Cube Numbers List

What this tool does

So, you want the cube numbers set out in a list. This tool builds them. Enter how many you want, and it lists the first n cube numbers, showing each as a number multiplied by itself three times alongside its result.

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

How to use it

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

What a cube number is

A cube number is what you get when you multiply a whole number by itself three times. So 8 is a cube number because it is 2 times 2 times 2, and 27 is a cube number because it is 3 times 3 times 3. Written with an exponent, the cube of a number is that number raised to the power 3. The first few cube numbers are 1, 8, 27, 64, 125, and they climb quickly. This tool works through the whole numbers in order, cubing each, and lists the results so you can watch the sequence grow.

Why they are called cubes

Just as a square number pictures a flat square, a cube number pictures a solid cube. If you take a cube number of little unit blocks, you can stack them into a perfect cube: eight blocks make a two-by-two-by-two cube, twenty-seven make a three-by-three-by-three cube, and so on. A cube number is the volume of a cube: a cube whose edge is 4 units holds 64 unit cubes, which is 4 cubed. Because a cube reaches into all three dimensions, cube numbers grow far faster than square numbers do, which is easy to see when the two lists sit side by side.

A remarkable fact about cubes

Cubes hide one of the most surprising patterns in all of number theory. Add up the first few cube numbers and something magical happens: the total is always a perfect square. Not just any square, but the square of the sum of the plain whole numbers you cubed. For example, 1 plus 8 plus 27 is 36, and 36 is 6 squared, where 6 is 1 plus 2 plus 3. So the sum of the first n cubes equals the square of the sum of the first n whole numbers. It is the kind of neat, unexpected link that jumps out only when you have the cubes listed in front of you.

A worked example

Ask for the first 5 cube numbers. The tool cubes each whole number in turn: 1 cubed is 1, 2 cubed is 8, 3 cubed is 27, 4 cubed is 64, and 5 cubed is 125. So the list is 1, 8, 27, 64, 125. Compare how much faster these rise than the squares 1, 4, 9, 16, 25, the difference the extra dimension makes.

Cubes keep their sign

There is one way cubes behave differently from squares that is worth knowing. A square is never negative, because multiplying a negative by a negative gives a positive. A cube, though, keeps the sign of the number you started with: a positive number cubed stays positive, and a negative number cubed comes out negative, since three negatives multiply to a negative. This list runs through the positive whole numbers, so every entry here is positive, but it is useful to remember that cubes, unlike squares, can be negative. To recover the edge length from a volume, the cube root calculator reverses cubing, and for the flat, two-dimensional relatives of these numbers, see the square numbers list.

Questions people ask

What is a cube number?

A whole number multiplied by itself three times. For example, 27 is a cube number because it is 3 times 3 times 3.

Why are they called cubes?

Because that many unit blocks stack into a perfect cube, and a cube number is the volume of a cube with a whole-number edge.

What is special about adding cubes?

The sum of the first n cubes equals the square of the sum of the first n whole numbers. For instance, 1 plus 8 plus 27 is 36, which is 6 squared.

Do cubes grow faster than squares?

Yes, much faster, because cubing reaches into three dimensions rather than two.

What is the two-dimensional version?

Square numbers, a number multiplied by itself. The square numbers list generates those.

References

On cube numbers. A cube number is a whole number of the form n cubed, and the first are 1, 8, 27, 64, 125.

1. Eric W. Weisstein, "Cubic Number," from MathWorld, a Wolfram resource, on cubic numbers as figurate numbers of the form n cubed.
2. "Sequence A000578 (the cubes)," The On-Line Encyclopedia of Integer Sequences, listing 1, 8, 27, 64, 125 and their properties.