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'cube of a number'
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| Title of object: |
cube of a number |
| Canonical Name: |
CubeOfANumber |
| Type: |
Definition |
| Created on: |
2005-03-08 19:08:41 |
| Modified on: |
2008-03-11 16:49:40 |
| Classification: |
msc:20-00 |
| Defines: |
cube function |
| Synonyms: |
cube of a number=cube
cube of a number=third power |
Revision comment (for changes between this and next version):
| Changes for correction #13631 ('Missing word'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
The {\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} hexahedron) with \PMlinkescapetext{edge length} $a$ is $a^3$; hence the name.
The {\em cube function}\, $x\mapsto x^3$\, from $\mathbb{R}$ to $\mathbb{R}$ 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. |
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