Question

Prove that root 2 is an irrational number

Answer 1

Let √2 be a rational number.

A rational number can be written in the form of p/q.

√2 = p/q

p = √2q

Squaring on both sides,

p²=2q²

2 divides p² then 2 also divides p.

So, p is a multiple of 2.

p = 2a (a is any integer)

Put p=2a in p²=2q²

(2a)² = 2q²

4a² = 2q²

2a² = q²

2 divides q² then 2 also divides q.

Therefore,q is also a multiple of 2.

So, q = 2b

Both p and q have 2 as a common factor.

But this contradicts the fact that p and q are co primes.

So our supposition is false.

Therefore, √2 is an irrational number.

Hence proved.

Hope it helps

Answer 2

Hey there !

Lets assume that √2 is rational.

let ,
√2 = p/q , where p and q are integers , q ≠ 0 , and " p and q are co prime "

squaring both the sides ;

2 = p²/q²
2q² = p²

here ,
2 dividies p².
so , 2 divides p                  -------> [1 ]

p = 2m

2q² = p²

2q² = [2m]²
2q² = 4m²
q² = 2m

here ,
2 divides q².
so ,
2 divides q       ----> [2]

from [1] and [2] its clear that 2 is a common factor of p and q .
this contradicts our assumption that p and q are co prime .

hence , 
our assumption was wrong .

∴ √2 is irrational