prove that √5 is an irrational numbers
Answers
ANSWER
We have to show that
5
is irrational.
We will prove this via the method of contradiction.
So let's assume
5
is rational.
Hence, we can write
5
in the form
b
a
, where a and b are co-prime numbers such that a,b,∈R and b
=0.
∴
5
=
b
a
squaring both sides we have
⇒5=
b
2
a
2
⇒5b
2
=a
2
⇒
5
a
2
=b
2
Hence, 5 divides a
2
Now, a theorem tells that if 'P' is a prime number and P divides a
2
then P should divide 'a', where a is a positive number.
Hence, 5 divides a ......(1)
∴ we can say that
5
a
=c
we already know that
⇒5b
2
=a
2
From (2), we know a=5c substituting that in the above equation we get,
⇒5b
2
=25c
2
⇒b
2
=5c
2
⇒
5
b
2
=c
2
Hence, 5 divides b
2
. And by the above mentioned theorem we can say that 5 divides b as well.
hence, 5 divides b .........(3)
So from (2) and (3) we can see that both a and b have a common factor 5. Therefore a&b are no co-prime. Hence our assumption is wrong. ∴ by contradiction
5
is irrational.
Hence, solved.
Step-by-step explanation:
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