Question

prove that √2 is a irrational number

Answer 1

let √2 = a / b wher a,b are integers b ≠ 0
we also let that a / b is written in the simplest form
⇒ √2 = a / b ⇒ 2 = a2 / b2 ⇒   2b2 = a2
∴ 2b2 is divisible by 2
⇒  a2 is divisible by 2    
⇒  a is divisible by 2  
∴ let a = 2c
a2 = 4c2 ⇒ 2b2 = 4c2 ⇒ b2 = 2c2
∴ 2c2  is divisible by 2
∴ b2  is divisible by 2
∴ b  is divisible by 2
∴a are b   are  divisible by 2 .
this contradicts our notion that a/b is written in the simplest form
as our notion is wrong
∴ √2 is irrational number.

Answer 2

Let √2 be a rational number
a rational number can be written in the form of p/q.
√2=p/q
squaring on both sides
2=p²/q²
2q²=p²
2 divides p² then 2 also divides p
p=2a. (definition of even number,a is a positive integer)
put in
2q²=p²
2q²=(2a)²
4a²=2q²
q²=2a²
2 divides q² then 2 also divides q
p and q have 2 as common factor.
But this contradicts the fact that p and q are co primes.
our supposition is false.
√2 is an irrational number
hence proved

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