prove √5 is irrational
Answers
Answer:
Let
5
be a rational number.
then it must be in form of
q
p
where, q
=0 ( p and q are co-prime)
5
=
q
p
5
×q=p
Suaring on both sides,
5q
2
=p
2
--------------(1)
p
2
is divisible by 5.
So, p is divisible by 5.
p=5c
Suaring on both sides,
p
2
=25c
2
--------------(2)
Put p
2
in eqn.(1)
5q
2
=25(c)
2
q
2
=5c
2
So, q is divisible by 5.
.
Thus p and q have a common factor of 5.
So, there is a contradiction as per our assumption.
We have assumed p and q are co-prime but here they a common factor of 5.
The above statement contradicts our assumption.
Therefore,
5
is an irrational number.
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