Before explaining the shortcuts, let's first understand ,how you will solve such two example as per normal process:
Example 1: Find Natural Numbers between 3^{2} and 4^{2}
Solution: As per the given question:
Given consecutive natural number are 3^{2} and 4^{2}
And to find natural numbers between them, we first need to solve:
3^{2} = 9
4^{2} = 16
Now, here between 9 and 16, we got following natural numbers:
10, 11, 12, 13, 14, 15
Hence, there are 6 natural numbers between 3^{2} and 4^{2}
Example 2: Find Natural Numbers between 6^{2} and 7^{2}
Solution: As per the given question:
Given consecutive natural number are 6^{2} and 7^{2}
And to find natural numbers between them, we first need to solve:
6^{2} = 36
7^{2} = 49
Now, here between 36 and 49, we got following natural numbers:
37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48
Hence, there are 12 natural numbers between 6^{2} and 7^{2}
ShortCut Method
Now, above two examples are solved below in the shortcut ways:
Example 3: Find Natural Numbers between 3^{2} and 4^{2}
Solution: As per the given question:
Given consecutive natural number are 3^{2} and 4^{2}
Or we can write is as:
1st consecutive Natural Number = n = 3 (statement 1)
2nd consecutive Natural Number = n + 1 = 4
Shortcut formula for this:
Number of natural numbers = 2n
= 2 X 3 (see statement 1)
= 6
Hence, there are 6 natural numbers between 3^{2} and 4^{2}
(You can match the answers of example 1 and example 3)
Example 4: Find Natural Numbers between 6^{2} and 7^{2}
Solution: As per the given question:
Given consecutive natural number are 6^{2} and 7^{2}
Or we can write is as:
1st consecutive Natural Number = n = 6 (statement 1)
2nd consecutive Natural Number = n + 1 = 7
Shortcut formula for this:
Number of natural numbers = 2n
= 2 X 6 (see statement 1)
= 12
Hence, there are 12 natural numbers between 6^{2} and 7^{2}
(You can match the answers of example 2 and example 4)
This shortcut formula refers that:
There are 2n natural numbers between squares of two consecutive natural numbers
In other words we can also say that:
There are 2n non perfect squares between squares of natural numbers; (n) & (n+1)."
From comparing examples you can easily observe that you are saved from expanding the power and also calculating each number one by one.
Shortcut is very useful in case of big natural number (as in example 5 below)
Example 5: Find Natural Numbers between 25^{2} and 26^{2}
(If you solve this with normal process, it would very long and time consuming. So use shortcut method only)
Solution: As per the given question:
Given consecutive natural number are 25^{2} and 26^{2}
Or we can write is as:
1st consecutive Natural Number = n = 25 (statement 1)
2nd consecutive Natural Number = 25 + 1 = 26
Shortcut formula for this:
Number of natural numbers = 2n
= 2 X 25 (see statement 1)
= 50
Hence, there are 50 natural numbers between 25^{2} and 26^{2}
