Question

prove that √5 is irrational number. ​

Answer 1

Answer:

Proof:

Let us assume that √5 is a rational number.

Sp it t 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

Answer 2

Answer:

To prove :

• √5 is an irrational number.

Proof :

Let √5 be a rational number.

Then,

p/q =√5 [ It can be put into p/q where p & q are integers, q ≠ 0]

➛ (p/q)² = (√5)²

➛ p²/q² = 5

➛ p²/q = 5q

➛ p² = 5 [ Equation holds true only when q = 1 ]

This is a contradiction since there is no integer whose square is 5.

.°. √5 is an irrational number.

Hence, Proved!