Question

Prove that root 5 is irrational number ​

Answer 1

Answer:

Let us assume that √5 is a rational number. So it can be expressed in the form p/q where p,q are co-prime integers and q≠0. ⇒ √5 = p/q. On squaring both the sides we get, ⇒5 = p²/q²

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Answer 2

Answer:

Let 5 be a rational number.

then it must be in form of  qp  where,  q=0     ( p and q are co-prime)

5=qp

5×q=p

Suaring on both sides,

5q2=p2           --------------(1)

p2 is divisible by 5.

So, p is divisible by 5.

p=5c

Suaring on both sides,

p2=25c2         --------------(2)

Put p2 in eqn.(1)

5q2=25(c)2

q2=5c2

So, q is divisible by 5.

.

Thus p and q have a common factor of 5.

So, there is a contradiction as per our assumption.

We have assumed p and q are co-prime but here they a common factor of 5.

The above statement contradicts our assumption.

Therefore, 5 is an irrational number.

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