Square Numbers List

What this tool does

So, you want the square numbers laid out in a list, not just one at a time. This tool builds them. Enter how many you want, and it lists the first n square numbers, showing each one as a number multiplied by itself alongside its result.

One input, and a clean run of squares comes back. It handles up to the first 100.

How to use it

1. Enter n, how many square numbers you want, up to 100.
2. Press Calculate.

What a square number is

A square number is what you get when you multiply a whole number by itself. So 4 is a square number because it is 2 times 2, and 9 is a square number because it is 3 times 3. Written with an exponent, the square of a number is that number raised to the power 2. Square numbers are also called perfect squares, and the first few are 1, 4, 9, 16, 25, and so on. This tool marches through the whole numbers in order, squaring each one, and lists the results so you can see the sequence build.

Why they are called square

The name is not decoration; it comes from a real picture. If you take a square number of dots, you can always arrange them into a perfect square grid. Four dots make a two-by-two square, nine dots make a three-by-three square, sixteen dots make a four-by-four square, and so on. In the same spirit, a square number is the area of an actual square: a square whose side is 5 units has an area of 25 square units, which is 5 squared. This link between multiplying a number by itself and the area of a square is exactly why the operation is called squaring.

The hidden odd-number pattern

There is a lovely pattern buried in the list of squares. Look at the gaps between them: from 1 to 4 is 3, from 4 to 9 is 5, from 9 to 16 is 7, from 16 to 25 is 9. The differences are the odd numbers, in order. Put the other way round, each square number is the sum of the odd numbers up to a point: 1 is 1, 1 plus 3 is 4, 1 plus 3 plus 5 is 9, 1 plus 3 plus 5 plus 7 is 16. So adding up the first n odd numbers always gives n squared. Seeing the list all at once makes this pattern jump out, which is one of the nicest reasons to view the squares together.

A worked example

Ask for the first 5 square numbers. The tool squares each whole number in turn: 1 squared is 1, 2 squared is 4, 3 squared is 9, 4 squared is 16, and 5 squared is 25. So the list is 1, 4, 9, 16, 25. Notice the gaps between them, 3, 5, 7, and 9, are the odd numbers in order.

Where square numbers matter

Square numbers turn up all over mathematics. They are the areas of squares, so they appear whenever you measure two-dimensional space. They sit at the heart of the Pythagorean theorem, where the squares of the two shorter sides of a right triangle add up to the square of the longest. They are the values a quadratic produces, and recognising a perfect square makes many algebra problems quicker. To go the other way and find the side length from an area, the square root calculator undoes squaring, and for the three-dimensional cousin of these numbers, see the cube numbers list.

Questions people ask

What is a square number?

A whole number multiplied by itself. For example, 25 is a square number because it is 5 times 5.

Why are they called squares?

Because that many dots form a perfect square grid, and a square number is the area of a square with a whole-number side.

What pattern do the squares follow?

Consecutive squares differ by consecutive odd numbers, so the sum of the first n odd numbers is n squared.

How many can I list?

Up to the first 100 square numbers.

What is the three-dimensional version?

Cube numbers, a number multiplied by itself three times. The cube numbers list generates those.

References

On square numbers. A square number is a whole number of the form n squared, and the first are 1, 4, 9, 16, 25.

1. Eric W. Weisstein, "Square Number," from MathWorld, a Wolfram resource, on square numbers, also called perfect squares, as figurate numbers.
2. "Sequence A000290 (the squares)," The On-Line Encyclopedia of Integer Sequences, listing 0, 1, 4, 9, 16, 25 and their properties.