prove that ✓5 irational
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Answer:
Step-by-step explanation:
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.
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