Difference between revisions of "Complex number"
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A '''complex number''' is a [[number]] composed of two parts - a [[Real number|real]] component and an [[Imaginary number|imaginary]] component, of the form <math>a + bi</math>, where ''a'' and ''b'' are real numbers and <math>i^2 = -1</math>. | A '''complex number''' is a [[number]] composed of two parts - a [[Real number|real]] component and an [[Imaginary number|imaginary]] component, of the form <math>a + bi</math>, where ''a'' and ''b'' are real numbers and <math>i^2 = -1</math>. | ||
| − | 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''). This representation is known as the Argand diagram. |
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]]. | 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]]. | ||
| − | Many functions used in real analysis can be extended in to complex numbers using [[Taylor series]] | + | Many functions used in real analysis can be extended in to complex numbers using [[Taylor series]]. This is the subject of [[complex analysis]]. |
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| + | ===Polar notation=== | ||
| + | The complex number <math>a+bi</math> can also be written in the form <math>\rho e^{i\theta}</math>, where | ||
| + | : <math>\rho^2=a^2+b^2</math> is the magnitude squared | ||
| + | : <math>\theta=\arctan\frac{b}{a}</math> is the phase | ||
| + | If a line is drawn on the Argand diagram from the origin to a given complex number, the length of that line will be <math>\rho</math> and the angle it makes to the real (horizontal) axis will be <math>\theta</math>. This leads to a straight-forward geometric interpretation for multiplication by a complex number: multiplying a complex number by <math>e^{i\theta}</math> is equivalent to an anticlockwise rotation through an angle <math>\theta</math> in the Argand diagram. | ||
===In popular culture=== | ===In popular culture=== | ||
Revision as of 18:07, May 11, 2008
A complex number is a number composed of two parts - a real component and an imaginary component, of the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a + bi} , where a and b are real numbers and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^2 = -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). This representation is known as the Argand diagram.
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.
Many functions used in real analysis can be extended in to complex numbers using Taylor series. This is the subject of complex analysis.
Polar notation
The complex number can also be written in the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho e^{i\theta}} , where
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^2=a^2+b^2} is the magnitude squared
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta=\arctan\frac{b}{a}} is the phase
If a line is drawn on the Argand diagram from the origin to a given complex number, the length of that line will be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} and the angle it makes to the real (horizontal) axis will be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} . This leads to a straight-forward geometric interpretation for multiplication by a complex number: multiplying a complex number by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{i\theta}} is equivalent to an anticlockwise rotation through an angle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} in the Argand diagram.
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.