Question

Prove that 1 / root 2 is an irrational number without using the fact that root 2 is irrational

Answer 1

Answer:

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Answer 2

Proof is given below.

Step-by-step explanation:

Suppose that the $\frac{1}{\sqrt{2} }$ is a rational no. Then, it can be written in $\frac{a}{b}$ form.

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

Now take the reciprocal of above equation.

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

But, we know that $\sqrt{2}$ is an irrational no.

So, a rational no. can not be equal to an irrational no.

So, we can not take the  $\frac{1}{\sqrt{2} }$ as an rational no.

Hence, it is an irrational no.