Question

how to prove root 2 is rational or irrerational

Answer 1

Let √2 be a rational number.A rational number can bewritten in the form of p/q.√2 = p/qp = √2qSquaring on both sides,p²=2q²2 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 = 2q²4a² = 2q²2a² = q²2 divides q² then 2 also divides q.Therefore,q is also a multiple of 2.So, q = 2bBoth 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

Sol:
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.