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| [[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.]]
| | Are we usually having problems with the PC? Are you usually interested in methods to increase PC performance? Next this is the article you have been seeking. Here we'll discuss a few of the many asked issues with regards to having we PC serve we well; how can I create my computer faster for free? How to make my computer run faster?<br><br>Windows Defender - this does come standard with countless Windows OS Machines, however, otherwise will be download from Microsoft for free. It usually aid protect against spyware.<br><br>System tray icon makes it convenient to launch the program plus displays "clean" status or the number of mistakes inside the last scan. The ability to obtain and remove the Invalid class keys and shell extensions is one of the leading advantages of the program. That is not usual function for the alternative Registry Cleaners. Class keys and shell extensions that are not working will really slow down your computer. RegCure scans to locate invalid entries and delete them.<br><br>There can be several factors why a computer may lose speed. Normal computer employ, like surfing the Internet will get the operating system inside a condition where it has no choice nevertheless to slow down. The constant entering and deleting of temporary files that happen when you surf the Web leave the registries with thousands of false indicators inside our running system's registry.<br><br>Another normal cause of PC slow down is a corrupt registry. The registry is a fairly important component of computers running on Windows platform. When this gets corrupted your PC will slowdown, or worse, not begin at all. Fixing the registry is simple with all the use of the program plus [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities 2014].<br><br>Windows relies heavily on this database, storing everything from your newest emails to your Internet favorites inside there. Because it's thus important, your computer is consistently adding and updating the files inside it. This is ok, yet it could create your computer run slow, when a computer accidentally breaks its crucial registry files. This really is a rather popular issue, and really makes the computer run slower each day. What happens is that since a computer is frequently utilizing 100's of registry files at when, it often gets confused and create a few of them unreadable. This then makes a computer run slow, because Windows takes longer to read the files it requires.<br><br>Most probably should you are experiencing a slow computer it will be a couple years aged. You moreover could not have been told which while you use a computer everyday; there are certain details that it requires to continue running in its best performance. We additionally will not even own any diagnostic tools that may receive your PC running like hot again. So never let which stop we from getting the system cleaned. With access to the web you can find the tools that will assist we get a system running like brand-new again.<br><br>By changing the means you utilize the web we can have access more of your precious bandwidth. This usually eventually provide you a quicker surfing experience. Here is a link to 3 techniques to personalize your PC speed found on the net. |
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| 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 + 4i and 3 − 4i are complex conjugates.
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| The conjugate of the complex number <math>z</math>
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| : <math> z=a+ib </math>,
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| where <math>a</math> and <math>b</math> are [[real number]]s, is
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| :<math>\overline{z} = a - ib.\,</math>
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| For example,
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| : <math> \overline{(3-2i)} = 3 + 2i</math>
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| : <math> \overline{7}=7</math>
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| : <math> \overline{i} = -i.</math>
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| 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.
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| 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.
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| 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]].
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| 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.
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| 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>
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| == Properties ==
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| 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''.
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| : <math>\overline{(z + w)} = \overline{z} + \overline{w} \!\ </math>
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| : <math>\overline{z-w} = \overline{z} - \overline{w} \!\ </math>
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| : <math>\overline{(zw)} = \overline{z}\; \overline{w} \!\ </math>
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| : <math>\overline{(z/w)} = \overline{z}/\overline{w} \!\ </math> if w is nonzero
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| : <math>\overline{z} = z \!\ </math> if and only if ''z'' is real
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| : <math>\overline{z^n} = \overline{z}^n </math> for any integer ''n''
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| : <math>\left| \overline{z} \right| = \left| z \right|</math>
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| : <math>{\left| z \right|}^2 = z\overline{z} = \overline{z}z</math>
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| : <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)
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| : <math>z^{-1} = \frac{\overline{z}}{{\left| z \right|}^2}</math> if ''z'' is non-zero
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| The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates. | |
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| : <math>\exp(\overline{z}) = \overline{\exp(z)}\,\!</math>
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| : <math>\log(\overline{z}) = \overline{\log(z)}\,\!</math> if ''z'' is non-zero
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| 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
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| : <math>\phi(\overline{z}) = \overline{\phi(z)}.\,\!</math>
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| 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]]).
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| 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.
