Ask HN: Math books that made you significantly better at math?

Out of this list, the books I am familiar with, are great (Hilbert-Courant, Spivak, Korner's books). At the same time, even with extensive mathematical training, I haven't read them from start to finish. I wouldn't even like to say "read". For someone who's not used to mathematical reading, some of these books require careful study. That means generating examples to understand results (theorems), trying your own conjectures, proving things yourself etc. Over time, one becomes familiar with most/all the material in a book but the knowledge might have been acquired through various books (and courses) over time.

Also, mathematics is a massive field. The first question would be what kinds of mathematics would you like to get better at. There are great books in analysis. If you are starting out with a solid calculus knowledge, try Abbott's Understanding Analysis [1] or Duren's Invitation to Classical Analysis [2]. For asymptotic methods in PDEs, try Bender and Orszag [3] which is a wonderful book. But again, this might not be your cup of tea at all and there are more abstract or formal books like Rudin's.

If you want to approach fields without a lot of machinery, graph theory books by Bollobas are great (but difficult). See his Modern Graph Theory book [4] as an example.

For linear algebra, one of my favorites (but it was after I already learned the subject) is Trefethen's Numerical Linear Algebra book [5]. Another beautiful topic is at the intersection of linear algebra and combinatorics. See Babai and Frankl's lectures freely available online.

Then there are wonderful topics in geometry. A massive mountain to climb would be algebraic geometry. For one starting point, see [6]. Differential geometry (Spivak's multi-volume work or Needham's differential forms book) is another wonderful area. I would recommend Crane's discrete differential geometry course at Carnegie Mellon [7] if you want a concrete introduction.

You might want to demystify a topic you have heard about. E.g. Galois theory and the unsolvability of quintic equations. You could look at [8] which guides your way through wonderful problems.

We haven't even touched huge swathes of mathematics including anything topological or number theory. Even within the topics mentioned above, once you start, your journey will take a life of its own and you'll encounter multiple books and papers opening up new sub-fields.

The only approach that worked well for me in the past was to get completely consumed by what one topic one was studying. This meant not getting distracted by multiple topics. Once one enters the workforce, this is very hard (or at least has been for me). Without knowing someone, it's hard to recommend anything but the advantage with topics like graph theory and combinatorics is that one needs less machinery (as opposed to something like algebraic geometry). These fields lead you to interesting problems very rapidly and one can wrestle with them part-time.

[1] https://www.amazon.com/Understanding-Analysis-Undergraduate-...

[2] https://www.amazon.com/Invitation-Classical-Analysis-Applied...

[3] https://www.amazon.com/Advanced-Mathematical-Methods-Scienti...

[4] https://www.amazon.com/Modern-Graph-Theory-Graduate-Mathemat...

[5] https://www.amazon.com/Numerical-Linear-Algebra-Lloyd-Trefet...

[6] https://www.amazon.com/Algebraic-Geometry-Approach-Mathemati...

[7] https://www.cs.cmu.edu/~kmcrane/Projects/DDG/

[8] https://www.amazon.com/Through-Exercises-Springer-Undergradu...

For what it's worth, version 3.0 of the RM's software now allows endlessly scrolling down.

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.

Seconded.

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.