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

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

I used this book and the sister book “advanced engineering mathematics” for my bachelors and masters degree in mechanical engineering.

It is probably the best pair of books ever written for what I like to call “plug and chug” maths. Strouds books cover the whole of engineering mathematics.

I turned to them for calculus in first year, and for Fourier and Laplace in my final year and masters.

But it will not teach you how to solve and develop proofs.

Generally agree. There are books, and some are better than others, but unless you have both the passion and the aptitude, there is no book that will magically make everything understandable. If you struggle with math, it's probably you and not the book you are using.

Edit: just to add, I struggle with math, lest there is any misinterpretation of my perspective. I went through phases where I thought I just needed to find the right books or the right teachers who could explain it in a way that meshed with my "learning style," but ultimately concluded I just don't have a very strong innate ability in the subject.

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/

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.