathematical Tapas: Volume 1 and Vol. 2.

Has anyone read "Common Sense Mathematics"? I really liked that book for shortcuts to elementary maths for mathematical analysis of everyday things.

* 4-Manifolds and Kirby Calculus by Andras I. Stipsicz and Robert E. Gompf

* Differential Manifolds by Antoni A. Kosinski

* Introduction to Smooth Manifolds by John M. Lee

Baby Rudin opened my eyes to what you can do with maths.

I'm going to cheat and combine a couple of books into "one book". The Manhattan Prep GMAT test prep math books were really good for me for everyday life. I learned a lot of shortcuts and quick heuristics to use and got better at estimating after going through those books. It doesn't help me "get better" in an academic sense, but those books pay dividends every day for me.

Not a book, but an animated short: https://en.wikipedia.org/wiki/Donald_in_Mathmagic_Land

“If you want to build a ship, don’t drum up the people to gather wood, divide the work, and give orders. Instead, teach them to yearn for the vast and endless sea.” --Antoine de Saint-Exupéry

The Art and Craft of Problem Solving https://archive.org/details/the-art-and-craft-of-problem-sol...

This is only math book I've ever read that teaches the mindset needed to work mathematics problems, rather than mathematical concepts or techniques.

Some favorites below. Books 0-3 are accessible. The remaining books are more difficult but I'd highly recommend them to math students.

0. Jan Gullberg, Mathematics, From the Birth of Numbers. A highly accessible popular survey on different branches of higher mathematics. I read this over the Summer between high school and starting my undergraduate degree. It's what made me want to study math. Previously I'd wanted to be a guitar player, but had to find a new ambition after an injury left me unable to play.

1. The high school mathematics series by Israel Gelfand. Algebra, Trigonometry, The Method of Coordinates, and Functions and Graphs. I didn't have much mathematics background in high school, but working through these really solidified my grasp on the basics.

2. George Polya. How to Solve it. A short book giving excellent high level advice on mathematical problem solving.

3. George E. Andrews, Number Theory. I worked through this freshman year contemporaneously with my first proof based class on simple logic and set theory. A very beautiful and accessible introduction to basic number theory. The combinatorial/geometric proofs of Fermat's Little Theorem and Wilson's Theorem are lovely. It also includes a very nice proof of Chebyshev's theorem on the asymptotic density of primes and even the Rogers-Ramanujan identities for integer partitions.

4. Vladimir Arnold, Ordinary Differential Equations: Undergrad ODE classes are often taught in a cookbook fashion and if so, don't offer much enlightenment. This book explains what's going on at geometrical level. I didn't appreciate ODEs until I read this. See https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html for Arnold's views on teaching mathematics.

5. E.C. Titchmarsh The Theory of Functions: Recommended by my undergraduate advisor because he noticed that I liked reading older books. It contains sections on complex analysis and real analysis with measure theory, but I've only read the complex analysis sections. It's not for everyone, if I recall correctly, there is not a single picture, but it is very lively and has a lot of material you won't find in a standard complex analysis book, including Dirichlet series. Excellent as a supplement to a standard complex analysis book.

6. George Polya. Mathematics and Plausible Reasoning. An excellent expansion on Polya's ideas on How to Solve it. While the goal is to seek rigorous proofs, to get there it's powerful to be able to think based on intuition, heuristics, and plausible reasoning. A lot of math exposition is theorem/proof based and doesn't help develop these skills. In a similar vein, see also Terence Tao's classic post There's more to mathematics than rigour and proofs https://terrytao.wordpress.com/career-advice/theres-more-to-....

7. H.S.M Coxeter, An Introduction to Geometry. A book of very beautiful classical geometry. Something typically not touched on at all in a typical mathematics curriculum.

I got a lot out of "Unknown Quantity" by John Derbyshire. Subtitle a real and imaginary history of algebra. I particularly enjoyed the lead up to the Chapter "Assault on the Quintic".

Also, I hold "The Dictionary of Curious and Interesting Numbers" close to my heart for the endless fun it brought me.

The Time/Life book "Mathematics" published 1969. I was in 2nd grade, liked mathematics and saw a book with that name and pictures. It was a high level survey of a lot of mathematical concepts, explaining things in a way I could understand at that age, but also in a way that wouldn't be talking down to me today.

The enjoyment of Math (https://www.barnesandnoble.com/w/enjoyment-of-math-hans-rade...). Best book ever. Made me fall in love with math as a teenager.

Mathematics, a Human Endeavor: A Book for Those Who Think They Don't Like the Subject by Harold R. Jacobs

Jeffreys, Harold, and Bertha Swirles Jeffreys. Methods Of Mathematical Physics. Cambridge At The University Press, 1950. http://archive.org/details/methodsofmathema031187mbp.

Shankar Basic Training in Mathematics

The answer is always Elementary Number Theory by David Burton. I read this book since 7th grade.

* 'The Joy of X' by Steven Strogatz. I really like this book, on my second read now.

What if by Randall Munroe Not rigorous but it changed my perspective that was instilled in me in middle school. Otherwise Feynman Notes changed me academically. Easier to pickup math through physics if you arent looking for deeply pure avenues.

