Question

Prove that 2^1/2 is irrational.(Please explain step wise because I am a bit confused and I am not able to understand the logic behind it.)

Answer 1

Given √2 is irrational number.

Let √2 = a / b wher a,b are integers b ≠ 0

we also suppose that a / b is written in the simplest form

Now √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 supposition that a/b is written in the simplest form

Hence our supposition is wrong

∴ √2 is irrational number.
Hope this helps.