Thread: Possible Rationalization?

I'm curious to know if there exists a way to rationalize this:


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

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 , okay- multiply numerator and denomonator by ! That gives the fraction . Of course, that is still equal to - you're not changing the whole number so it is still irrational but now the numerator is irrational, and the denominator, 1, is rational.

Just a few comments.

1. If we rule out the option that SHY GUY is a complete idiot, then obviously he means "rationalize the denominator". Maybe we should give students the benefit of the doubt.

2. If, when asked to "simplify" , a student were to answer:

, how would that fare for him? Let's try and understand the motivation a little here, folks...

Now for my answer:
"NO".
Pi is transcendental. Our methods for rationalizing squares, cubes, etc relies on the factorization of certain polynomials with INTEGER coefficients.