Question

√5is irrational prove it​

Answer 1

Answer:

hope it is clear to you please mark as brilliant answer

Step-by-step explanation:

Given: √5

We need to prove that √5 is irrational

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:

p=5m for some positive integer m . Now p>q>m , so q,m is a smaller pair of integers whose quotient is √5 , contradicting our hypothesis. So our hypothesis that √5 can be represented by p/q for some integers p and q is false. That is, √5 is irrational.