# Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude, but opposite in sign. Given a complex number $z=a+bi$ (where a and b are real numbers), the complex conjugate of $z$ , often denoted as ${\overline {z}}$ , is equal to $a-bi.$  Geometric representation (Argand diagram) of z and its conjugate in the complex plane. The complex conjugate is found by reflecting z across the real axis.

In polar form, the conjugate of $re^{i\varphi }$ is $re^{-i\varphi }$ . This can be shown using Euler's formula.

The product of a complex number and its conjugate is a real number: $a^{2}+b^{2}$ (or $r^{2}$ in polar coordinates).

Complex conjugates are important for finding roots of polynomials. According to the complex conjugate root theorem, if a complex number is a root to a polynomial in one variable with real coefficients (such as a quadratic or a cubic equation), then so is its conjugate.

## Notation

The complex conjugate of a complex number $z$  is written as ${\overline {z}}$  or $z^{*}\!$ . The first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate. The second is preferred in physics, where dagger (†) is used for the conjugate transpose, while the bar-notation is more common in pure mathematics. If a complex number is represented as a 2×2 matrix, the notations are identical. In some texts, the complex conjugate of a previous known number is abbreviated as "c.c.". For example, writing $e^{i\varphi }+{\text{c.c.}}$  means $e^{i\varphi }+e^{-i\varphi }$ .

## Properties

The following properties apply for all complex numbers z and w, unless stated otherwise, and can be proved by writing z and w in the form a + bi.

For any two complex numbers w,z, conjugation is distributive over addition, subtraction, multiplication and division.

{\begin{aligned}{\overline {z+w}}&={\overline {z}}+{\overline {w}}\\{\overline {z-w}}&={\overline {z}}-{\overline {w}}\\{\overline {zw}}&={\overline {z}}\;{\overline {w}}\\{\overline {\left({\frac {z}{w}}\right)}}&={\frac {\overline {z}}{\overline {w}}},\quad {\text{if }}w\neq 0\\\end{aligned}}

Real numbers are the only fixed points of conjugation. A complex number is equal to its complex conjugate if its imaginary part is zero.

{\begin{aligned}{\overline {z}}&=z~\Leftrightarrow ~z\in \mathbb {R} \\\end{aligned}}

Composition of conjugation with the modulus is equivalent to the modulus alone.

{\begin{aligned}\left|{\overline {z}}\right|&=\left|z\right|\\\end{aligned}}

Conjugation is an involution; the conjugate of the conjugate of a complex number z is z.

{\begin{aligned}{\overline {\overline {z}}}&=z\\\end{aligned}}

The product of a complex number with its conjugate is equal to the square of the number's modulus. This allows easy computation of the multiplicative inverse of a complex number given in rectangular coordinates.

{\begin{aligned}z{\overline {z}}&={\left|z\right|}^{2}\\z^{-1}&={\frac {\overline {z}}{{\left|z\right|}^{2}}},\quad \forall z\neq 0\end{aligned}}

Conjugation is commutative under composition with exponentiation to integer powers, with the exponential function, and with the natural logarithm for nonzero arguments.

{\begin{aligned}{\overline {z^{n}}}&=\left({\overline {z}}\right)^{n},\quad \forall n\in \mathbb {Z} \\\end{aligned}}
$\exp \left({\overline {z}}\right)={\overline {\exp(z)}}\,\!$
$\log \left({\overline {z}}\right)={\overline {\log(z)}}\,\!$  if z is non-zero

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

In general, if $\varphi \,$  is a holomorphic function whose restriction to the real numbers is real-valued, and $\varphi (z)\,$  is defined, then

$\varphi \left({\overline {z}}\right)={\overline {\varphi (z)}}.\,\!$

The map $\sigma (z)={\overline {z}}\,$  from $\mathbb {C} \,$  to $\mathbb {C}$  is a homeomorphism (where the topology on $\mathbb {C}$  is taken to be the standard topology) and antilinear, if one considers $\mathbb {C} \,$  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 automorphism. As it keeps the real numbers fixed, it is an element of the Galois group of the field extension $\mathbb {C} /\mathbb {R}$ . This Galois group has only two elements: $\sigma \,$  and the identity on $\mathbb {C}$ . Thus the only two field automorphisms of $\mathbb {C}$  that leave the real numbers fixed are the identity map and complex conjugation.

