Irrationals forming rationals Can we obtain every rational number from the multiplication of two irrational numbers? If not, which ones can we not obtain?
Let $r$ be a non-zero rational number.  Then $\pi$ and $r/\pi$ are irrational, and $\pi \cdot \frac{r}{\pi}=r$ is rational.  To see why $r/\pi$ is irrational, suppose for the sake of a contradiction that it were rational, so $r/\pi=r'$, where $r'\in \mathbb{Q}$; in fact, $r'$ is non-zero since the quotient of two non-zero numbers is non-zero.  But then $r=r'\pi$ and $r/r'=\pi$.  The quotient of two rational numbers is rational, so this would show that $\pi$ is rational, which is not the case.  Thus, $r/\pi$ must be irrational.

However, the same cannot be said about $0$, for if $ab=0$, then at least one of $a$ and $b$ must also be $0$.  The above proof that $r/\pi$ is irrational fails because in this case $r'=0$, and we could not divide by it.
Take the square root of a rational.  It's likely irrational.  Multiply that square root by itself, you have a rational.  Thus you have a big set of irrational * irrational = rational numbers.