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.