|
|
| (616 intermediate revisions by more than 100 users not shown) |
| Line 1: |
Line 1: |
| [[File:Complex conjugate picture.svg|right|thumb|Geometric representation of ''z'' and its conjugate ''z̅'' in the complex plane. The complex conjugate is found by [[reflection symmetry|reflecting]] ''z'' across the real axis.]]
| | This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users. |
|
| |
|
| In [[mathematics]], '''complex conjugates''' are a pair of [[complex number]]s, both having the same [[real number|real]] part, but with [[imaginary number|imaginary]] parts of equal magnitude and opposite [[sign (mathematics)|sign]]s.<ref>{{MathWorld|ComplexConjugate|Complex Conjugates}}</ref><ref>{{MathWorld|ImaginaryNumber|Imaginary Numbers}}</ref> For example, 3 + 4''i'' and 3 − 4''i'' are complex conjugates.
| | If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]] |
| | * Only registered users will be able to execute this rendering mode. |
| | * Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere. |
|
| |
|
| The conjugate of the complex number <math>z</math>
| | Registered users will be able to choose between the following three rendering modes: |
|
| |
|
| : <math> z=a+ib </math>, | | '''MathML''' |
| | :<math forcemathmode="mathml">E=mc^2</math> |
|
| |
|
| where <math>a</math> and <math>b</math> are [[real number]]s, is
| | <!--'''PNG''' (currently default in production) |
| | :<math forcemathmode="png">E=mc^2</math> |
|
| |
|
| :<math>\overline{z} = a - ib.\,</math> | | '''source''' |
| | :<math forcemathmode="source">E=mc^2</math> --> |
|
| |
|
| For example,
| | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
| : <math> \overline{(3-2i)} = 3 + 2i</math> | |
| : <math> \overline{7}=7</math> | |
| : <math> \overline{i} = -i.</math> | |
|
| |
|
| An alternative notation for the complex conjugate is <math>z^*\!</math>. However, the <math>\bar z</math> notation avoids confusion with the notation for the [[conjugate transpose]] of a [[matrix (mathematics)|matrix]], which can be thought of as a generalization of complex conjugation. The star-notation is preferred in [[physics]], where [[Dagger_(typography)|dagger]] is used for the conjugate transpose, while the bar-notation is more common in pure [[mathematics]]. If a complex number is [[Complex_number#Matrix_representation_of_complex_numbers|represented as a 2×2 matrix]], the notations are identical.
| | ==Demos== |
|
| |
|
| Complex numbers are considered points in the [[complex plane]], a variation of the [[Cartesian coordinate system]] where both axes are real number lines that cross at the origin, however, the ''y''-axis is a product of real numbers multiplied by <math>\pm i</math>. On the illustration, the ''x''-axis is called the ''real axis'', labeled ''Re'', while the ''y''-axis is called the ''imaginary axis'', labeled ''Im''. The plane defined by the ''Re'' and ''Im'' axes represents the space of all possible complex numbers. In this view, complex conjugation corresponds to reflection of a complex number at the ''x''-axis, equivalent to a 180 degree rotation of the complex plane about the ''Re'' axis.
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
|
| |
|
| In [[Polar coordinate system#Complex numbers|polar form]], the conjugate of <math>r e^{i \phi}</math> is <math>r e^{-i \phi}</math>. This can be shown using [[Euler's formula]].
| |
|
| |
|
| Pairs of complex conjugates are significant because the [[imaginary unit]] <math>i</math> is qualitatively indistinct from its additive and multiplicative inverse <math>-i</math>, as they both satisfy the definition for the imaginary unit: <math> x^2=-1</math>. Thus in most "natural" settings, if a complex number provides a solution to a problem, so does its conjugate, such as is the case for complex solutions of the [[quadratic formula]] with real coefficients.
| | * accessibility: |
| | ** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)] |
| | ** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]]. |
| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
|
| |
|
| In some texts, the complex conjugate of a previous known number is abbreviated as "c.c.". For example, writing <math>e^{i \phi}+c.c.</math> means <math>e^{i \phi}+e^{-i \phi}</math>
| | ==Test pages == |
|
| |
|
| == Properties ==
| | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
| These properties apply for all complex numbers ''z'' and ''w'', unless stated otherwise, and can be proven by writing ''z'' and ''w'' in the form ''a'' + ''ib''.
