Question

Show that 2- root 2 is irrational number

Answer 1

Let 2-√2 be rational.

Then, 2-√2 = p/q, where p and q are integers, q≠0, p and q are co-primes.

Squaring both sides,

(2-√2)² = (p/q)²

2² + (2√2)² - 2(2)(2√2) = p²/q²

4 + 8 - 8√2 = p²/q²

-8√2 = p²/q² - 4 - 8

-8√2 = p²/q² - 12 = (p² - 12)/(q²)

√2 = (p² - 12)/(q²)(-8) = (-p² + 12)/(8q²)

In the above equation, clearly LHS is an irrational value while RHS is a rational value. Thus, LHS ≠ RHS.

Thus, our supposition is wrong.

∴ 2 - √2 is irrational.