Freshman year of undergraduate math required

  How to Solve it -- Polya
  The Art of Problem Posing -- Brown and Walter

I'm not sure it made me any _better_ at math, but I did always enjoy

  How to Lie With Statistics -- Huff

All the Math You Missed by Thomas A. Garrity. I have not read it, but it looks interesting and is on my list. It is aimed at new graduate students who need a quick refresher that is still detailed enough to be useful for postgraduate math.

A better question: What books made you significantly better at math that are also genuinely FUN to learn from?

I find that my biggest barrier to learning math has always been how unengaging, excessively contrived, and unfun the learning material has been.

Nonlinear Dynamics and Chaos by Strogatz - does a great job at explaining very complex mathematical topics with great examples. Not for beginners. Every applied math student should read this cover to cover.

Optimization by Jan Brinkhuis

https://books.google.com/books/about/Optimization.html?id=UW...

Not one book, but this: https://github.com/TalalAlrawajfeh/mathematics-roadmap

Advanced engineering mathematics by Kreyszig

Advanced Engineering Mathematics, 10Ed, Isv https://a.co/d/axcq9nk

The Art of Approximation gave me far more intuition than any class

It may be sacrilege but I learned a lot from "Numerical Recipes". Not in much depth, but enough to wet my feet in a number of areas that were new to me.

Polya: How to solve it

https://en.wikipedia.org/wiki/How_to_Solve_It

“The Bones”, technically by Euclid.

An amazing translation of Euclid’s elements that contains diagrams and commentary that actually make it clear what he’s talking about.

I had a print of Euclid's Elements as a kid.

My mom was really into mathematical proofs and I being a huge loser kid with no friends naturally took to this book as well.

I really liked “Excursions in Modern Mathematics” and “The Shape of Space”.

Not the most technical — but really influenced how I thought about mathematics.

Quantum Mechanics: The Theoretical Minimum --> Great book to learn about quantum mechanics and as a side effect math

Abstract Algebra: A Student-Friendly Approach by Dos Reis

Paperback layout feels like a workbook, not overwhelming-- beginner friendly

For me, it was just practice with whatever the problem that I was doing, and the book which was in front of me.

On a similar note: Does anyone have book they would recommend to teach Algebra or pre-algebra to young kid?

From Mathematics to Generic Programming - Stepanov & Rose

Gödel, Escher, Bach: an Eternal Golden Braid - Hofstadter

Euclid's Elements

Baldor's series of books (Spanish)

Best "math book" for kindling interest in mathematical thinking in an elementary school kid?

The Cauchy-Schwarz Master Class by J. Michael Steele

Linear Algebra by A.O. Morris (out of print and tricky to find)

Math magic

Https://www.amazon.com/Math-Magic-Calculator-Everyday-Problems/dp/0688104762

Baby Rudin for analysis.

Dummit and Foote for algebra.

take math classes, start a level lower than you think you need to, do lots of practice problems

The correct answer is, of course, Beast Academy, the lead-in to the Art of Problem Solving.

Algebra Baldor !

College Algebra Heineman

Discrete Math Rosen !

Linear algebra D lay

Calculus Stewart

Nonlinear Dynamics Strogatz +

Combinatorics Mazur +

Statistics *

ESLR +

Maybe not "made me better at math" per-se, but definitely "made me more enthusiastic about math":

The Universe Speaks in Numbers[1] by Graham Farmelo

I found this very motivating and insightful, in terms of developing even more of an appreciation for how much math underpins other branches of science. Not that that is a novel insight by any means... but the details of the incidents where breakthroughs in mathematics allowed further advances in physics, etc. and looking at the "back and forth" between the domains, that was wildly interesting to me. Reading this book definitely helped motivate me to get serious about committing more time / focus to studying mathematics.

I also enjoyed the "counterpoint" book by Sabine Hosenfelder, Lost in Math[2]. I think these two books complement each other nicely.

Then the handful of additional (no pun intended) books that jump to mind would be:

- How Mathematicians Think by William Byers[3]

- How to Think Like a Mathematician by Kevin Houston[4]

- Discrete Mathematics with Applications[5] by Susanna Epp

- How Not To Be Wrong[6] by Jordan Ellenberg

- Introduction to Mathematical Thinking[7] by Keith Devlin

- How to Measure Anything[8] by Douglas Hubbard

[1]: https://www.amazon.com/Universe-Speaks-Numbers-Reveals-Natur...

[2]: https://www.amazon.com/Lost-Math-Beauty-Physics-Astray/dp/15...

[3]: https://www.amazon.com/How-Mathematicians-Think-Contradictio...

[4]: https://www.amazon.com/How-Think-Like-Mathematician-Undergra...

[5]: https://www.amazon.com/Susanna-S-Epp-Mathematics-Application...

[6]: https://www.amazon.com/How-Not-Be-Wrong-Mathematical/dp/0143...

[7]: https://www.amazon.com/Introduction-Mathematical-Thinking-Ke...

[8]: https://www.amazon.com/How-Measure-Anything-Intangibles-Busi...

Nonlinear Dynamics and Chaos by Strogatz is a work of art

A decade of the berkeley math circle and concrete mathematics

A Programmer's Introduction to Mathematics by Jeremy Kun

Apostol's Calculus changed my life when I was 18

"How to use Calculators" by R. U. Kiddinme

_Mathematics for the Million_ by Lancelot Hogben.

linear algebra done right - sheldon axler

something with gamedev probably

spivak calculus