Last updated at July 17, 2019 by Teachoo

Transcript

Ex 1.3 , 3 Prove that the following are irrationals : 1/โ2 We have to prove 1/โ2 is irrational Let us assume the opposite, i.e., 1/โ2 is rational Hence, 1/โ2 can be written in the form ๐/๐ where a and b (bโ 0) are co-prime (no common factor other than 1) Hence, 1/โ2 = ๐/๐ (๐ )/๐= โ2 " " Here, (๐ )/๐ is a rational number But โ2 is irrational Since, Rational โ Irrational This is a contradiction โด Our assumption is incorrect Hence 1/โ2 is irrational Hence proved Ex 1.3 , 3 Prove that the following are irrationals : (ii) 7โ5 We have to prove 7โ5 is irrational Let us assume the opposite, i.e., 7โ5 is rational Hence, 7โ5 can be written in the form ๐/๐ where a and b (bโ 0) are co-prime (no common factor other than 1) Hence, 7โ5 = ๐/๐ โ5 " = " 1/7 " ร " (๐ )/๐ " " โ5 " = " (๐ )/7๐ Here, (๐ )/7๐ is a rational number But โ5 is irrational Since, Rational โ Irrational This is a contradiction โด Our assumption is incorrect Hence 7โ5 is irrational Hence proved Ex 1.3 , 3 Prove that the following are irrationals : (iii) 6 + โ2 We have to prove 6 + โ2 is irrational Let us assume the opposite, i.e., 6 + โ2 is rational Hence, 6 + โ2 can be written in the form ๐/๐ where a and b (bโ 0) are co-prime (no common factor other than 1) Hence, 6 + โ2 = ๐/๐ โ2 = ๐/๐ - 6 โ2 = (๐ โ6๐)/๐ Here, (๐ โ6๐)/๐ is a rational number But โ5 is irrational Since, Rational โ Irrational This is a contradiction โด Our assumption is incorrect Hence, 6 + โ2 is irrational Hence proved.

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.