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Ex 1.3, 2 - Prove that 3 + 2 root 5 is irrational - Ex 1.3

Ex 1.3, 2 - Chapter 1 Class 10 Real Numbers - Part 2

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Ex 1.3 , 2 Prove that 3 + 2 root 5 √5 is irrational. We have to prove 3 + 2 root 5√5 is irrational Let us assume the opposite, i.e., 3 + 2√5 is rational Hence, 3 + 2√5 can be written in the form 𝑎/𝑏 where a and b (b≠ 0) are co-prime (no common factor other than 1) Hence, 3 + 2√5 = 𝑎/𝑏 2√5 = 𝑎/𝑏 - 3 2√5 = (𝑎 − 3𝑏)/𝑏 √5 = 1/2 × (𝑎 − 3𝑏)/𝑏 √5 = (𝑎 − 3𝑏)/2𝑏 Here, (𝑎 − 3𝑏)/2𝑏 is a rational number But √5 is irrational Since, Rational ≠ Irrational This is a contradiction ∴ Our assumption is incorrect Hence 3 + 2√5 is irrational Hence proved

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.