1. ## Possible Rationalization?

I'm curious to know if there exists a way to rationalize this:
$\displaystyle \frac {2}{\pi}$

But without dividing it by 1 or some number.

I always knew about rationalizing fractions with square roots, cube roots, etc, but that question never crossed my mind.

Is there a way to rationalize that fraction? If not, is there a way to prove that you can't?

Thank you

2. What do you mean by rationalizing it? You talk about rationalizing fractions but you mean rationalizing the numerator or denominator- you can't just change an irrational number to a rational number. If you change the denominator of a fraction from an irrational number to a rational number, the numerator has to become irrational.

If you mean "rationalize the denominator" in $\displaystyle \frac{2}{\pi}$, okay- multiply numerator and denomonator by $\displaystyle \frac{1}{\pi}$! That gives the fraction $\displaystyle \frac{\frac{2}{\pi}}{1}$. Of course, that is still equal to $\displaystyle \frac{2}{\pi}$- you're not changing the whole number so it is still irrational but now the numerator $\displaystyle \frac{2}{\pi}$ is irrational, and the denominator, 1, is rational.

2. If, when asked to "simplify" $\displaystyle 1/\sqrt2$, a student were to answer:
$\displaystyle \frac{1/\sqrt2}{1}$, how would that fare for him? Let's try and understand the motivation a little here, folks...