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    Question

    Prove that √2 is irrational. (NCERT Class 10, Real Numbers – Very Important)

    Answer: We prove by contradiction. Assume √2 is rational, so √2 = a/b where a, b are integers, b ≠ 0, and gcd(a,b)=1. Squaring: 2 = a²/b² ⇒ a² = 2b². Thus a² is even, so a is even. Let a=2k. Then 4k²=2b² ⇒ b²=2k², so b is even. Both a and b even contradict gcd(a,b)=1. Hence √2 is irrational.

    Question

    Prove that √2 is irrational. (NCERT Class 10, Real Numbers – Very Important)

    Answer: We prove by contradiction. Assume √2 is rational, so √2 = a/b where a, b are integers, b ≠ 0, and gcd(a,b)=1. Squaring: 2 = a²/b² ⇒ a² = 2b². Thus a² is even, so a is even. Let a=2k. Then 4k²=2b² ⇒ b²=2k², so b is even. Both a and b even contradict gcd(a,b)=1. Hence √2 is irrational.