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| ==Use as a variable==
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| 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:
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| *Real part: <math>x = \operatorname{Re}\,(z) = \dfrac{z + \overline{z}}{2}</math>
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| *Imaginary part: <math>y = \operatorname{Im}\,(z) = \dfrac{z - \overline{z}}{2i}</math>
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| *[[Absolute value|Modulus]]/[[absolute value]]: <math>\rho = \left| z \right| = \sqrt {z \overline{z}}</math>
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| *[[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>
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| 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:
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| :<math> \{z \mid z \overline{r} + \overline{z} r = 0 \}</math>
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| 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:
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| :<math> \frac{z - z_0}{\overline{z} - \overline{z_0}} = u</math>
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| determines the line through <math>z_0\,</math> in the direction of u.
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| 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.
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| ==Generalizations==
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| The other planar real algebras, [[dual numbers]], and [[split-complex number]]s are also explicated by use of complex conjugation.
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| 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>.
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| 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.
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| 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>.
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| Note that all these generalizations are multiplicative only if the factors are reversed:
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| :<math>{\left(zw\right)}^* = w^* z^*.</math>
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| Since the multiplication of planar real algebras is [[commutative]], this reversal is not needed there.
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| There is also an abstract notion of conjugation for [[vector spaces]] <math>V</math> over the [[complex number]]s. In this context,
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| any [[antilinear map]] <math>\phi: V \rightarrow V\,</math> that satisfies
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| # <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>,
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| # <math>\phi(zv) = \overline{z} \phi(v)</math> for all <math>v\in V\,</math>, <math>z\in{\mathbb C}\,</math>, and
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| # <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>,
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| 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>.
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| 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>
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| 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.
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| ==See also==
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| * [[Complex conjugate vector space]]
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| * [[Real structure]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Spinger-Verlag, 1988. ISBN 0-387-19078-3. (antilinear maps are discussed in section 3.3).
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| {{DEFAULTSORT:Complex Conjugate}}
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| [[Category:Complex numbers]]
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| [[ru:Комплексное число#Сопряжённые числа]]
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Are we usually having problems with the PC? Are you usually interested in methods to increase PC performance? Next this is the article you have been seeking. Here we'll discuss a few of the many asked issues with regards to having we PC serve we well; how can I create my computer faster for free? How to make my computer run faster?
Windows Defender - this does come standard with countless Windows OS Machines, however, otherwise will be download from Microsoft for free. It usually aid protect against spyware.
System tray icon makes it convenient to launch the program plus displays "clean" status or the number of mistakes inside the last scan. The ability to obtain and remove the Invalid class keys and shell extensions is one of the leading advantages of the program. That is not usual function for the alternative Registry Cleaners. Class keys and shell extensions that are not working will really slow down your computer. RegCure scans to locate invalid entries and delete them.
There can be several factors why a computer may lose speed. Normal computer employ, like surfing the Internet will get the operating system inside a condition where it has no choice nevertheless to slow down. The constant entering and deleting of temporary files that happen when you surf the Web leave the registries with thousands of false indicators inside our running system's registry.
Another normal cause of PC slow down is a corrupt registry. The registry is a fairly important component of computers running on Windows platform. When this gets corrupted your PC will slowdown, or worse, not begin at all. Fixing the registry is simple with all the use of the program plus tuneup utilities 2014.
Windows relies heavily on this database, storing everything from your newest emails to your Internet favorites inside there. Because it's thus important, your computer is consistently adding and updating the files inside it. This is ok, yet it could create your computer run slow, when a computer accidentally breaks its crucial registry files. This really is a rather popular issue, and really makes the computer run slower each day. What happens is that since a computer is frequently utilizing 100's of registry files at when, it often gets confused and create a few of them unreadable. This then makes a computer run slow, because Windows takes longer to read the files it requires.
Most probably should you are experiencing a slow computer it will be a couple years aged. You moreover could not have been told which while you use a computer everyday; there are certain details that it requires to continue running in its best performance. We additionally will not even own any diagnostic tools that may receive your PC running like hot again. So never let which stop we from getting the system cleaned. With access to the web you can find the tools that will assist we get a system running like brand-new again.
By changing the means you utilize the web we can have access more of your precious bandwidth. This usually eventually provide you a quicker surfing experience. Here is a link to 3 techniques to personalize your PC speed found on the net.