Hello ladies and gentlemen! A friend of mine and I have been thinking about this particular issue: under what circumstances is the product of two irrational numbers rational?
For example, multiplying $\sqrt{2}$ by any nonzero rational multiple of $\sqrt{2}$ or its inverse we obtain a rational number. Moreover, whenever we multiply an irrational number by a rational multiple of its inverse we obviously obtain a rational number.
I've also thought of the following case: if $q= \frac{m}{n}$ is in lowest terms where m and n are not k-th powers of integers, $\large q^{\frac{k-l}{k}} \cdot q^{\frac{l}{k}}$ is also rational.
In that direction, what is the best we know? i.e., is it known exactly when the product of two irrational numbers is rational? Or even, is it known exactly when the product of two transcendental numbers is algebraic?
I am a freshman and lessons started just one week ago so please be tolerant :) References are welcome and appreciated. Thanks in advance.
If $a$ is any rational number, and $b$ is any irrational number, then $c=a/b$ is irrational (it's pretty easy to prove that; I can give specifics if necessary) so the product of the two irrational numbers $b$ and $c$ is rational. And every case of a product of two irrational numbers being rational is an instance of exactly that situation, as you'll see if you think it through.