Question

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

Answer 1

LET 1/√2 BE RATIONAL NO. A
SO,1/√2=A
1/A=√2
SO,1 IS A RATIONAL NO. AND A IS ALSO A RATIONAL NO.AS WE ASSUME.
BUT ON ANOUTHER SIDE IS √2 WHICH IS A
IRRATIONAL NO.
SO, IT IS A CONTRADICTION SO OUR ASSUMTION THAT 1/√2 IS A RATIONAL NO. IS
WRONG.
SO,1/√2 IS IRRATIONAL.

Answer 2

$to \: prove \: \frac{1}{ \sqrt{2} } \: is \: irrational \: number$
let us assume that 1/root 2 is rational.
1/root2 = p/q ( p and q are co prime )
q/p = root2
q= (root2)p
squaring on bothsides
q^2 = 2p^2 ...... (i)

by theorem, q is divisible by 2,
so , q = 2c ( where c is an integer )

put q =2c in (i)

(2c)^2 = 2p^2
4c^2 = 2p^2
2c^2 = p^2
p^2 = 2c^2
p^2/2 = c^2

by therom p is also divisible by 2
But p and q are co prime .
This is contradiction. Hence our assumption is wrong.
Therefore , 1/root 2 is irrational number.