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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?

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Excuse the question, but what is there to "investigate" in this question further?? –  DonAntonio Mar 24 at 17:50
I suggest getting Ivan Niven's book Numbers: Rational and Irrational. –  Dave L. Renfro Mar 24 at 18:06
The set of irrational numbers is simply nowhere near closed under multiplication. If you give me any irrational numbers $x_1,\dots,x_k$, either their product is already rational, or I can give you an irrational number $y$ such that $x_1\cdots x_ky$ is rational (for example, $y=1/x_1\cdots x_k$). The question is like asking "can the concatenation of two non-words be a word?". –  Greg Martin Mar 24 at 18:23
@DonAntonio I think the OP is curious about mathematics and asking how they can educate themselves to get to the point where they can conclude there is nothing to investigate. (This is a much better question in my opinion than the standard copy and paste of homework along with the instruction to show our working neatly.) –  TooTone Mar 24 at 20:04
@TooTone Exactly! I'm merely interested in looking in to the matter myself and seeing what I find. Unfortunately I didn't now exactly where to start with my somewhat limited understanding of mathematics (which I hope to improve on) –  Tephra Mar 24 at 22:15

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