Prove that 1 2 is irrational. - Vidyadip

Question

Prove that $\frac{1}{\sqrt{2}}$ is irrational.

✓

Answer

Let us assume, to the contrary, that is $\frac{1}{\sqrt{2}}$ rational.
So, we can find coprime integers a and $b (\neq 0)$ such that
$\frac{1}{\sqrt{2}}=\frac{a}{b} \Rightarrow \sqrt{2}=\frac{b}{a}$
Since, a and b are integers, $\frac{b}{a}$ is rational, and so is $\sqrt{2}$ rational
But this contradicts the fact that is $\sqrt{2}$ irrational.
So, we conclude that is $\frac{1}{\sqrt{2}}$ irrational.

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