| | *[[Displaystyle]] |
| | *[[MathAxisAlignment]] |
| | *[[Styling]] |
| | *[[Linebreaking]] |
| | *[[Unique Ids]] |
| | *[[Help:Formula]] |
|
| |
|
| : <math>\overline{(z + w)} = \overline{z} + \overline{w} \!\ </math>
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| | | *[[Url2Image|Url2Image (private Wikis only)]] |
| : <math>\overline{z-w} = \overline{z} - \overline{w} \!\ </math>
| | ==Bug reporting== |
| | | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
| : <math>\overline{(zw)} = \overline{z}\; \overline{w} \!\ </math>
| |
| | |
| : <math>\overline{(z/w)} = \overline{z}/\overline{w} \!\ </math> if w is nonzero
| |
| | |
| : <math>\overline{z} = z \!\ </math> if and only if ''z'' is real
| |
| | |
| : <math>\overline{z^n} = \overline{z}^n </math> for any integer ''n''
| |
| | |
| : <math>\left| \overline{z} \right| = \left| z \right|</math>
| |
| | |
| : <math>{\left| z \right|}^2 = z\overline{z} = \overline{z}z</math>
| |
| | |
| : <math>\overline{\overline{z}} = z \!\ </math>, [[Involution (mathematics)|involution]] (i.e., the conjugate of the conjugate of a complex number ''z'' is again that number)
| |
| | |
| : <math>z^{-1} = \frac{\overline{z}}{{\left| z \right|}^2}</math> if ''z'' is non-zero
| |
| | |
| The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.
| |
| | |
| : <math>\exp(\overline{z}) = \overline{\exp(z)}\,\!</math>
| |
| | |
| : <math>\log(\overline{z}) = \overline{\log(z)}\,\!</math> if ''z'' is non-zero
| |
| | |
| In general, if <math>\phi\,</math> is a [[holomorphic function]] whose restriction to the real numbers is real-valued, and <math>\phi(z)\,</math> is defined, then
| |
| | |
| : <math>\phi(\overline{z}) = \overline{\phi(z)}.\,\!</math>
| |
| | |
| Consequently, if <math>p</math> is a [[polynomial]] with [[real number|real]] coefficients, and <math>p(z) = 0</math>, then <math>p(\overline{z}) = 0</math> as well. Thus, non-real roots of real polynomials occur in complex conjugate pairs (''see'' [[Complex conjugate root theorem]]).
| |
| | |
| The map <math>\sigma(z) = \overline{z}\,</math> from <math>\mathbb{C}\,</math> to <math>\mathbb{C}</math> is a [[homeomorphism]] (where the topology on <math>\mathbb{C}</math> is taken to be the standard topology) and [[antilinear]], if one considers <math>{\mathbb C}\,</math> as a complex [[vector space]] over itself. Even though it appears to be a [[well-behaved]] function, it is not [[holomorphic]]; it reverses orientation whereas holomorphic functions locally preserve orientation. It is [[bijective]] and compatible with the arithmetical operations, and hence is a [[field (mathematics)|field]] [[automorphism]]. As it keeps the real numbers fixed, it is an element of the [[Galois group]] of the [[field extension]] <math>\mathbb{C}/\mathbb{R}</math>. This Galois group has only two elements: <math>\sigma\,</math> and the identity on <math>\mathbb{C}</math>. Thus the only two field automorphisms of <math>\mathbb{C}</math> that leave the real numbers fixed are the identity map and complex conjugation.
| |
| | |
| ==Use as a variable== | |
| Once a complex number <math>z = x + iy</math> or <math>z = \rho e^{i\theta}</math> is given, its conjugate is sufficient to reproduce the parts of the z-variable:
| |
| *Real part: <math>x = \operatorname{Re}\,(z) = \dfrac{z + \overline{z}}{2}</math>
| |
| *Imaginary part: <math>y = \operatorname{Im}\,(z) = \dfrac{z - \overline{z}}{2i}</math>
| |
| *[[Absolute value|Modulus]]/[[absolute value]]: <math>\rho = \left| z \right| = \sqrt {z \overline{z}}</math>
| |
| *[[Argument (complex analysis)|Argument]]: <math>e^{i\theta} = e^{i\arg z} = \sqrt {\dfrac{z}{\overline z}}</math>, so <math>\theta = \arg z = \dfrac{1}{i}\ln \sqrt{\frac{z}{\overline z}} = \dfrac{\ln z - \ln \overline z}{2i}</math>
| |
| Thus the pair of variables <math>z\,</math> and <math>\overline{z}</math> also serve up the plane as do ''x,y'' and <math>\rho \,</math> and <math>\theta</math>. Furthermore, the <math>\overline{z}</math> variable is useful in specifying lines in the plane:
| |
| | |
| :<math> \{z \mid z \overline{r} + \overline{z} r = 0 \}</math>
| |
| | |
| is a line through the origin and perpendicular to <math>\overline{r}</math> since the real part of <math>z\cdot\overline{r}</math> is zero only when the cosine of the angle between <math>z\,</math> and <math>\overline{r}</math> is zero. Similarly, for a fixed complex unit ''u'' = exp(''b'' i), the equation:
| |
| | |
| :<math> \frac{z - z_0}{\overline{z} - \overline{z_0}} = u</math>
| |
| | |
| determines the line through <math>z_0\,</math> in the direction of u.
