Editing Complex number (section)

==Applications==
Complex numbers have applications in many scientific areas, including [[signal processing]], [[control theory]], [[electromagnetism]], [[fluid dynamics]], [[quantum mechanics]], [[cartography]], and [[Vibration#Vibration analysis|vibration analysis]]. Some of these applications are described below.

Complex conjugation is also employed in [[inversive geometry]], a branch of geometry studying reflections more general than ones about a line. In the [[Network analysis (electrical circuits)|network analysis of electrical circuits]], the complex conjugate is used in finding the equivalent impedance when the [[maximum power transfer theorem]] is looked for.

===Geometry===
====Shapes====
Three [[collinearity|non-collinear]] points <math>u, v, w</math> in the plane determine the [[Shape#Similarity classes|shape]] of the triangle <math>\{u, v, w\}</math>. Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic as
<math display=block>S(u, v, w) = \frac {u - w}{u - v}. </math>
The shape <math>S</math> of a triangle will remain the same, when the complex plane is transformed by translation or dilation (by an [[affine transformation]]), corresponding to the intuitive notion of shape, and describing [[similarity (geometry)|similarity]]. Thus each triangle <math>\{u, v, w\}</math> is in a [[shape#Similarity classes|similarity class]] of triangles with the same shape.<ref>{{cite journal |last=Lester |first=J.A. |title=Triangles I: Shapes |journal=[[Aequationes Mathematicae]] |volume=52 |pages=30–54 |year=1994 |doi=10.1007/BF01818325 |s2cid=121095307}}</ref>

====Fractal geometry====
[[File:Mandelset hires.png|right|thumb|The Mandelbrot set with the real and imaginary axes labeled.]]
The [[Mandelbrot set]] is a popular example of a fractal formed on the complex plane. It is defined by plotting every location <math>c</math> where iterating the sequence <math>f_c(z)=z^2+c</math> does not [[diverge (stability theory)|diverge]] when [[Iteration|iterated]] infinitely. Similarly, [[Julia set]]s have the same rules, except where <math>c</math> remains constant.

====Triangles====
Every triangle has a unique [[Steiner inellipse]] – an [[ellipse]] inside the triangle and tangent to the midpoints of the three sides of the triangle. The [[Focus (geometry)|foci]] of a triangle's Steiner inellipse can be found as follows, according to [[Marden's theorem]]:<ref>{{cite journal |last1=Kalman|first1=Dan|title=An Elementary Proof of Marden's Theorem |url=http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3338&pf=1 |journal=[[American Mathematical Monthly]] |volume=115 |issue=4 |pages=330–38 |year=2008a |doi=10.1080/00029890.2008.11920532 |s2cid=13222698 |issn=0002-9890 |access-date=1 January 2012 |archive-url=https://web.archive.org/web/20120308104622/http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3338&pf=1 |archive-date=8 March 2012|url-status=live}}</ref><ref>{{cite journal |last1=Kalman |first1=Dan |title=The Most Marvelous Theorem in Mathematics |url=http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=1663 |journal=[[Journal of Online Mathematics and Its Applications]] |year=2008b |access-date=1 January 2012|archive-url=https://web.archive.org/web/20120208014954/http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=1663 |archive-date=8 February 2012 |url-status=live}}</ref> Denote the triangle's vertices in the complex plane as {{math|1=''a'' = ''x''<sub>''A''</sub> + ''y''<sub>''A''</sub>''i''}}, {{math|1=''b'' = ''x''<sub>''B''</sub> + ''y''<sub>''B''</sub>''i''}}, and {{math|1=''c'' = ''x''<sub>''C''</sub> + ''y''<sub>''C''</sub>''i''}}. Write the [[cubic equation]] <math>(x-a)(x-b)(x-c)=0</math>, take its derivative, and equate the (quadratic) derivative to zero. Marden's theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse.

