Question

prove that root 5 is an irrational number​

Answer 1

Answer:

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.

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

Answer 2

Answer:

let be assume that √5 is a rational no.

∴ it can be written in the form of a/b

√5 = a/b (where a and b are co prime

                 positive integers b ≠ 0)

√5b = a

squaring both side

5b² = a² ⇒ (1)

a² is divisible by 5

∴ a is also divisible by 5

5c = a

squaring both side

25c² = a² ⇒ (2)

from (1) and (2)

25c² = 5b²

5c² = b²

b² is divisible by 5

∴ b is also divisible by 5

∴ 5 is a common factor of a and b

   but it is contradict that a and b are

   co prime therefore our assumption

   is wrong and √5 is an irrational no.

HENCE PROVED

Step-by-step explanation: