Question
Prove that $\frac{1}{{\sqrt 2 }}$ is irrational.
✓
Answer
We can prove $\frac{1}{{\sqrt 2 }}$ irrational by contradiction.
Let us suppose that $\frac{1}{{\sqrt 2 }}$is rational.
It means we have some co-prime integers a and b (b ≠ 0)
Such that
$\frac{1}{{\sqrt 2 }}$= $\frac ab$
$\Rightarrow $ $\sqrt 2 = \frac{b}{a}$..........(1)
R.H.S of (1) is rational but we know that is$\sqrt 2 $ irrational.
It is not possible which means our supposition is wrong.
Therefore,$\frac{1}{{\sqrt 2 }}$can not be rational.
Hence, it is irrational.
Let us suppose that $\frac{1}{{\sqrt 2 }}$is rational.
It means we have some co-prime integers a and b (b ≠ 0)
Such that
$\frac{1}{{\sqrt 2 }}$= $\frac ab$
$\Rightarrow $ $\sqrt 2 = \frac{b}{a}$..........(1)
R.H.S of (1) is rational but we know that is$\sqrt 2 $ irrational.
It is not possible which means our supposition is wrong.
Therefore,$\frac{1}{{\sqrt 2 }}$can not be rational.
Hence, it is irrational.
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