Proofs that √2 and √3 are Irrational

• Prove √2 is irrational using contradiction (NCERT proof).
• Extend the method to show √3 is irrational.
• Understand why terminating/non-terminating repeating decimals are rational.

1. Irrational number — cannot be written as p/q in lowest terms (q ≠ 0).
2. Proof by contradiction — assume √2 = a/b in lowest terms; square → both a, b even — contradiction.
3. √3 proof — assume √3 = a/b; 3b² = a² → 3 divides a; follow similar steps.
4. Sum/product — sum of rational and irrational is irrational (e.g. 5 + √2).
5. Decimal view — irrational decimals are non-terminating non-repeating.

Prove √2 is irrational (NCERT outline)

Assume √2 = a/b in lowest terms, b ≠ 0.
Then 2b² = a² → a² even → a even → a = 2c.
2b² = 4c² → b² = 2c² → b even.
Both a, b even contradicts lowest terms. Hence √2 is irrational.

• Assuming √2 = a/b without requiring coprime a, b.
• Claiming √4 is irrational (√4 = 2 is rational).
• Thinking all square roots are irrational (√9 = 3 is rational).

• Define irrational number.
• Why must a/b be in lowest terms in the proof?
• Is 2 + √5 rational or irrational? Why?

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