Question

prove that 1/root 2 is irrational

Answer 1

To prove 1/√2 is irrational

Let us assume that √2 is irrational 

1/√2 = p/q (where p and q are co prime)

q/p = √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 =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

Answer 2

GIven,

A number 1/√2.

To Prove,

1/√2 is an irrational number.

Solution,

Let us assume that √2 is a rational number.

1/√2 = p/q (where p and q are coprime)

q/p = √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 =4c² /2 = 2c²

p²/2 = c²

by theorem, p is also divisible by 2

But p and q are coprime

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

Hence, 1/√2 is an irrational number.