Prove that √7 is an irrational number.

Prove that √7 is an irrational number.

Solution:

Let us consider √7 be a rational number, then
√7 = p/q, where ‘p’ and ‘q’ are integers, q ≠ 0 and p, q have no common factors (except 1).
So,
7 = p² / q²
p² = 7q² …. (1)
As we know, ‘7’ divides 7q², so ‘7’ divides p² as well. Hence, ‘7’ is prime.
So 7 divides p
Now, let p = 7k, where ‘k’ is an integer
Square on both sides, we get
p² = 49k²
7q² = 49k² [Since, p² = 7q², from equation (1)]
q² = 7k²
As we know, ‘7’ divides 7k², so ‘7’ divides q² as well. But ‘7’ is prime.
So 7 divides q
Thus, p and q have a common factor 7. This statement contradicts that ‘p’ and ‘q’ has no common factors (except 1).
We can say that, √7 is not a rational number.
√7 is an irrational number.
Hence proved.

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