Ask HN: Recommend a maths book for a teenager?

I would be absolutely flabbergasted if a layman could get through Road to Reality. Even those with an engineering or computer science background would struggle considerably.

That said, it serves as a great overview of physics for mathematicians and perhaps as very casual intro to twistor theory for physicists.

Just wanted to chime in regarding Concepts of modern mathematics.

Really enjoyed reading it when I was in college. It's not a textbook, just a prose book for enjoyable reading, but it's inspirational and a very interesting overview of the field of mathematics.

The princeton companion is nice to have around, but you do not really read it end to end.

Spivak's calculus you bring to the beach and read it between swim and swim.

EDIT: Also, some books by Hilbert are breathtakingly beautiful: Geometry and the Imagination (just the chapter on synthetic differential geometry is worth more than 10 other great books), and the Methods of Mathematical Physics is also great. It begins by giving three proofs of cauchy-schwartz inequality, and then goes on to give several different definitions of the eigenvectors of a matrix. Both of those make great beach readings for this summer.

Concrete Mathematics by Knuth and Patashnik (already mentioned for u/pmiller2) if the kid likes numbers. That's perhaps the guiding thread of the book -- it's about the beautiful (yet usually very elementary and natural) things you can do with numbers.

Geometry Revisited by Coxeter and Greitzer and/or Episodes in Nineteenth and Twentieth Century Euclidean Geometry by Honsberger if the kid is into plane geometry. It's an idyllic subject, great for independent exploration, and the books shouldn't take long to read. Not very deep, though (at least Honsberger).

Anything by Tom Körner, just because of the writing. Seriously, he can make the axiomatic construction of the real number system read like a novel; open https://web.archive.org/web/20190813160507/https://www.dpmms... on any page and you will see.

Proofs from the BOOK by Aigner and Ziegler is a cross-section of some of the nicest proofs in reasonably elementary (read: undergrad-comprehensible) maths. Might be a bit too advanced, though (the writing is terse and a lot of ground is covered).

Problems from the BOOK by Andreescu and Dospinescu (a play on the previous title, which itself is a play on an Erdös quote) is an olympiad problem book; it might be one of the best in its genre.

Oystein Ore has some nice introductory books on number theory (Number Theory and its History) and on graphs (Graphs and their uses); they should be cheap now due to their age, but haven't gotten any less readable.

Kvant Selecta by Serge Tabachnikov is a 3(?)-volume series of articles from the Kvant journal translated into English. These are short expositions of elementary mathematical topics written for talented (and experienced) high-schoolers.

I wouldn't do Princeton Companion; it's a panorama shot from high orbit, not a book you can really read and learn from.

>What would you have appreciated having been given at that age?

I remember getting God Created The Integers when I was a teenager and... not finishing it. I also got a copy of Brown & Churchill's Complex Variables and Applications and spent hundreds of hours on it. As a teenager, I preferred textbooks with problem sets to popularizations. (I still do.) Of course, this was [complex] analysis, so it doesn't qualify.

One book which is fully technical but also entertaining by way of the subject matter, and which was inspiring to me around 14-15, was Kenneth Falconer's Fractal Geometry:

https://www.amazon.com/Fractal-Geometry-Mathematical-Foundat...

Of course, at that age, I didn't understand what Falconer meant by describing the Cantor set as "uncountable", or what a "topological dimension" was, but I was able to grasp the gist of many of the arguments in the book because it is very well illustrated and does not rely too much on abstruse algebra techniques. Some people don't enjoy reading a book if they don't fully understand it, but I liked that kind of thing. As I got older and learned more, I started to be able to understand the technical arguments in the book as well.