Last updated at March 23, 2020 by Teachoo

Transcript

Ex 1.3 , 2 Prove that 3 + 2 root 5 โ5 is irrational. We have to prove 3 + 2 root 5โ5 is irrational Let us assume the opposite, i.e., 3 + 2โ5 is rational Hence, 3 + 2โ5 can be written in the form ๐/๐ where a and b (bโ 0) are co-prime (no common factor other than 1) Hence, 3 + 2โ5 = ๐/๐ 2โ5 = ๐/๐ - 3 2โ5 = (๐ โ 3๐)/๐ โ5 = 1/2 ร (๐ โ 3๐)/๐ โ5 = (๐ โ 3๐)/2๐ Here, (๐ โ 3๐)/2๐ is a rational number But โ5 is irrational Since, Rational โ Irrational This is a contradiction โด Our assumption is incorrect Hence 3 + 2โ5 is irrational Hence proved

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.