Editing Complex conjugate (section)

== Properties ==
The following properties apply for all complex numbers <math>z</math> and <math>w,</math> unless stated otherwise, and can be proved by writing <math>z</math> and <math>w</math> in the form <math>a + b i.</math>

For any two complex numbers, conjugation is [[distributive property|distributive]] over addition, subtraction, multiplication and division:<ref name = fis>{{citation|title = Linear Algebra | first1 = Stephen | last1 = Friedberg | first2 = Arnold | last2 = Insel | first3 = Lawrence | last3 =Spence | edition = 5 | year = 2018 | publisher = Pearson | isbn = 978-0134860244}}, Appendix D</ref>
<math display="block">\begin{align}
                     \overline{z + w} &= \overline{z} + \overline{w}, \\
                     \overline{z - w} &= \overline{z} - \overline{w}, \\
                        \overline{zw} &= \overline{z} \; \overline{w}, \quad \text{and} \\
  \overline{\left(\frac{z}{w}\right)} &= \frac{\overline{z}}{\overline{w}},\quad \text{if } w \neq 0.
\end{align}</math>

A complex number is equal to its complex conjugate if its imaginary part is zero, that is, if the number is real.  In other words, real numbers are the only [[Fixed point (mathematics)|fixed point]]s of conjugation. 

Conjugation does not change the modulus of a complex number: <math>\left| \overline{z} \right| = |z|.</math>

Conjugation is an [[involution (mathematics)|involution]], that is, the conjugate of the conjugate of a complex number <math>z</math> is <math>z.</math>  In symbols, <math>\overline{\overline{z}} = z.</math><ref name = fis />

The product of a complex number with its conjugate is equal to the square of the number's modulus: <math display="block">z\overline{z} = {\left| z \right|}^2.</math> This allows easy computation of the [[multiplicative inverse]] of a complex number given in rectangular coordinates: <math display="block">z^{-1} = \frac{\overline{z}}{{\left| z \right|}^2},\quad \text{ for all } z \neq 0.</math>

Conjugation is [[commutative]] under composition with exponentiation to integer powers, with the exponential function, and with the natural logarithm for nonzero arguments:
<math display="block">\overline{z^n} = \left(\overline{z}\right)^n,\quad \text{ for all } n \in \Z </math><ref group =note>See [[Exponentiation#Non-integer powers of complex numbers]].</ref>
<math display="block">\exp\left(\overline{z}\right) = \overline{\exp(z)}</math> 
<math display="block">\ln\left(\overline{z}\right) = \overline{\ln(z)} \text{ if } z \text{ is not zero or a negative real number }</math>

If <math>p</math> is a [[polynomial]] with [[real number|real]] coefficients and <math>p(z) = 0,</math> then <math>p\left(\overline{z}\right) = 0</math> as well. Thus, non-real roots of real polynomials occur in complex conjugate pairs (''see'' [[Complex conjugate root theorem]]).

In general, if <math>\varphi</math> is a [[holomorphic function]] whose restriction to the real numbers is real-valued, and <math>\varphi(z)</math> and <math>\varphi(\overline{z})</math> are defined, then
<math display="block">\varphi\left(\overline{z}\right) = \overline{\varphi(z)}.\,\!</math>

The map <math>\sigma(z) = \overline{z}</math> from <math>\Complex</math> to <math>\Complex</math> is a [[homeomorphism]] (where the topology on <math>\Complex</math> is taken to be the standard topology) and [[antilinear]], if one considers <math>\Complex</math> as a complex [[vector space]] over itself. Even though it appears to be a [[well-behaved]] function, it is not [[holomorphic function|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>\Complex/\R.</math> This Galois group has only two elements: <math>\sigma</math> and the identity on <math>\Complex.</math> Thus the only two field automorphisms of <math>\Complex</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 + yi</math> or <math>z = re^{i\theta}</math> is given, its conjugate is sufficient to reproduce the parts of the <math>z</math>-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 (or absolute value)]]: <math>r= \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>

Furthermore, <math>\overline{z}</math> can be used to specify lines in the plane: the set
<math display="block">\left\{z : z \overline{r} + \overline{z} r = 0 \right\}</math>
is a line through the origin and perpendicular to <math>{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>{r}</math> is zero. Similarly, for a fixed complex unit <math>u = e^{i b},</math> the equation
<math display="block">\frac{z - z_0}{\overline{z} - \overline{z_0}} = u^2</math>
determines the line through <math>z_0</math> parallel to the line through 0 and <math>u.</math>

These uses of the conjugate of <math>z</math> as a variable are illustrated in [[Frank Morley]]'s book ''Inversive Geometry'' (1933), written with his son Frank Vigor Morley.