Ask HN: Recommend a maths book for a teenager?

I worked through a lot of this partial differential equations book during downtime while working in a gas station after my freshman year of college:

https://www.amazon.com/Differential-Equations-Scientists-Eng...

Might be a little advanced for most teenagers (I was 19 that summer), but I love the book and still refer to it from time to time. I did have experience with ordinary differential equations at the time, but I haven't found an ODE book that's quite the same.

If you want to get away from analysis, I've found that cryptography can be quite an engaging subject. If you have the right book, it can have the rigors of more mathematical subjects, while being accessible without extensive background and having visible real-world applications. I unfortunately don't have much experience with books in this area, but I do like https://files.boazbarak.org/crypto/lnotes_book.pdf (plus it's free ;) ).

[EDIT: Previously I recommended Calculus on Manifolds here also, but on further reflection and reading some of the other responses I think I both misremembered the difficulty level of the book and overestimated what early-undergrad level means]

God Created the Integers: The Mathematical Breakthroughs That Changed History. Stephen Hawking. I bought mine for cheap on alibris (https://www.alibris.com/God-Created-the-Integers-The-Mathema...)

From the blurb:

"...includes landmark discoveries spanning 2500 years and representing the work of mathematicians such as Euclid, Georg Cantor, Kurt Godel, Augustin Cauchy, Bernard Riemann and Alan Turing. Each chapter begins with a biography of the featured mathematician, clearly explaining the significance of the result, followed by the full proof of the work, reproduced from the original publication, many in new translations."

What's great about this book for a teenager is that they get to read original sources for the stuff they've already learned! And indeed, as they learn more they can keep coming back for more original sources. Personally, reading Descartes original words in Geometry was awe-inspiring, not because every word was so perfect, but because he comes across as just so damn human, the ideas he presents are subtle and profound, and yet presented with an interesting combination of humility and pride that is instantly recognizable. I truly wish I'd had something like that book before embarking on my own journey through math - we stand on the shoulders of giants, but we so rarely look down to see their faces.

In response to the question about the best book to learn [subject] from, the best answer I ever received was: "the third book".

The point being, of course, that it may take a few different expositions before something 'clicks'. I think this observation is particularly important for self study.

So, in answer to your question: maybe more than one book?

I think "Gödel, Escher, Bach" is the perfect book.