Question

prove that square root 5 is not a rational number​

Answer 1

Let us consider √5 is rational.

So,

√5 = p/q.

(where p and q are co-prime number and q ≠ 0)

Squaring on both sides give,

5 = p²/q²

5q² = p²

From this we can say that 5 divides p² so 5 will also divide p.

So, 5 is one of the factor of p.

So we can write,

p = 5a

Therefore,

5q² = (5a)²

5q² = 25a²

q² = 5a²

From this we can say that 5 divides q² so 5 will also divide q.

So, 5 is one of the factor of q.

As, we know p and q are co-prime so it cannot have common factor. But here a contradiction arise that 5 is factor of both p and q.

So, by this we can say that √5 is not rational which means √5 is irrational.

Hope

this

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u

mate..

Answer 2

let root 5 be rational then it must in the form of p/q [q is not equal to 0][p and q are co-prime] root 5=p/q => root 5 * q = p squaring on both sides => 5*q*q = p*p ------> 1 p*p is divisible by 5 p is divisible by 5 p = 5c [c is a positive integer] [squaring on both sides ] p*p = 25c*c --------- > 2 sub p*p in 1 5*q*q = 25*c*c q*q = 5*c*c => q is divisble by 5 thus q and p have a common factor 5 there is a contradiction as our assumsion p &q are co prime but it has a common factor so √5 is an irrational

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