Ex 1.2, 3 (i) - Chapter 1 Class 10 Real Numbers (Important Question)

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Ex 1.2, 3 Prove that the following are irrationals : 1/√2 We have to prove 1/√2 is irrational Let us assume the opposite, i.e., 𝟏/√𝟐 is rational Hence, 1/√2 can be written in the form 𝑎/𝑏 where a and b (b≠ 0) are co-prime (no common factor other than 1) Hence, 𝟏/√𝟐 = 𝒂/𝒃 (𝑏 )/𝑎= √2 " " (𝒃 )/𝒂= √𝟐 " " Here, (𝑏 )/𝑎 is a rational number But √2 is irrational Since, Rational ≠ Irrational This is a contradiction ∴ Our assumption is incorrect Hence 1/√2 is irrational Hence proved