Question
Prove that $\frac{2}{\sqrt{7}}$ is irrational.
✓
Answer
Let us assume that $\frac{2}{\sqrt{7}}$ is rational.
Then, there exist positive co$-$primes a and $b$ such that
$\frac{2}{\sqrt{7}}=\frac{a}{b}$
$\sqrt{7}=\frac{2 b}{a}$
As $2b$ and $a$ are rational numbers.
Then $\frac{2 b}{a}$ is rational number.
But $\sqrt{7}$ is not a rational number.
Since a rational number cannot be equal to an irrational number.
Our assumption that $\frac{2}{\sqrt{7}}$ is rational number is wrong .
Hence $\frac{2}{\sqrt{7}}$ is an irrational number
Then, there exist positive co$-$primes a and $b$ such that
$\frac{2}{\sqrt{7}}=\frac{a}{b}$
$\sqrt{7}=\frac{2 b}{a}$
As $2b$ and $a$ are rational numbers.
Then $\frac{2 b}{a}$ is rational number.
But $\sqrt{7}$ is not a rational number.
Since a rational number cannot be equal to an irrational number.
Our assumption that $\frac{2}{\sqrt{7}}$ is rational number is wrong .
Hence $\frac{2}{\sqrt{7}}$ is an irrational number
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