Question

prove that 1/√2 is irrational​

Answer 1

To prove:

• 1/√2 is irrational

Proof:

Let us assume that √2 is irrational 

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

$\frac{q}{p} = \sqrt{2} \\ \\ q = \sqrt{2} p$

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

$2p² = q² = 2c² =4c² \\ \\ p² =4c² /2 = 2c² \\ \\ p²/2 = c²$

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

$\: \: \: \: \: \: \: \: \: \: \huge \bold{@WildCat7083 }$