In response to your actual question, I’d actually guess that the construction of the real numbers might not be the best thing to talk about with 15-year-olds. When I started this blog, I might have felt differently, but it’s since become clear to me that a lot of things I felt like blogging about then may not have been truly interesting enough to be worth the effort. The things that get people interested in math are not its technical constructions, tedious fact-checking, or eternal search for ‘correct’ definitions — these all have their place, but appreciating them requires to some extent thinking like a mathematician. Instead, people are captivated by its surprises (every polynomial over the complex numbers has a root) and its metaphors (the complex numbers behave like a plane, which can be translated by addition and dilated and rotated by multiplication).

I’d advise instead showing your fifteen-year-olds Cantor’s proof that the real numbers are uncountable. This is a beautiful argument with a lot going for it: they learn that we can talk about infinity rigorously, but that it doesn’t work the way we might expect it to; that the very way things are written down can have surprising consequences; that sometimes we have to put things in weird places or orderings to study them correctly; and that the real numbers are a lot more complicated than we might give them credit for. It’s also a good example of a proof by contradiction (explain carefully that the proof doesn’t give you a new real number to add to the list, but that it shows that no such list can exist), and of a non-constructive proof (it shows that irrational numbers exist without actually constructing any; as a bonus, if you show that the algebraic numbers are countable, the same argument shows that transcendental numbers exist, but actually finding them is much harder than finding algebraic irrational numbers). And as some additional cheap entertainment, you can throw in the gory details of Cantor’s life, how he was blackballed by his own advisor for his controversial work, went in and out of insane asylums, etc.

Your book sounds interesting and I’d love to read it when it comes out (has it already?). I don’t know about the other sciences, but math still seems to have a hard time attracting women, which is a true disservice to both math and the women who could be doing it. I’d be curious to hear if you have any ideas on how to fix this problem.