Difference between revisions of "Complex number"
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Whereas the real numbers can be represented as all the possible points on an infinitely extended [[number line]], to represent all the complex numbers requires the use of a two dimensional coordinate system, usually with the real components on the horizontal axis (the ''abscissa'') and the imaginary components on the vertical axis (the ''ordinate''). | Whereas the real numbers can be represented as all the possible points on an infinitely extended [[number line]], to represent all the complex numbers requires the use of a two dimensional coordinate system, usually with the real components on the horizontal axis (the ''abscissa'') and the imaginary components on the vertical axis (the ''ordinate''). | ||
| − | The complex numbers form an [[algebraic closure|algebraically closed]] [[field (mathematics)|field]] but do not permit a non-trivial ordering that is preserved under operations. | + | The complex numbers form an [[algebraic closure|algebraically closed]] [[field (mathematics)|field]] but do not permit a non-trivial ordering that is preserved under operations. They are the algebraic closure of the [[real numbers|real numbers]]. |
===In popular culture=== | ===In popular culture=== | ||
Revision as of 15:53, July 1, 2007
A complex number is a number composed of two parts - a real component and and imaginary component, of the form <math>a + bi</math>, where a and b are real numbers and i is the square root of negative 1.
Whereas the real numbers can be represented as all the possible points on an infinitely extended number line, to represent all the complex numbers requires the use of a two dimensional coordinate system, usually with the real components on the horizontal axis (the abscissa) and the imaginary components on the vertical axis (the ordinate).
The complex numbers form an algebraically closed field but do not permit a non-trivial ordering that is preserved under operations. They are the algebraic closure of the real numbers.
In popular culture
In Yvgeny Zamyatin's satirical novel We, the narrator's psychological distress at contemplating the concept of complex numbers becomes a metaphor for the limitations of totalitarian systems of thought.