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The old high school algebra proof that the square root of 2 is irrational can be easily beaten. 

What we want to do is show that any whole positive number that has an integer root that is fractional, actually has an irrational, rather than a rational fraction root. Thus if the nth root of positive integer a is fractional, or not a whole number it is actually irrational.

First assume that any such root when expressed as a fraction must be reduced to lowest terms since that is its simplest form. Thus we have a=b/c. Now express b and c as unique prime factors.

We have b/c= pf(b)/pf(c). None of the prime factors of b can be equal to any of the prime factors of c or b/c would not be reduced to lowest terms. Thus taking b/c to any positive whole integer power leaves the resulting fraction reduced to lowest terms, or not a whole number. From this we can conclude that taking b/c to any positive whole power can never produce a positive whole number, it must always be fractional. Therefore, the root expressed by b/c has to be irrational in order to produce a whole number when taken to a positive whole power.

This seems a lot simpler, and yet much more encompassing than the popular school book proof that the square root of 2 is irrational. It shows, in a few lines that an infinity of these roots are irrational, and uses a very simplistic argument. It also shows that a rational fraction, reduced to lowest terms, and taken to an infinite power, is still never exactly a whole number. In addition a fraction reduced to lowest terms can have an infinite numerator and an infinite denominator and is still not equivalent to an irrational number. This indicates that there are infinitely many more reals than rationals, and this simple thought process might have added to Georg Cantor's original motivation to develop the transfinite numbers.

This. is probably an old well known method, but since it occurred to me while trying to remember the old high school proof and it was so simple and easy to understand I thought that others might be interested and amused to see it resurrected here. It would be wonderful if something simple like this could be found for Fermat's Last Theorem.
Copyright 2008  by Douglas A. Engel
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