# 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 ${\displaystyle z=a+bi}$ (where a and b are real numbers), the complex conjugate of ${\displaystyle z}$, often denoted as ${\displaystyle {\overline {z}}}$, is equal to ${\displaystyle a-bi.}$[1][2][3]

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 ${\displaystyle re^{i\varphi }}$ is ${\displaystyle re^{-i\varphi }}$. This can be shown using Euler's formula.

The product of a complex number and its conjugate is a real number: ${\displaystyle a^{2}+b^{2}}$ (or ${\displaystyle r^{2}}$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 ${\displaystyle z}$  is written as ${\displaystyle {\overline {z}}}$  or ${\displaystyle z^{*}\!}$ .[1][2] 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 ${\displaystyle e^{i\varphi }+{\text{c.c.}}}$  means ${\displaystyle e^{i\varphi }+e^{-i\varphi }}$ .

## 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.[2]

{\displaystyle {\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.

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

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

{\displaystyle {\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.[2]

{\displaystyle {\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.

{\displaystyle {\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.

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

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

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

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

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

## Use as a variable

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

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

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

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

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

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

determines the line through ${\displaystyle z_{0}}$  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 ${\displaystyle \mathbf {A} }$ .[4] 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:

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

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

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

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 ${\displaystyle V}$ .[5]

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.