Difference between revisions of "Complex number"
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: <math>\rho^2=a^2+b^2</math> is the square of the number's [[Absolute value|magnitude]] | : <math>\rho^2=a^2+b^2</math> is the square of the number's [[Absolute value|magnitude]] | ||
: <math>\theta=\arctan\frac{b}{a}</math> is the phase | : <math>\theta=\arctan\frac{b}{a}</math> is the phase | ||
| − | If a line is drawn on the [[complex plane]] 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 complex plane. | + | If a line is drawn on the [[complex plane]] (also known as an 'Argand diagram' or the 'Argand plane') 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 complex plane. |
===Complex Numbers as Matrices=== | ===Complex Numbers as Matrices=== | ||
The field F on complex numbers is isomorphic to the field F' of 2x2 matrices of the form | The field F on complex numbers is isomorphic to the field F' of 2x2 matrices of the form | ||
Revision as of 18:34, July 3, 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 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} 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 square of the number's magnitude
- 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 complex plane (also known as an 'Argand diagram' or the 'Argand plane') 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 complex plane.
Complex Numbers as Matrices
The field F on complex numbers is isomorphic to the field F' of 2x2 matrices of the form
- [a b]
- [-b a],
with 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} mapping as a function f to the above matrix. We can see that F and F' are isomorphic because: The function f is clearly 1-to-1 and onto, 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 f(x+y)=f(x)+f(y)} , 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 f(x*y)=f(x)*f(y)} .
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.