Question

$\frac{1}{\sqrt{2}}$ is

(a) a fraction (b) a rational number

(c) an irrational number (d) none of these

Answer 1

To prove $\frac{1}{\sqrt{2}}$ is irrational

Let us assume that √2 is irrational

$\frac{1}{\sqrt{2}}$ = $\frac{p}{q}$(where p and q are co prime)

$\frac{p}{q}$ = √2
q = √2p

squaring both sides

q² = 2p² .....................(1)

By theorem
q is divisible by 2

∴ q = 2c ( where c is an integer)

putting the value of q in equitation 1

2p² = q² = 2c² =4c²
p² =$\frac{4c^2}{2}$ = 2c²
$\frac{p^2}{2}$ = c²

by theorem p is also divisible by 2

But p and q are coprime

This is a contradiction which has arisen due to our wrong assumption

∴1/√2 is irrational