Solution: Real numbers are simply the combination of rational and irrational numbers, in the number system. In general, all the arithmetic operations can be performed on these numbers and they can be represented in the number line.

Let ‘a’ be any positive integer.

Then,

According to Euclid’s division lemma,

a=bq+r

According to the question, when b = 4.

a = 4k + r, n < r < 4

When r = 0, we get, a = 4k 

a² = 16k² = 4(4k²) = 4q, where q = 4k²

When r = 1, we get, a = 4k + 1

a² = (4k + 1)² = 16k² + 1 + 8k = 4(4k + 2) + 1 = 4q + 1, where q = k(4k + 2)

When r = 2, we get, a = 4k + 2

a² = (4k + 2)² = 16k² + 4 + 16k = 4(4k² + 4k + 1) = 4q, where q = 4k² + 4k + 1

When r = 3, we get, a = 4k + 3

a² = (4k + 3)² = 16k² + 9 + 24k = 4(4k² + 6k + 2) + 1

= 4q + 1, where q = 4k² + 6k + 2

Therefore, the square of any positive integer is either of the form 4q or 4q + 1 for some integer q.