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 to calculate the area of a square. They are also used in algebra 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?
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.
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, multiply a number by itself.
For example:
- $5 \times 5 = 25$ → $25$ is a square number
- $9 \times 9 = 81$ → $81$ is a square number
Steps:
- Choose a number
- Multiply it by itself
- The result is a square number
You can also use a calculator. Most calculators have a square ($x^2$) button to compute square values directly.
Properties of Square Numbers
- Square numbers always end in $0, 1, 4, 5, 6, 9$
- The number of zeros at the end is always even
- If a number ends in $1$ or $9$, its square ends in $1$
- If a number ends in $4$ or $6$, its square ends in $6$
Applications of Square Numbers
Square numbers appear in many real-world contexts:
Geometry
To calculate the area of a square, multiply side by side:
- Side length = $5$, Area = $5 \times 5 = 25$
Architecture and Engineering
Square shapes are often more efficient and aesthetically pleasing. Designers use square numbers for layout and structure planning.
Computer Science
Square numbers are used in:
- Optimising data structures
- Algorithms (e.g., binary search variants)
- Memory layouts
Physics
In wave and vibration analysis:
- Energy of a wave ∝ square of its amplitude
Conclusion
Now that you know what a square number is, you can recognise patterns and use them to solve problems.
Examples of square numbers: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, \ldots$
Square numbers also help when:
- Solving quadratic equations
- Understanding square roots
For example:
- $\sqrt{16} = 4$ since $4 \times 4 = 16$
Understanding square numbers is a foundational maths skill that will support you in many areas of mathematics. Keep practicing and exploring!