Question

$prove \: that \: rooot5 \: is \: irrational$
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Answer 1

Answer:

heya mate...

There u go :

Given: √5

We need to prove that √5 is irrational

Proof:

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

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,

On squaring both the sides we get,⇒5=p²/q² 

On squaring both the sides we get,⇒5=p²/q² ⇒5q²=p² —————–(i)

On squaring both the sides we get,⇒5=p²/q² ⇒5q²=p² —————–(i)p²/5= q²

So 5 divides p

So 5 divides pp is a multiple of 5

So 5 divides pp is a multiple of 5⇒p=5m

So 5 divides pp is a multiple of 5⇒p=5m⇒p²=25m² ————-(ii)

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

From equations (i) and (ii), we get,5q²=25m²

From equations (i) and (ii), we get,5q²=25m²⇒q²=5m²

From equations (i) and (ii), we get,5q²=25m²⇒q²=5m²⇒q² is a multiple of 5

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

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 .

hope it would help u mate ♡

Answer 2

hope it helps you dear.......

thanks dear a lot...♥️♥️♥️♥️

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