Question

Prove that 5√2 is irrational.

Answer 1

Let 5√2 is a rational number.

∴ 5√2 = \(\frac{p}{q}\), where p and q are some integers and HCF(p, q) = 1 …(1)

⇒5√2q = p 

⇒(5√2q)² = p² 

⇒ 2(25q²) = p² 

⇒ p² is divisible by 2 

⇒ p is divisible by 2 ….(2) 

Let p = 2m, where m is some integer. 

∴5√2q = 2m 

⇒(5√2q)² = (2m)² 

⇒2(25q²) = 4m² 

⇒25q² = 2m² 

⇒ q² is divisible by 2 

⇒ q is divisible by 2 ….(3) 

From (2) and (3) is a common factor of both p and q, which contradicts (1). 

Hence, our assumption is wrong. 

Thus, 5√2 is irrational.