Question

Prove that √5 is irrational

Answer 1

Let
√5 a rational no.
so we have a and b as co prime no.
That is.
√5=a/b
b√5=a
Squaring both sides we get
5bsq.=a sq.
Therefore. a sq.is divisible 5
Take a=5c
we get
5b sq.=25c sq.
b sq.=5c sq.
That means b sq.is divisible by 5
so b is also divisible by 5
That means a and. b have at least 5 as common factor.

But this contradicts the fact that a and b are co prime
This contradiction has arisen because of our incorrect assumption that √5 is rational.

So √5 is irrational.

Hope you helpful. ...

Answer 2

$\huge \red{\bf Answer}$

We need to prove that √5 is irrational

$\pink{\rm Proof:}$

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

ANSWER

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

$\green{ \bf \sqrt{5} \: is \: a \: irrational \: number}$