Question

prove that 1 upon root 5 is irrational

Answer 1

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 2

Step-by-step explanation:

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