How Many Numbers Lie Between Squares of 12 and 13

Introduction

Before we can find out how many numbers are there between squares 12 and 13, we need to first understand what a square is and how to find it. After doing so, we can look into all the different ways we can find the numbers between squares 12 and 13.

Understanding the Square of a Number

When we multiply a number by itself, the number we get is considered a square of the original number. When two numbers are multiplying, then it is represented as follows

\begin{aligned}

&\text{n}^2 = \text{n} \times \text{n} = \text{m}

\end{aligned}

In the above equation, squaring a number is represented by the number itself with a superscript of 2 on it (n2). In the above equation, ‘m’ is the square of the number ‘n’. Inversely, ‘n’ is the square root of the number ‘m’.

Let's take a look at some examples.

Example 1 - Find the square of number 2.

\begin{aligned}

&2^2 = 2 \times 2 = 4

\end{aligned}

Example 2 - What is the square of number 100?

\begin{aligned}

&100^2 = 100 \times 100 = 10000

\end{aligned

Following is a list of squares of numbers from 1-20

Calculating total numbers between squares of 12 and 13

Before calculating the total numbers between the squares of 12 and 13, let us find the square first. According to the table above, we can see that the square of 12 is 144 and of 13 is 169.

We can also find it using mathematics. It would look as follows

\begin{aligned}

&12^2 = 12 \times 12 = 144\\

\\

&13^2 = 13 \times 13 = 169

\end{aligned}

Now that we know the squares of 12 and 13 we can find the numbers between them. There are three methods of doing this.

Method 1

We can simply count the numbers between 144 and 169. Following are the numbers between 144 and 169

\begin{aligned}

&145, 146, 147, 148, 149, 150, 151,152, 153, 154, 155, 156,\\

&157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168

\end{aligned}

Notice that 144 and 169 are not on the list because we are looking for the numbers in between them, which will not include them.

If we count them individually, we will find out that there are 24 numbers in between the squares 12 and 13.

Method 2

The second method is using basic mathematics. We will be using the following formula

\begin{aligned}

& \text{n} -\text{m}-1

\end{aligned}

If we substitute the square of 12 and the square of 13 with ‘n’ and ‘m’, we get the following.

\begin{aligned}

&13^2 -12^2-1\\

&= (13\times13)-(12\times12) - 1\\

&= 169 -144 -1\\

&=25 - 1\\

&=24

\end{aligned}

Method 3

For the third method lets solve a small fun equation.

Let's think of the number ‘a’. The number which comes after it is going to be ‘a+1’.

In method 2 we saw that we can find total numbers between two numbers by the following formula

\begin{aligned}

& \text{n} -\text{m}-1

\end{aligned}

If we substitute ‘n’ with square ‘a+1’ and ‘m’ with the square of ‘a’, we get the following equation.

\begin{aligned}

&\text{(a+1)}^2 -\text{a}^2-1\\

&= (\text{a}^2 + 1 + \text{2a}) -\text{a}^2 - 1\\

&= 2 \text{a}

\end{aligned}

So we can conclude that if you want to find the total numbers between two numbers which are next to each other, multiply the smaller number by 2.

In the case of the total number between the squares 12 and 13, 12 is the smaller number.

\begin{aligned}

& 12 \times 2 = 24

\end{aligned}

We again end up with the number 24.

NOTE: Remember that method 3 only applies to consecutive numbers that come one after the other. For example, 12 and 13, 56 and 57, 89 and 90, etc.