Question

prove that ✓5 is irrational

Answer 1

let us assume that √5 is rational number
√5=a/b. where a, b are Co primes
√5=a/b,b√5=a. ....1
squaring on both sides
(b√5)2=a2
b2*5=a2
b2=a2/5
5 divides a2,and also a
put a=5c
b√5=5c
squaring on both sides
b2*5=25c2
b2=5c2
c2=b2/5
5 divides b2 and b also
this is a contradiction
our assumption is wrong
given statement is true

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

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