How to study the irrational numbers with a high school math background?

Question

Recently a friend posed the question "can the product of two irrational numbers be rational?" We the trivial answers like for example $\sqrt{2}\sqrt{8} = 4$. I have become somewhat obsessed with the question and I would like to ask if anyone would have an idea on what field(s) of mathematics that one could pursue in order to reason and investigate this question further?

Answer 1

From the standpoint of research math this is too basic to fall into any one area, but if you're curious, I would call it algebraic number theory ultimately.

But that's just for curiosity's sake. Here are some concrete directions to go that will both be extremely rewarding and challenging for your mathematical pursuits, and squarely hit the question you are asking:

• Elementary number theory - there are a lot of books. It's a huge field and intro books are more concerned with the integers, but these are the tools to investigate this question. Even some challenging books are accessible with just high school math. I find number theory extremely rewarding because it's starting point is the thing you have the most real-world intuition for, and it's shocking how deep and rich and mathematical studying something so obvious as 1,2,3... can be. It's a good choice.
• Galois Theory by Ian Stewart. It's surprisingly accessible given the field, but you will need to know linear algebra (and I believe nothing else past high school math). It's worth mentioning for its relevance to the topic, and for the fact that you can get to it relatively soon in your mathematical career, but it's too difficult for a high school math background. Possibly cool to just flip through and admire that if you use lots and lots of complicated polynomials, you can in fact prove $\pi^2$ is irrational.
• Calculus by Michael Spivak. Yes, that exact book. This seems unrelated but is an introduction to rigorous math in general, including rigorously talking about real numbers, polynomials and rational numbers. Probably a very good choice, and probably the best topically for your subsequent study in math.

Basically, go number theory for something more "elective" feeling, Calculus for something more "core track" feeling. I strongly recommend whichever you find more fascinating but just barely able to absorb. You can also likely find a less difficult number theory book than Calculus is, so feel free to explore that direction.