If you want to recall what are perfect square numbers then please refer to my article on What are the perfect square numbers. This article asks a question that how many natural numbers lie between two consecutive perfect square numbers. You can also ask the same question as how many non-perfect square numbers lie between two consecutive perfect square numbers? Can we come up with a general formula for that?

Lets write some perfect square numbers.

There are 2n natural numbers lying between two consecutive perfect square numbers and . We can see bunch of examples.

Lets choose two consecutive perfect square numbers. Lets say that we have chosen and . We can see that there are two numbers (2 and 3) which lie between and . Note that

If, we choose and . We can see that there are 4 numbers (5, 6, 7, 8) which lie between and . Note that .

If, we choose and . We can see that there are 6 numbers (10, 11, 12, 13, 14, 15) which lie between and . Note that

How many natural numbers lie between and ?

(A) 2257

(B) 2222

(C) 2356

(D) 2500

(E) None of these

We know that there are 2n natural numbers which lie between two consecutive perfect square numbers and . Therefore, between and , there are natural numbers. Therefore, the answer is (B).

**Practice Problems:**

- How many natural numbers lie between and ?
- How many non-perfect square numbers lie between and ?
- Do you know how many natural numbers lie between any two given natural numbers? How many natural numbers are present between 1555 and 3520?

**Answer:** For any two given natural numbers n and m where n>m. There are (n-m-1) natural numbers between n and m. Therefore, between 1555 and 3520, there are 3520-1555-1=1964 natural numbers.