===Algebraic number theory===
[[File:Pentagon construct.gif|right|thumb|Construction of a regular pentagon [[compass and straightedge constructions|using straightedge and compass]].]]
As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in <math>\mathbb{C}</math>. ''[[Argumentum a fortiori|A fortiori]]'', the same is true if the equation has rational coefficients. The roots of such equations are called [[algebraic number]]s – they are a principal object of study in [[algebraic number theory]]. Compared to <math>\overline{\mathbb{Q}}</math>, the algebraic closure of <math>\mathbb{Q}</math>, which also contains all algebraic numbers, <math>\mathbb{C}</math> has the advantage of being easily understandable in geometric terms. In this way, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, more specifically applying the machinery of [[field theory (mathematics)|field theory]] to the [[number field]] containing [[root of unity|roots of unity]], it can be shown that it is not possible to construct a regular [[nonagon]] [[compass and straightedge constructions|using only compass and straightedge]] – a purely geometric problem.

Another example is the [[Gaussian integer]]s; that is, numbers of the form {{math|''x'' + ''iy''}}, where {{mvar|x}} and {{mvar|y}} are integers, which can be used to classify [[Fermat's theorem on sums of two squares|sums of squares]].

===Analytic number theory===
{{main|Analytic number theory}}
Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the [[Riemann zeta function]] {{math|ζ(''s'')}} is related to the distribution of [[prime number]]s.

===Improper integrals===
In applied fields, complex numbers are often used to compute certain real-valued [[improper integral]]s, by means of complex-valued functions. Several methods exist to do this; see [[methods of contour integration]].

===Dynamic equations===
In [[differential equation]]s, it is common to first find all complex roots {{mvar|r}} of the [[Linear differential equation#Homogeneous equation with constant coefficients|characteristic equation]] of a [[linear differential equation]] or equation system and then attempt to solve the system in terms of base functions of the form {{math|1=''f''(''t'') = ''e''<sup>''rt''</sup>}}. Likewise, in [[difference equations]], the complex roots {{mvar|r}} of the characteristic equation of the difference equation system are used, to attempt to solve the system in terms of base functions of the form {{math|1=''f''(''t'') = ''r''<sup>''t''</sup>}}.

=== Linear algebra ===
Since <math>\C</math> is algebraically closed, any non-empty complex [[square matrix]] has at least one (complex) [[eigenvalue]]. By comparison, real matrices do not always have real eigenvalues, for example [[rotation matrix|rotation matrices]] (for rotations of the plane for angles other than 0° or 180°) leave no direction fixed, and therefore do not have any ''real'' eigenvalue. The existence of (complex) eigenvalues, and the ensuing existence of [[Eigendecomposition of a matrix|eigendecomposition]] is a useful tool for computing matrix powers and [[matrix exponential]]s. 

Complex numbers often generalize concepts originally conceived in the real numbers. For example, the [[conjugate transpose]] generalizes the [[transpose]], [[Hermitian matrix|hermitian matrices]] generalize [[Symmetric matrix|symmetric matrices]], and [[Unitary matrix|unitary matrices]] generalize [[Orthogonal matrix|orthogonal matrices]].

===In applied mathematics===

====Control theory====
{{see also|Complex plane#Use in control theory}}

In [[control theory]], systems are often transformed from the [[time domain]] to the complex [[frequency domain]] using the [[Laplace transform]]. The system's [[zeros and poles]] are then analyzed in the ''complex plane''. The [[root locus]], [[Nyquist plot]], and [[Nichols plot]] techniques all make use of the complex plane.

In the root locus method, it is important whether zeros and poles are in the left or right half planes, that is, have real part greater than or less than zero. If a linear, time-invariant (LTI) system has poles that are

* in the right half plane, it will be [[unstable]],
* all in the left half plane, it will be [[BIBO stability|stable]],
* on the imaginary axis, it will have [[marginal stability]].

If a system has zeros in the right half plane, it is a [[nonminimum phase]] system.

====Signal analysis====
Complex numbers are used in [[signal analysis]] and other fields for a convenient description for periodically varying signals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a [[sine wave]] of a given [[frequency]], the absolute value {{math|{{!}}''z''{{!}}}} of the corresponding {{mvar|z}} is the [[amplitude]] and the [[Argument (complex analysis)|argument]] {{math|arg ''z''}} is the [[phase (waves)|phase]].

