## Square Root of 5 is Irrational

\[\sqrt{5}\]

is rational. Let \[\sqrt{5}= \frac{m}{n}\]

, where this fraction cannot be simplified.. Then \[\sqrt{5}= \frac{ \sqrt{5}( \sqrt{5} -2)}{\sqrt{5}-2}= \frac{5-2 \sqrt{5}}{\sqrt{5}-2}=\frac{5-2m/n}{m/n-2}=\frac{5m-2n}{m-2n}\]

Then

\[m-2n=( \sqrt{5}-2)n\]

and \[0 \lt \sqrt{5}=2 \lt 1 \]

so \[0 \lt m-2n \lt n\]

. The assumption that \[\sqrt{5}\]

is rational leads to s simpler rational fraction, but we assumed that \[\frac{m}{n}\]

was already as simple as possible - a contradiction, so \[\sqrt{5}\]

is irrational.