how to prove  square root of 2 is a irrational no. ?

Asked by Prachi | 5th Jul, 2017, 08:44: PM

Expert Answer:

begin mathsize 16px style Let space square root of 2 space is space straight a space rational space number.
rightwards double arrow square root of 2 equals straight a over straight b
rightwards double arrow 2 equals straight a squared over straight b squared
rightwards double arrow straight a squared equals 2 straight b squared
rightwards double arrow straight a squared space is space divisible space by space 2
rightwards double arrow straight a space is space also space divisible space by space 2 space space space space space space space space space..... left parenthesis straight I right parenthesis
Let space straight a equals 2 straight c
rightwards double arrow straight a squared equals 4 straight c squared
rightwards double arrow 2 straight b squared equals 4 straight c squared
rightwards double arrow straight b squared equals 2 straight c squared
rightwards double arrow straight b squared space is space divisible space by space 2
rightwards double arrow straight b space is space also space divisible space by space 2 space space space space space space space space space..... left parenthesis straight I right parenthesis
From space left parenthesis straight I right parenthesis space and space left parenthesis II right parenthesis comma space we space get space straight a space and space straight b space both space are space divisible space by space 2.
straight i. straight e. space straight a space and space straight b space have space straight a space common space factor space 2.
This space contradicts space our space assumption space that space straight a over straight b space is space rational comma
straight i. straight e. space straight a space and space straight b space do space not space have space any space common space factor space other space than space unity.
rightwards double arrow straight a over straight b space is space not space rational
rightwards double arrow square root of 2 space is space irrational. end style

Answered by Rashmi Khot | 6th Jul, 2017, 10:07: AM