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Thread: Possible Rationalization?

  1. #1
    Member >_<SHY_GUY>_<'s Avatar
    Jan 2008

    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
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  2. #2
    MHF Contributor

    Apr 2005
    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.
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  3. #3
    Super Member TheChaz's Avatar
    Nov 2010
    Northwest Arkansas
    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" $\displaystyle 1/\sqrt2$, 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...

    Now for my answer:
    Pi is transcendental. Our methods for rationalizing squares, cubes, etc relies on the factorization of certain polynomials with INTEGER coefficients.
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