Question
Prove that $5 \sqrt{2}$ is irrational.
✓
Answer
Let us assume that $5 \sqrt{2}$ is rational.
Then, there exist positive co$-$primes $a$ and $b$ such that
$5 \sqrt{2}=\frac{a}{b}$
$\sqrt{2}=\frac{a}{b}-5$
$\sqrt{2}=\frac{a-5 b}{b}$
As $a-5 b$ and $b$ are integers .
So, $\frac{a-5 b}{b}$ is rational number .
But $\sqrt{2}$ is not rational number.
Since a rational number cannot be equal to an irrational number.
Our assumption that $5 \sqrt{2}$ is rational wrong.
Hence, $5 \sqrt{2}$ is irrational.
Then, there exist positive co$-$primes $a$ and $b$ such that
$5 \sqrt{2}=\frac{a}{b}$
$\sqrt{2}=\frac{a}{b}-5$
$\sqrt{2}=\frac{a-5 b}{b}$
As $a-5 b$ and $b$ are integers .
So, $\frac{a-5 b}{b}$ is rational number .
But $\sqrt{2}$ is not rational number.
Since a rational number cannot be equal to an irrational number.
Our assumption that $5 \sqrt{2}$ is rational wrong.
Hence, $5 \sqrt{2}$ is irrational.
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