rationalize the denominatorIs there any theorem saying that in below fraction we can't rationalize the denominator 2 π I couldn't find any way and I don't think there is but I was wondering if this actually proved? for example for 1 2 we say 1 2 = 1 × 2 2 × 2 = 2 2 result have meaning if we look at fraction like Division

Question

rationalize the denominatorIs there any theorem saying that in below fraction we can&#039;t rationalize the denominator      2    π  I couldn&#039;t find any way and I don&#039;t think there is but I was wondering if this actually proved? for example for       1          2       we say      1          2        =            1      ×              2                            2            ×              2              =            2        2  result have meaning if we look at fraction like Division

benedictazk · Accepted Answer

Are you speaking of rationalizing the denominator and the numerator at the same time? If not, a trivial solution would be             2              /            π        1  , but that doesn&#039;t seem very interesting. So perhaps you are speaking about some sort of manipulation that creates       2    π    =      p    q  , where both   p and   q  are rational. But, I would find that greatly disturbing, because then it would be the case that       p    q   is also rational... in which case       2    π   would be rational as well.EDIT: Your example with       2   is similar to the trivial solution above, but, in the case of a radical, might have some more use.So, yes, you can rationalize the denominator, if you find such action useful. But, would you, in the case of       2    π  ?