Square Numbers in Maths: A Simple Explanation

If you’re studying maths, you’re likely to come across the term “ square number ” at some point.

But what exactly is a square number?

Put simply, a square number is a number that is the result of an integer being multiplied to itself.

For example, 3 \times 3 = 9 , so 9 is a square number .

Understanding square numbers is important in maths, as they are used in a variety of calculations.

For instance, they are often used in geometry, where they can be used to calculate the area of a square. They are also used in algebra, where they can be used to solve equations . By learning about square numbers, you’ll be able to better understand these and other mathematical concepts.

What Is A Square Number?

Simply put, a square number is the product of multiplying an integer by itself.

For example, 4 is a square number because it is the product of 2 \times 2 , while 9 is a square number because it is the product of 3 \times 3 .

Square numbers get their name from the fact that they can be arranged in the shape of a square.

For instance, if you have 4 square tiles, you can arrange them in a square shape with 2 tiles on each side. Similarly, if you have 9 square tiles, you can arrange them in a square shape with 3 tiles on each side. This is why square numbers are often referred to as perfect squares.

Here are the first twenty square numbers:

Square Number Product of Integer
1 1 \times 1
4 2 \times 2
9 3 \times 3
16 4 \times 4
25 5 \times 5
36 6 \times 6
49 7 \times 7
64 8 \times 8
81 9 \times 9
100 10 \times 10
121 11 \times 11
144 12 \times 12
169 13 \times 13
196 14 \times 14
225 15 \times 15
256 16 \times 16
289 17 \times 17
324 18 \times 18
361 19 \times 19
400 20 \times 20
First twenty square numbers

Square numbers are important in maths because they have many practical applications. For example, they are used in geometry to calculate the area of a square or rectangle. They are also used in algebra to solve equations and in computer science to calculate the size of data structures.

How To Find Square Numbers

To find a square number, you need to multiply a number by itself.

For example, to find the square of 5 , you would multiply 5 \times 5 , which gives you 25 . This means that 25 is a square number.

Here are some steps to help you find square numbers:

  • Choose a number that you want to square.
  • Multiply the number by itself.
  • The result is the square number.

For example, to find the square of 9 , you would multiply 9 by 9 , which gives you 81 . This means that 81 is a square number.

You can also use a calculator to find square numbers. Most calculators have a square button that you can press to find the square of a number.

Properties of Square Numbers

When you multiply an integer by itself, the resulting product is called a square number or a perfect square. Square numbers have some unique properties that make them easy to identify.

  • Square numbers always end with the digits 0, 1, 4, 5, 6, 9 . Have a look at the list of the first 20 square numbers above, notice the last numbers?
  • The number of zeros at the end of a square number is always even.
  • If a number has 1 or 9 in the unit’s place, then its square ends in 1.
  • The square of a number ending with 4 and 6 will always end with 6 at the unit place.

Applications of Square Numbers

If you think square numbers are just a concept you learn in math class and never use again, think again! Square numbers have many real-world applications in fields like architecture, engineering, and computer science.

One use of square numbers is in determining the area of a square. As we learned earlier, the area of a square is the product of its two sides, which are always equal. So, if you know the length of one side of a square, you can find its area by squaring that length. For example, if a square has a side length of 5 meters, its area is 5 \times 5 = 25 square meters.

Square numbers also come into play in the construction of buildings and other structures. When designing a building, architects often use square numbers to determine the dimensions of rooms and other spaces. This is because square shapes are often more aesthetically pleasing and easier to work with than irregular shapes.

In computer science, square numbers are used in algorithms that involve searching through large amounts of data. For example, a common algorithm for searching through a sorted list of numbers involves comparing the target number to the square of the middle number in the list. This allows the algorithm to quickly narrow down the search space and find the target number more efficiently.

Finally, square numbers also have applications in physics, particularly in the study of waves and vibrations. In these fields, square numbers are used to represent the energy of a wave or vibration, which is proportional to the square of its amplitude.


Now that you know what a square number is, you can use this knowledge to solve various mathematical problems. Square numbers are the results you get when you multiply any integer by itself.

For example, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 , and so on, are all square numbers.

Knowing what a square number is can be particularly helpful when working with quadratic equations, as you can use this knowledge to complete the square and solve for the roots of the equation.

Additionally, understanding square numbers is essential for understanding square roots. A square root is the inverse of a square number, meaning it is the number that, when multiplied by itself, gives the original square number.

For example, the square root of 16 is 4 , because 4 multiplied by itself equals 16 .

Overall, understanding square numbers is a fundamental concept in mathematics that can help you solve many problems. Keep practicing and exploring the world of mathematics to continue building your knowledge and skills.