# (日本語) 互いに素なピタゴラス数が無限に存在することの証明

Sorry, this entry is only available in Japanese.

# Proof: $$\sqrt{2}$$ is an irrational number

I suppose that you know definition of rational number. Irrational number is real number but is not rational number.

Here, I prove that $$\sqrt{2}$$ is a irrational number.

## Proof

Suppose $$\sqrt{2}$$ a rational number, then it can be represented with coprime natural numbers , $$m, n$$ , as the following

$\sqrt{2} = \frac{m}{n}$

Now,

$2n^2 = m^2 .$

Left hand $$2 n^2$$ is even, so $$m^2$$ is also even. Therefore $$m$$ can be represented as $$m = 2k$$ , with the natural number $$k$$ , and

\begin{array}[crcl] & & 2 n ^2 & = & ( 2k )^2 \\ & & = & 4k^2 \\ \Leftrightarrow & n^2 & = & 2k^2 \end{array}

Therefore $$n$$ is also even, but it is against the supposition that $$m, n$$ are coprime. Above all, $$\sqrt{2}$$ is a irrational number.