Ex 1.3, 3 - Prove irrational :(i) 1/root(2) (ii) 7root(5) - Ex 1.3

  1. Chapter 1 Class 10 Real Numbers
  2. Serial order wise

Transcript

Ex 1.3 , 3 Prove that the following are irrationals : 1/โˆš2 We have to prove 1/โˆš2 is irrational Let us assume the opposite, i.e., 1/โˆš2 is rational Hence, 1/โˆš2 can be written in the form ๐‘Ž/๐‘ where a and b (bโ‰  0) are co-prime (no common factor other than 1) Hence, 1/โˆš2 = ๐‘Ž/๐‘ (๐‘ )/๐‘Ž= โˆš2 " " Here, (๐‘ )/๐‘Ž is a rational number But โˆš2 is irrational Since, Rational โ‰  Irrational This is a contradiction โˆด Our assumption is incorrect Hence 1/โˆš2 is irrational Hence proved Ex 1.3 , 3 Prove that the following are irrationals : (ii) 7โˆš5 We have to prove 7โˆš5 is irrational Let us assume the opposite, i.e., 7โˆš5 is rational Hence, 7โˆš5 can be written in the form ๐‘Ž/๐‘ where a and b (bโ‰  0) are co-prime (no common factor other than 1) Hence, 7โˆš5 = ๐‘Ž/๐‘ โˆš5 " = " 1/7 " ร— " (๐‘Ž )/๐‘ " " โˆš5 " = " (๐‘Ž )/7๐‘ Here, (๐‘Ž )/7๐‘ is a rational number But โˆš5 is irrational Since, Rational โ‰  Irrational This is a contradiction โˆด Our assumption is incorrect Hence 7โˆš5 is irrational Hence proved Ex 1.3 , 3 Prove that the following are irrationals : (iii) 6 + โˆš2 We have to prove 6 + โˆš2 is irrational Let us assume the opposite, i.e., 6 + โˆš2 is rational Hence, 6 + โˆš2 can be written in the form ๐‘Ž/๐‘ where a and b (bโ‰  0) are co-prime (no common factor other than 1) Hence, 6 + โˆš2 = ๐‘Ž/๐‘ โˆš2 = ๐‘Ž/๐‘ - 6 โˆš2 = (๐‘Ž โˆ’6๐‘)/๐‘ Here, (๐‘Ž โˆ’6๐‘)/๐‘ is a rational number But โˆš5 is irrational Since, Rational โ‰  Irrational This is a contradiction โˆด Our assumption is incorrect Hence, 6 + โˆš2 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 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.