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Viewing Version
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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: |
2005-03-27 06:04:50 |
| Classification: |
msc:20-00 |
| Synonyms: |
cube of a number=cube
cube of a number=third power |
Revision comment (for changes between this and next version):
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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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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