Prove that, √5 is an irrational number.

Prove that, √5 is an irrational number.

Solution:

Let us consider √5 be a rational number, then
√5 = p/q, where ‘p’ and ‘q’ are integers, q ≠ 0 and p, q have no common factors (except 1).
So,
5 = p² / q²
p² = 5q² …. (1)
As we know, ‘5’ divides 5q², so ‘5’ divides p² as well. Hence, ‘5’ is prime.
So 5 divides p
Now, let p = 5k, where ‘k’ is an integer
Square on both sides, we get
p² = 25k²
5q² = 25k² [Since, p² = 5q², from equation (1)]
q² = 5k²
As we know, ‘5’ divides 5k², so ‘5’ divides q² as well. But ‘5’ is prime.
So 5 divides q
Thus, p and q have a common factor 5. This statement contradicts that ‘p’ and ‘q’ has no common factors (except 1).
We can say that, √5 is not a rational number.
√5 is an irrational number.
Hence proved.

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