| |
| | |
| These uses of the conjugate of ''z'' as a variable are illustrated in [[Frank Morley]]'s book ''Inversive Geometry'' (1933), written with his son Frank Vigor Morley.
| |
| | |
| ==Generalizations== | |
| The other planar real algebras, [[dual numbers]], and [[split-complex number]]s are also explicated by use of complex conjugation.
| |
| | |
| For matrices of complex numbers <math>\overline{\mathbf{AB}} = (\overline{\mathbf{A}}) (\overline{\mathbf{B}})</math>, where <math>\overline{\mathbf{A}}</math> represents the element-by-element conjugation of <math>\mathbf{A}</math>.<ref>Arfken, ''Mathematical Methods for Physicists'', 1985, pg. 201</ref> Contrast this to the property <math>(\mathbf{AB})^*=\mathbf{B}^* \mathbf{A}^* </math>, where <math>\mathbf{A}^*</math> represents the [[conjugate transpose]] of <math>\mathbf{A}</math>.
| |
| | |
| Taking the [[conjugate transpose]] (or adjoint) of complex [[matrix (mathematics)|matrices]] generalizes complex conjugation. Even more general is the concept of [[adjoint operator]] for operators on (possibly infinite-dimensional) complex [[Hilbert space]]s. All this is subsumed by the *-operations of [[C*-algebra]]s.
| |
| | |
| One may also define a conjugation for [[quaternion]]s and [[coquaternion]]s: the conjugate of <math>a + bi + cj + dk</math> is <math>a - bi - cj - dk</math>.
| |
| | |
| Note that all these generalizations are multiplicative only if the factors are reversed:
| |
| | |
| :<math>{\left(zw\right)}^* = w^* z^*.</math>
| |
| | |
| Since the multiplication of planar real algebras is [[commutative]], this reversal is not needed there.
| |
| | |
| There is also an abstract notion of conjugation for [[vector spaces]] <math>V</math> over the [[complex number]]s. In this context,
| |
| any [[antilinear map]] <math>\phi: V \rightarrow V\,</math> that satisfies
| |
| | |
| # <math>\phi^2 = \operatorname{id}_V\,</math>, where <math>\phi^2=\phi\circ\phi</math> and <math>\operatorname{id}_V\,</math> is the [[identity map]] on <math>V\,</math>,
| |
| # <math>\phi(zv) = \overline{z} \phi(v)</math> for all <math>v\in V\,</math>, <math>z\in{\mathbb C}\,</math>, and
| |
| # <math>\phi(v_1+v_2) = \phi(v_1)+\phi(v_2)\,</math> for all <math>v_1\in V\,</math>, <math>v_2\in V\,</math>,
| |
| | |
| is called a ''complex conjugation'', or a [[real structure]]. As the involution <math>\operatorname{\phi}</math> is [[antilinear]], it cannot be the identity map on <math>V</math>.
| |
| Of course, <math>\operatorname{\phi}</math> is a <math>\mathbb{R}</math>-linear transformation of <math>V</math>, if one notes that every complex space ''V'' has a real form obtained by taking the same [[vector (mathematics and physics)|vector]]s as in the original space and restricting the scalars to be real. The above properties actually define a [[real structure]] on the complex vector space <math>V</math>.<ref>Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Spinger-Verlag, 1988, p. 29</ref>
| |
| One example of this notion is the conjugate transpose operation of complex matrices defined above. It should be remarked that on generic complex vector spaces there is no ''[[canonical form|canonical]]'' notion of complex conjugation.
| |
| | |
| ==See also==
| |
| * [[Complex conjugate vector space]]
| |
| * [[Real structure]]
| |
| | |
| ==Notes==
| |
| {{Reflist}}
| |
| | |
| ==References==
| |
| | |
| * Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Spinger-Verlag, 1988. ISBN 0-387-19078-3. (antilinear maps are discussed in section 3.3).
| |
| | |
| {{DEFAULTSORT:Complex Conjugate}}
| |
| [[Category:Complex numbers]]
| |
| | |
| [[ru:Комплексное число#Сопряжённые числа]]
| |
