Class 10 Mathematics Chapter 1Real Numbers

Prove √2 is Irrational

1. Assume √2 = a/b (Rational).
2. Square it: 2b² = a². So 2 divides a.
3. Let a = 2c. Then 2b² = 4c² → b² = 2c².
4. So 2 divides b.
5. Contradiction! 2 divides both. Assumption Wrong.

Prove 5 – √3 is Irrational

• 1 Assume 5 – √3 = a/b (Rational).
• 2 Rearrange: √3 = 5 – a/b.
• 3 RHS (5 – a/b) is Rational. LHS (√3) is Irrational.
• 4 Impossible. Thus, Irrational.