cube of a numberCubeOfANumber2005-03-08 19:08:412008-10-14 10:42:50Definitiongeneral associativitycube function% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe {\em cube of a number} $x$ is the third \PMlinkname{power}{GeneralAssociativity} $x^3$ of $x$.\, Similarly one may speak of the cube of an element $x$ in any semigroup with the operation denoted multiplicatively (cf. general associativity).
The volume of a cube (i.e. \PMlinkname{regular}{RegularPolyhedron} \PMlinkname{hexahedron}{Hexahedron}) with \PMlinkescapetext{edge length} $a$ is $a^3$; hence the name.
The {\em cube function}\, $x\mapsto x^3$\, from $\mathbb{R}$ to $\mathbb{R}$ is injective, but not as a mapping from $\mathbb{C}$ to $\mathbb{C}$; one has\, $x^3 = y^3$\, always when\, $\frac{x}{y} = \frac{-1\pm i\sqrt{3}}{2}$,\, the \PMlinkescapetext{primitive} third root of unity.