Question

5
$\sqrt{5} is irrational prove$

Answer 1

Let us assume that √5 is a rational


where a and b are co prime,

so,

b=a

So,

squaring both sides, we get5b2 = a2.


Therefore a2 is dividable by 5 and alsoa is divisible by 5.

Now we can write a=5c for some integer c.

Substituting this value of for a, in (1) we get

5.


i.e :

This means that is divisible by 5 and also b is divisible by 5. Therefore a and b have 5 as their common factor, but this contradicts the fact that a and b are co prime. The contradiction arises because of our wrong assumption that is a rational.

So, we conclude that is an irrational.