Question

Prove that $4-5\sqrt{2}$ is an irrational number.

Answer 1

Consider, 4−5√2
Let 4−5√2 = (a/b) a rational number
⇒ −5√2 = (a/b) − 4
⇒ −5√2 = (a − 4b)/b
⇒ √2 = (a − 4b)/(−5b)
Since a, b are integers, then (a − 4b)/(−5b) represents a rational number.
But this is a contradiction since RHS is a rational number where as LHS (√2) is an irrational number
Hence our assumption that " 4−5√2 = (a/b) is a rational number" is incorrect.
Thus 4−5√2 is an irrational number