>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:
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.
I would give God Created The Integers as a reference to read up to where you are in math, not as something to get through. So, you could get a feel for what Euclid wrote, or what Descartes wrote, either after or during learning those lessons. As you move through your education, you can keep moving through the book, Cauchy, Galois, Riemann, etc. Anyway, that's the context in which I would give it. BTW Cantor's original diagonal proof is in GCTI. :)
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.