Tag Archives: Coprime

(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

Prove by contradiction.

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.