Ask HN: Recommend a maths book for a teenager?

Anything by Iam Stewart is a good read.

My father (and me) would always recommend Zeldovich's "Higher mathematics for beginners" for learning analysis at the upper high-school level. This particular book does not seem to be available in translation, instead there is a reworked version with Yaglom (who was a brilliant science educator himself):

https://archive.org/details/MIRZeldovichYAndYaglomIHigherMat...

Zeldovich's book with Myshkis on applied mathematics is also excellent: https://archive.org/details/ZeldovichMyskisElementsOfApplied...

Subjectively, I prefer typesetting of the latter, but that is because it is closer to the original Russian edition. Zeldovich was a physicist, so these take an engineer's/physicist's approach, which is, in my opinion, the right entry point to analysis. The reader effectively has to follow the historic development of the subject, starting with some intuitive observations, and eventually developing quite delicate insights.

I read HMFB when I was about 17, and it was great. I remember making up questions of the sort "What level of soda in a can makes it the most stable", and the like, inspired by the book.

A little off-topic (and perhaps more useful for younger students) but you could do worse than introduce them to the achievements of Gauss (though they are probably somewhat familiar), who as a teenager had discovered and rediscovered several important theorems - his foundational Disquisitiones Arithmeticae was written at 21.

https://www.storyofmathematics.com/19th_gauss.html

This book is aimed at a young audience, though I haven't read it and cannot say whether it is age-appropriate for late-teens.

https://www.goodreads.com/book/show/837010.The_Prince_of_Mat...

If I could give my high school self only one math book, it would have to be Seven Sketches in Compositionality by Fong and Spivak. Did every exercise over winter break in college and realized along the way that I had been hustling through math courses and olympiad problems without appreciating any beauty in the structure of mathematics. It completely changed my life and, at least in my eyes, dissolved the assumption that “applied” math must be less rigorous or “pure” math must be less practical. Not only did it immediately recast my basic intuition about what math “is” (and what numbers “are” or what processes “do”) but with a bit more effort toward studying category theory, I came to see my previous encounters with more advanced topics like forcing in set theory or the Legendre-Fenchel transform used in physics/economics in a completely new light. What is truly wild to me is that Seven Sketches has no real prerequisites, and I could have just as easily read it when I was 14. This book should be the basis of a mandatory course for a math-loving high schooler. Instead of rushing to learn linear algebra and real analysis in high school, I wish I had gained the wonderful perspective of Fong and Spivak—I would have fallen truly in love with math much sooner, found a deeper perspective in my courses much faster, and enjoyed all of it so much more along the way.

Hope someone sees this and shares the book with a high schooler—it’s also available for free online!