[Math] when product of irrational numbers = rational number

Question

let $a$ and $b$ be irrational numbers. when do we have $ a \cdot b $ = rational number?

for example $\sqrt{2} \cdot \sqrt{2}=2$. I was wondering if there some conditions for the product to be a rational number.

Answer 1

I'm not sure what you're hoping for in terms of an answer, but: exactly when one is a rational multiple of the other's reciprocal!

Ex: $\sqrt{2}\times {3\over \sqrt{2}}$ is rational, but $\sqrt{2}\times{\pi\over\sqrt{2}}$ is irrational.

You might find this unsatisfying - unfortunately, I'm not really sure a satisfying characterization exists!

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A somewhat abstract necessary condition can be stated using the language of fields: $\mathbb{Q}(x)=\mathbb{Q}(y)$. However, this is definitely not sufficient, since e.g. $\pi^2$ isn't rational.