Question

prove that 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.

Answer 2

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