Question

1. Prove that 5 is irrational. ​

Answer 1

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.

Answer 2

Prove that root 5 is irrational number

Given: √5

We need to prove that √5 is irrational

Proof:

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

⇒5q² = p² —————–(i)

p²/5 = q²

So 5 divides p

p is a multiple of 5

⇒ p = 5m

⇒ p² = 25m² ————-(ii)

From equations (i) and (ii), we get,

5q² = 25m²

⇒ q² = 5m²

⇒ q² is a multiple of 5

⇒ q is a multiple of 5

Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√5 is an irrational number.

Hence proved