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:A cube number, or a perfect cube, or sometimes just a cube, is a number which is the cube of an integer.The perfect cubes up to 603 are (sequence A000578 in the OEIS):Every positive rational number is the sum of three positive rational cubes, and there are rationals that are not the sum of two rational cubes.In real numbers, the cube function preserves the order: larger numbers have larger cubes. In other words, cubes (strictly) monotonically increase. Also, its codomain is the entire real line: the function x ↦ x3 : R → R is a surjection (takes all possible values). Only three numbers are equal to their own cubes: −1, 0, and 1. If −1 < x < 0 or 1 < x, then x3 > x. If x < −1 or 0 < x < 1, then x3 < x. All aforementioned properties pertain also to any higher odd power (x5, x7, …) of real numbers. Equalities and inequalities are also true in any ordered ring.Determination of the cubes of large numbers was very common in many ancient civilizations. Mesopotamian mathematicians created cuneiform tablets with tables for calculating cubes and cube roots by the Old Babylonian period (20th to 16th centuries BC). Cubic equations were known to the ancient Greek mathematician Diophantus. Hero of Alexandria devised a method for calculating cube roots in the 1st century CE. Methods for solving cubic equations and extracting cube roots appear in The Nine Chapters on the Mathematical Art, a Chinese mathematical text compiled around the 2nd century BCE and commented on by Liu Hui in the 3rd century CE. The Indian mathematician Aryabhata wrote an explanation of cubes in his work Aryabhatiya. In 2010 Alberto Zanoni found a new algorithm to compute the cube of a long integer in a certain range, faster than squaring-and-multiplying.