Question

Prove that (4 - 5√2) is irrational.

Answer 1

Let x = 4 - 5√2 be a rational number. 

x = 4 - 5√2 

⇒ x² = (4 - 5√2)² 

⇒ x² = 4² + (5√2)² – 2(4) (5√2) 

⇒ x² = 16 + 50 – 40√2 

⇒ x² – 66 = – 40√2

⇒ \(\frac{66-x^2}{40}\) = √2

 Since x is a rational number, x² is also a rational number. 

⇒ 66 - x² is a rational number 

⇒ \(\frac{66-x^2}{40}\) is a rational number 

⇒√2 is a rational number 

But √2 is an irrational number, which is a contradiction. 

Hence, our assumption is wrong. 

Thus, (4 - 5√2) is an irrational number.