Question

Prove that 1÷√2 is irrational​

Answer 1

$\huge\underline\bold\pink{Solution:-}$

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

Let assume $\dfrac{1}{\sqrt2}$ is rational.

So, We can Write as:

$\frac{1}{ \sqrt{2} } = \frac{a}{b} \: .... (1)$

Here a and b are two Co-prime numbers and b is not equal to zero.

Now,

Simplifying the equation (1) multiply by √2 both sides, we get -

$1 = \frac{a \sqrt{2} }{b}$

Dividing by b, we get:-

$b = a \sqrt{2} \: \implies \frac{b}{a} = \sqrt{2}$

Here, a & b are integers so , $\dfrac{b}{a}$ is a rational number, so √2 must be a rational number.

But,

√2 is a irrational number , so it's contradictory.

$\therefore \frac{1}{\sqrt{2} }\:is\:a\:irrational\: number.$

Hence, Proved !