If [[Fourier analysis]] is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as complex-valued functions of the form

<math display=block>x(t) = \operatorname{Re} \{X( t ) \} </math>

and

<math display=block>X( t ) = A e^{i\omega t} = a e^{ i \phi } e^{i\omega t} = a e^{i (\omega t + \phi) } </math>

where ω represents the [[angular frequency]] and the complex number ''A'' encodes the phase and amplitude as explained above.

This use is also extended into [[digital signal processing]] and [[digital image processing]], which use digital versions of Fourier analysis (and [[wavelet]] analysis) to transmit, [[Data compression|compress]], restore, and otherwise process [[Digital data|digital]] [[Sound|audio]] signals, still images, and [[video]] signals.

Another example, relevant to the two side bands of [[amplitude modulation]] of AM radio, is:

<math display=block>\begin{align}
  \cos((\omega + \alpha)t) + \cos\left((\omega - \alpha)t\right)
    & = \operatorname{Re}\left(e^{i(\omega + \alpha)t} + e^{i(\omega - \alpha)t}\right) \\
    & = \operatorname{Re}\left(\left(e^{i\alpha t} + e^{-i\alpha t}\right) \cdot e^{i\omega t}\right) \\
    & = \operatorname{Re}\left(2\cos(\alpha t) \cdot e^{i\omega t}\right) \\
    & = 2 \cos(\alpha t) \cdot \operatorname{Re}\left(e^{i\omega t}\right) \\
    & = 2 \cos(\alpha t) \cdot \cos\left(\omega t\right).
\end{align}</math>

===In physics===
====Electromagnetism and electrical engineering====
{{Main|Alternating current}}
In [[electrical engineering]], the [[Fourier transform]] is used to analyze varying [[electric current]]s and [[voltage]]s. The treatment of [[resistor]]s, [[capacitor]]s, and [[inductor]]s can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the [[Electrical impedance|impedance]]. This approach is called [[phasor]] calculus.

In electrical engineering, the imaginary unit is denoted by {{mvar|j}}, to avoid confusion with {{mvar|I}}, which is generally in use to denote electric current, or, more particularly, {{mvar|i}}, which is generally in use to denote instantaneous electric current.

Because the voltage in an AC circuit is oscillating, it can be represented as

<math display=block> V(t) = V_0 e^{j \omega t} = V_0 \left (\cos\omega t + j \sin\omega t \right ),</math>

To obtain the measurable quantity, the real part is taken:

<math display=block> v(t) = \operatorname{Re}(V) = \operatorname{Re}\left [ V_0 e^{j \omega t} \right ] = V_0 \cos \omega t.</math>

The complex-valued signal {{math|''V''(''t'')}} is called the [[analytic signal|analytic]] representation of the real-valued, measurable signal {{math|''v''(''t'')}}.
<ref>{{cite book |last1=Grant |first1=I.S. |title=Electromagnetism |year=2008|edition=2 |publisher=Manchester Physics Series |isbn=978-0-471-92712-9 |last2=Phillips |first2=W.R.}}</ref>

====Fluid dynamics====
In [[fluid dynamics]], complex functions are used to describe [[potential flow in two dimensions]].

====Quantum mechanics====
The complex number field is intrinsic to the [[mathematical formulations of quantum mechanics]], where complex [[Hilbert space]]s provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the [[Schrödinger equation]] and Heisenberg's [[matrix mechanics]] – make use of complex numbers.

====Relativity====
In [[special relativity]] and [[general relativity]], some formulas for the metric on [[spacetime]] become simpler if one takes the time component of the spacetime continuum to be imaginary. (This approach is no longer standard in classical relativity, but is [[Wick rotation|used in an essential way]] in [[quantum field theory]].) Complex numbers are essential to [[spinor]]s, which are a generalization of the [[tensor]]s used in relativity.