Let us assume to the contrary that √5 is rational.
Since √5 is rational, it has to be in the form of 'a/b' where 'a' and 'b' are integers such that b ≠ 0.
or
we can assume that
√5 = a/b
Let 'a' and 'b' and have a common factor other than 1 which is 'c'.
Now, divide both numerator and denominator of the fraction a/b by the common factor 'c' and assume the result as x/y.
√5 = (a ÷ c)/(b ÷ c)
√5 = x/y
where 'x' and 'y' are relative prime. That is, 'x' and 'y' don't have a common factor other than 1.
So,
y√5 = x
Squaring both sides,
(y√5)^{2} = x^{2}
5y^{2} = x^{2 }----(1)
Clearly x^{2 }is a multiple of 5.
Since x^{2 }is a multiple of 5, x also should be a multiple of 5.
For example, 225 is a perfect square and it is a multiple of 5.
225 = 5(45)
15^{2} = 5(45)
When 225 is a multiple of 5, its square root value 15 is also is a multiple of 5.
15 = 5(3)
So, x = 5k, for some integer k.
Substitute x = 5k in (1).
5y^{2} = (5k)^{2}
5y^{2} = 25k^{2}
Divide both sides by 5.
y^{2} = 5k^{2}
Clearly y^{2 }is a multiple of 5.
Since y^{2 }is a multiple of 5, y also should be a multiple of 5.
So, y = 5r, for some integer r.
Since both 'x' and 'y' are multiples of 5, it is clear that both 'x' and 'y' have the common factor 5.
But this contradicts the fact that 'x' and 'y' have no common factor other than 1.
This contradiction has arisen because of our incorrect assumption that √5 is rational.
So, we conclude that √5 is irrational.
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