Ex 1.2, 3 (iii) - Chapter 1 Class 10 Real Numbers

Transcript

Ex 1.2, 3 Prove that the following are irrationals : (iii) 6 + √2 We have to prove 6 + √2 is irrational Let us assume the opposite, i.e., 6 + √𝟐 is rational Hence, 6 + √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, 6 + √𝟐 = 𝒂/𝒃 √2 = 𝑎/𝑏 − 6 √𝟐 = (𝒂 − 𝟔𝒃)/𝒃 Here, (𝑎 −6𝑏)/𝑏 is a rational number But √5 is irrational Since, Rational ≠ Irrational This is a contradiction ∴ Our assumption is incorrect Therefore, 6 + √𝟐 is irrational Hence proved.