Square number facts for kids

A square number, also known as a perfect square, is a number you get when you multiply an integer (a whole number) by itself. For example, 1, 4, 9, 16, and 25 are the first five square numbers.

When we write a square number in a formula, we use a small "2" above the number, like n². This is called an exponentiation and is usually read as "n squared". The name "square number" comes from the shape of a square, as you'll see below.

Square numbers are always positive or zero. Another way to think about a square number is that its square root is also a whole number. For instance, the square root of 9 is 3 (√9 = 3), so 9 is a square number.

What are Square Numbers?

Square numbers are the result of multiplying a whole number by itself. Imagine you have a certain number of small blocks. If you can arrange these blocks to form a perfect square shape, then the total number of blocks you have is a square number!

Here are some examples of square numbers:

• 0 multiplied by 0 is 0 (0² = 0)
• 1 multiplied by 1 is 1 (1² = 1)
• 2 multiplied by 2 is 4 (2² = 4)
• 3 multiplied by 3 is 9 (3² = 9)
• 4 multiplied by 4 is 16 (4² = 16)
• 5 multiplied by 5 is 25 (5² = 25)

There are endless square numbers because there are endless natural numbers (counting numbers) that you can multiply by themselves.

How Square Numbers Look

The name "square number" makes sense when you see them! You can arrange a square number of items into a perfect square shape.

1 = 12
4 = 22
9 = 32
16 = 42
25 = 52

If a square shape has a side length of n units, its area (the space it covers) is n² square units.

Interesting Facts About Square Numbers

Sum of Odd Numbers

You can also get a square number by adding up a series of odd numbers. For example:

• 1 = 1²
• 1 + 3 = 4 = 2²
• 1 + 3 + 5 = 9 = 3²
• 1 + 3 + 5 + 7 = 16 = 4²
• 1 + 3 + 5 + 7 + 9 = 25 = 5²

This pattern shows that if you add the first n odd numbers together, you will always get n².

Last Digits of Square Numbers

When you look at square numbers, you'll notice that their last digit can only be 0, 1, 4, 5, 6, or 9.

• If a number ends in 0, its square will end in at least two 0s (like 10² = 100).
• If a number ends in 1 or 9, its square will end in 1 (like 11² = 121 or 19² = 361).
• If a number ends in 2 or 8, its square will end in 4 (like 12² = 144 or 18² = 324).
• If a number ends in 3 or 7, its square will end in 9 (like 13² = 169 or 17² = 289).
• If a number ends in 4 or 6, its square will end in 6 (like 14² = 196 or 16² = 256).
• If a number ends in 5, its square will end in 25 (like 15² = 225).

Even and Odd Square Numbers

• Even numbers squared are always even. In fact, they are always divisible by 4. For example, (2 × 3)² = 6² = 36, and 36 is divisible by 4.
• Odd numbers squared are always odd. For example, (2 × 3 + 1)² = 7² = 49, which is an odd number.

This means that if you have an even square number, its square root must be an even number. If you have an odd square number, its square root must be an odd number.

Special Ways to Calculate Squares

There are some cool tricks for squaring numbers, especially those ending in 0 or 5, or numbers around 50.

Numbers Ending in 5

If a number ends in 5, like 65, you can find its square easily:

1. Take the digit(s) before the 5 (in 65, this is 6).
2. Multiply that digit by the next whole number (6 × 7 = 42).
3. Put "25" at the end of that result.

So, 65² = 4225.

Numbers Ending in 0

If a number ends in 0, like 70, you can find its square quickly:

1. Take the digit(s) before the 0 (in 70, this is 7).
2. Square that number (7² = 49).
3. Put "00" at the end of that result.

So, 70² = 4900.

Two-Digit Numbers Starting with 5

If a two-digit number starts with 5, like 57, you can find its square:

1. Take the last digit (in 57, this is 7).
2. Add it to 25 (25 + 7 = 32). This gives you the first two digits of the answer.
3. Square the last digit (7² = 49). This gives you the last two digits of the answer.

So, 57² = 3249.

Images for kids

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  This animation shows how the sum of the first n odd numbers equals n². For example, 1 + 3 + 5 + ... + (2n − 1) = n².

See also

In Spanish: Cuadrado perfecto para niños