Question: Prove that \\sqrt5) is irrational.
Explanatory Answer
Let us assume the contrary. i.e., \\sqrt5) is rational.
So, \\sqrt5) = \\frac{a}{b}) where a and b are co-prime integers and b ≠ 0
∴ a = b\\sqrt5)
Squaring both sides, a2 = 5b2 ----------- (1)
Inference 1: So, we can infer that 5 divides a2
Because ‘a’ is a positive integer, if 5 divides a2, 5 will also divide a.
∴ ‘a’ can be written as a multiple of 5 i.e., a = 5c for some integer c
Substitute a = 5c in (1)
25c2 = 5b2 or b2 = 5c2
Inference 2: 5 divides b2.
Because ‘b’ is a positive integer, if 5 divides b2, 5 will also divide b.
5 divides a. 5 divides b
a and b have 5 as common factor.
But we picked ‘a’ and ‘b’ as co-prime integers. So, 'a' and 'b' cannot have 5 as a common factor.
The contradiction is because of our incorrect assumption that \\sqrt5) is rational.
So, we can conclude that \\sqrt5) is irrational.