prove that 1/√5 is irrational
Answers
Answer:
Let, 1/√5 be a rational number,
Therefore, 1/√5 = a/b ---- ( where a and b are co primes , b is not equal to 0 )
=> √5 = b/a ---- ( b/a is rational as a and b are integers )
here,
Let , √5 be a rational number.
Therefore , √5 = a/b ----(a and be are co primes and b is not equal to 0)
squaring both sides,
=> 5 = a square/b square
=> 5 × b square = a square
=> a square / 5
=> a/5 (i) -------( let p and m be any two natural no. , therefore, if p divides m square then it will divide m also )
Similarly,
let , a = 5c
squaring both sides ,
=> a square = 25 c square
=> 5 b square = 25 c square
dividing both sides by 5,
=> b square = 5 c square
=> b square / 5
=> b / 5 (ii)
From (i) and (ii) we get that 5 is also a factor of a and b , which contradicts that a and b are co primes . So our assumption is incorrect . Hence √5 is an irrational number .
From above we conclude that LHS contradicts RHS , as √5 is an irrational number and b/a is rational. Hence , 1/√5 is an irrational number
Answer:
Therefore it is an irrational number