## Use as a variable

Once a complex number $z=x+yi$  or $z=re^{i\theta }$  is given, its conjugate is sufficient to reproduce the parts of the z-variable:

• Real part: $x=\operatorname {Re} (z)={\dfrac {z+{\overline {z}}}{2}}$
• Imaginary part: $y=\operatorname {Im} (z)={\dfrac {z-{\overline {z}}}{2i}}$
• Modulus (or absolute value): $r=\left|z\right|={\sqrt {z{\overline {z}}}}$
• Argument: $e^{i\theta }=e^{i\arg z}={\sqrt {\dfrac {z}{\overline {z}}}}$ , so $\theta =\arg z={\dfrac {1}{i}}\ln {\sqrt {\frac {z}{\overline {z}}}}={\dfrac {\ln z-\ln {\overline {z}}}{2i}}$

Furthermore, ${\overline {z}}$  can be used to specify lines in the plane: the set

$\left\{z\mid z{\overline {r}}+{\overline {z}}r=0\right\}$

is a line through the origin and perpendicular to ${r}$ , since the real part of $z\cdot {\overline {r}}$  is zero only when the cosine of the angle between $z$  and ${r}$  is zero. Similarly, for a fixed complex unit u = exp(b i), the equation

${\frac {z-z_{0}}{{\overline {z}}-{\overline {z_{0}}}}}=u^{2}$

determines the line through $z_{0}$  parallel to the line through 0 and 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 numbers are also analyzed using complex conjugation.

For matrices of complex numbers, ${\textstyle {\overline {\mathbf {AB} }}=\left({\overline {\mathbf {A} }}\right)\left({\overline {\mathbf {B} }}\right)}$ , where ${\textstyle {\overline {\mathbf {A} }}}$  represents the element-by-element conjugation of $\mathbf {A}$ . Contrast this to the property ${\textstyle \left(\mathbf {AB} \right)^{*}=\mathbf {B} ^{*}\mathbf {A} ^{*}}$ , where ${\textstyle \mathbf {A} ^{*}}$  represents the conjugate transpose of ${\textstyle \mathbf {A} }$ .

Taking the conjugate transpose (or adjoint) of complex matrices generalizes complex conjugation. Even more general is the concept of adjoint operator for operators on (possibly infinite-dimensional) complex Hilbert spaces. All this is subsumed by the *-operations of C*-algebras.

One may also define a conjugation for quaternions and split-quaternions: the conjugate of ${\textstyle a+bi+cj+dk}$  is ${\textstyle a-bi-cj-dk}$ .

All these generalizations are multiplicative only if the factors are reversed:

${\left(zw\right)}^{*}=w^{*}z^{*}.$

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 ${\textstyle V}$  over the complex numbers. In this context, any antilinear map ${\textstyle \varphi :V\rightarrow V\,}$  that satisfies

1. $\varphi ^{2}=\operatorname {id} _{V}\,$ , where $\varphi ^{2}=\varphi \circ \varphi$  and $\operatorname {id} _{V}$  is the identity map on $V\,$ ,
2. $\varphi (zv)={\overline {z}}\varphi (v)$  for all $v\in V\,$ , $z\in \mathbb {C} \,$ , and
3. $\varphi \left(v_{1}+v_{2}\right)=\varphi \left(v_{1}\right)+\varphi \left(v_{2}\right)\,$  for all $v_{1}\in V\,$ , $v_{2}\in V\,$ ,

is called a complex conjugation, or a real structure. As the involution $\varphi$  is antilinear, it cannot be the identity map on $V$ .

Of course, ${\textstyle \varphi }$  is a ${\textstyle \mathbb {R} }$ -linear transformation of ${\textstyle V}$ , if one notes that every complex space V has a real form obtained by taking the same vectors 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 $V$ .

One example of this notion is the conjugate transpose operation of complex matrices defined above. Note that on generic complex vector spaces, there is no canonical notion of complex conjugation.