Cube (arithmetic)

From Wikipedia, the free encyclopedia

y=x³, for integer values of 1≤x≤25.

A 3x3x3 Rubik's Cube composed of 27 (26) smaller cubes

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 of side length n, giving rise to the name.

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 (sequence A000578 in OEIS):

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...

Different from a square number, there is no smallest cube number, 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 and 75 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 it 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.

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.

Every positive integer can be written as the sum of nine 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 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:

$1^3+2^3+...+n^3 = (\frac{n(n+1)}{2})^2$

For example, the sum of the first five perfect cubes, 1³ + 2³ + 3³ + 4³ + 5³, is equal to sum of the 5th triangular number squared, namely 15² which is 225.

[edit] 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.