Question

prove that
$\sqrt{5 \: }$
Is irrational ​

Answer 1

Answer:

To prove root5 is  irrational

  we would prove it using contradictory  method

contradictory  method: In this method ,first we would assume something and later we prove that our assumption was wrong.This method is known as  contradictory method .

         Let root 5 be a rational number .Hence according to this condition we can say that

root5 = a/b

(where  a and b are integers and co-primes )

thus, a/b = root 5

or, a= b*root5

(square on both sides )

therefore,  a^2 = 5*b^2--------------(equ1)

or, a^2/5 =b^2

if p divides a^2 , then p also divides a

=>  a/5 = c [ where 'c' is another integer]

or, a = 5c ---------------equ(2)

substitute values of equation 2 in equ 1

=> (5c)^2 =5 b^2

=> 25c^2 = 5b^2

or, 5c^2 = b^2

or,c^2 = b^2/5

=> d = b/5 [where 'd' is an integer]

This indicates the fact that 5 is a common factor .This contradicts the fact 'a' and 'b' are not co primes .

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Note:

CO-PRIME : a integer that does not have factors more than 1 and the number itself.

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This case have arisen because of our incorrect assumption that root5 is rational .Hence,it is irrational.

Thanks for reading

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