Question

$prove \: that \: \sqrt{5} \: is \: irrational$
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Answer 1

Answer:

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

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Lets assume √5 is a rational number.

Rational number can be written in the format

$\frac{p}{q}$.

Therefore,

√5 = $\frac{p}{q}$

√5q = p

p is a rational number.

But,

√5q is not a rational number as √5 is irrational number and q is rational number.

Therefore,

it is proved that √5 is a irrational number.

$\huge\underline\mathfrak\orange{Hope\:it\:helps\:you.}$