*p*/

*q*, where

*p*and

*q*are relatively prime. Then

(

*p*/

*q*)

^{2}= 2

so

*p*

^{2}= 2

*q*

^{2}

so

*p*

^{2}has 2 as a factor, which means that

*p*must have 2 has a factor. So

*p*=2

*r*for some

*r*.

We now have

(2*r*)^{2} = 2 *q*^{2}

so

4*r*^{2} = 2 *q*^{2}

so

2*r*^{2} = *q*^{2}

and therefore *q* has 2 as a factor.

So both *p* and *q* have 2 as a factor, but we assumed that they were relatively prime, which is a contradiction.

The kids didn't really think much of this one. They seem take it for granted that √2 is irrational, so why do they need a proof. Or maybe they don't like proof by contradiction. (Maybe they are constructionist mathematicians...)

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