Question

Prove that the root 5 is an irrational number​

Answer 1

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

Answer 2

Answer:

Step-by-step explanation:

let √5 is rational

√5= a/b    ( a,b are the co-prime factores of √5)

squaring on both sides

5=a²/b²

b²=a²/5

if a² is divided by 5 then a also divided by

so let a =5k

b²=(5k)²/5

b²=25k²/5

b²=5k²

k²=b²/5

if b² is divided by 5 then  b also divided by 5

so it is not possible

it contradicts that our assumption is wrong

so √5 is an irrational number

hope you understand

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