Cube (algebra)

Cube (algebra)

In arithmetic and algebra, the cube of a number n is its third power — the result of the number multiplied by itself twice:

n3 = n × n × n.

This is also the volume formula for a geometric cube with sides of length n, giving rise to the name. The inverse operation of finding a number whose cube is n is called extracting the cube root of n. It determines the side of the cube of a given volume. It is also n raised to the one-third power.

The cube of a number or any other mathematical expression is denoted by a superscript 3, for example 23 = 8 or (x+1)3.

A perfect cube (also called a cube number, or sometimes just a cube) is a number which is the cube of an integer.

The positive perfect cubes up to 603 are (sequence A000578 in OEIS):

Geometrically speaking, a positive number m is a perfect cube if and only if one can arrange m solid unit cubes into a larger, solid cube. For example, 27 small cubes can be arranged into one larger one with the appearance of a Rubik's Cube, since 3 × 3 × 3 = 27.

The pattern between every perfect cube from negative infinity to positive infinity is as follows,

n3 = (n − 1)3 + 3(n − 1)n + 1.

or

n3 = (n + 1)3 − 3(n + 1)n − 1.