Prove there is no rational number which squares to 2.

by RoRi

July 1, 2015

Prove that there is no $r \in \mathbb{Q}$ such that $r^2 = 2$ .

Proof. Suppose otherwise, that there is some rational number $r \in \mathbb{Q}$ such that $r^2 = 2$ . Then, since $r \in \mathbb{Q}$ , we know there exist integers $a,b \in \mathbb{Z}$ not both even (I.3.12, Exercise #10 (e) such that $r = \frac{a}{b}$ . Then, since $r^2 = 2$ we have

$2 = \frac{a^2}{b^2} \quad \implies \quad 2b^2 = a^2.$

But, by I.3.12, Exercise #10 (d), we know $2b^2 = a^2$ implies both $a$ and $b$ are even, contradicting our choice of $a$ and $b$ not both even. Hence, there can be no such rational number $. \qquad \blacksquare$

2 comments

February 11, 2016 at 12:05 pm

martin says:

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- February 11, 2016 at 12:13 pm
  
  RoRi says:
  
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Stumbling Robot

Prove there is no rational number which squares to 2.

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