# User Contributed Dictionary

### Verb

cubed- past of cube

# Extensive Definition

In arithmetic and algebra, the cube of a number n
is its third power — the
result of multiplying it by itself three times:

- 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.

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

The sequence of non-negative perfect cubes starts
:

- 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 59319, 64000, 68921, 74088, 79507, 85184, 91125, 97736, 103823, 110592, 117649, 125000, 132651, 140608, 148877, 157464, 166375, 175616, 185193, 195112, 205379, 216000, 226981, 238328...

There is no smallest perfect cube, since negative
integers are included. For example,
(−4) × (−4) × (−4) = −64.
For any n, (-n)3 = -(n3).

Unlike perfect
squares, perfect cubes do not have a small number of
possibilities for the last two digits. Except for cubes divisible
by 5, where only 25, 75 and 00 can be the last two digits, any pair
of digits with the last digit odd can be a perfect cube. With
even cubes, there is
considerable restriction, for only 00, o2, e4, o6 and e8 can be the
last two digits of a perfect cube (where o stands for any odd digit and e for any even digit).
Some cube numbers are also square numbers, for example 64 is a
square
number (8 × 8) and a cube number
(4 × 4 × 4);
this happens if and only if the number is a perfect sixth
power.

It is, however, easy to show that most numbers
are not perfect cubes because all perfect cubes must have digital root
1, 8 or 9. Moreover, the digital root of any number's cube can be
determined by the remainder the number gives when divided by 3:

- If the number is divisible by 3, its cube has digital root 9;
- If it has a remainder of 1 when divided by 3, its cube has digital root 1;
- If it has a remainder of 2 when divided by 3, its cube has digital root 8.

Every positive integer can be written as the sum
of nine positive cubes or fewer; see Waring's
problem. This upper limit of nine cubes cannot be reduced
because, for example, 23 cannot be written as the sum of fewer than
nine positive cubes:

- 23 = 23 + 23 + 13 + 13 + 13 + 13 + 13 + 13 + 13.

The number m is a perfect cube if and only if one
can arrange m points in a cube, for example
3 × 3 × 3 =
27.

The sum of the first n perfect cubes is the nth
triangle
number squared:

- 1^3+2^3+...+n^3 = \left(\frac\right)^2.

## History

Determination of the Cube of large numbers was very common in many ancient civilizations. Aryabhatta, the ancient Indian mathematician in his famous work Aryabhatiya explains about the mathematical meaning of cube (Aryabhatiya, 2-3), as "the continuous product of three equals as also the (rectangular) solid having 12 equal edges are called cube". Similar definitions can be seen in ancient texts such as Brahmasphuta Siddhanta (XVIII. 42) , Ganitha sara sangraha (II. 43) and Siddhanta sekhara (XIII. 4). It is interesting that in modern mathematics too, the term "Cube" stands for two mathematical meanings just like in Sanskrit , where the word Ghhana means a factor of power with the number, multiplied by itself three times and also a cubical structure.## External links

cubed in German: Kubikzahl

cubed in Esperanto: Kubo (algebro)

cubed in Spanish: Cubo (aritmética)

cubed in French: Cube (algèbre)

cubed in Korean: 세제곱수

cubed in Japanese: 立方数

cubed in Polish: ³

cubed in Portuguese: Cubo (aritmética)

cubed in Russian: Куб (алгебра)

cubed in Finnish: Kuutio (algebra)

cubed in Swedish: Kub (aritmetik)

cubed in Chinese: 立方數

# Synonyms, Antonyms and Related Words

cube-shaped, cubic, cubiform, cuboid, diced, foursquare, oblong, orthogonal, quadrangular, quadrate, quadriform, quadrilateral, rectangular, rhombic, rhomboid, square, tetragonal, tetrahedral, trapezohedral, trapezoid