TheInfoList In
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, the complex conjugate of a
complex number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
is the number with an equal
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
part and an
imaginary Imaginary may refer to: * Imaginary (sociology), a concept in sociology * The Imaginary (psychoanalysis), a concept by Jacques Lacan * Imaginary number, a concept in mathematics * Imaginary time, a concept in physics * Imagination, a mental faculty ...
part equal in magnitude but opposite in
sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or ...
. That is, (if $a$ and $b$ are real, then) the complex conjugate of $a + bi$ is equal to $a - bi.$ The complex conjugate of $z$ is often denoted as $\overline.$ In
polar form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, the conjugate of $r e^$ is $r e^.$ This can be shown using
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler's ... . The product of a complex number and its conjugate is a real number: $a^2 + b^2$ (or $r^2$ in
polar coordinates In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... ). If a root of a
univariateIn mathematics, a univariate object is an expression, equation In mathematics, an equation is a statement that asserts the equality (mathematics), equality of two Expression (mathematics), expressions, which are connected by the equals sign "=". ...
polynomial with real coefficients is complex, then its complex conjugate is also a root.

Notation

The complex conjugate of a complex number $z$ is written as $\overline z$ or $z^*.$ The first notation, a
vinculum Vinculum may refer to: * Vinculum (ligament), a band of connective tissue, similar to a ligament, that connects a flexor tendon to a phalanx bone * Vinculum (symbol), a horizontal line used in mathematical notation for a specific purpose * Vinculum ...
, avoids confusion with the notation for the
conjugate transpose In mathematics, the conjugate transpose (or Hermitian transpose) of an ''m''-by-''n'' matrix (mathematics), matrix \boldsymbol with complex number, complex entries is the ''n''-by-''m'' matrix obtained from \boldsymbol by taking the transpose and ...
of a
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
, which can be thought of as a generalization of the complex conjugate. The second is preferred in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scien ... , where
dagger A dagger is a knife A knife (plural knives; from Old Norse'' knifr'', "knife, dirk") is a tool or weapon with a cutting edge or blade, often attached to a handle or hilt. One of the earliest tools used by humanity, knives appeared at least ...
(†) is used for the conjugate transpose, as well as electrical engineering and
computer engineering Computer engineering (CoE or CpE) is a branch of engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and ...
, where bar notation can be confused for the
logical negation In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argume ...
("NOT")
Boolean algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
symbol, while the bar notation is more common in
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...
. If a complex number is represented as a $2 \times 2$ matrix, the notations are identical.

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 $a + b i.$ For any two complex numbers, conjugation is distributive over addition, subtraction, multiplication and division:, Appendix D $\begin \overline &= \overline + \overline, \\ \overline &= \overline - \overline, \\ \overline &= \overline \; \overline, \quad \text \\ \overline &= \frac,\quad \text w \neq 0. \end$ A complex number is equal to its complex conjugate if its imaginary part is zero, or equivalently, if the number is real. In other words, real numbers are the only fixed points of conjugation. Conjugation does not change the modulus of a complex number: $\left, \overline \ = , z, .$ Conjugation is an
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * ''Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inputs ...
, that is, the conjugate of the conjugate of a complex number $z$ is $z.$ In symbols, $\overline = z.$ 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 Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ... of a complex number given in rectangular coordinates. $\begin z\overline &= ^2\\ z^ &= \frac,\quad \text z \neq 0 \end$ Conjugation is
commutative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
under composition with exponentiation to integer powers, with the exponential function, and with the natural logarithm for nonzero arguments: :$\overline = \left\left(\overline\right\right)^n,\quad \text n \in \Z$ :$\exp\left\left(\overline\right\right) = \overline$ :$\ln\left\left(\overline\right\right) = \overline \text z \text$ If $p$ is a
polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtra ... with
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
coefficients and $p\left(z\right) = 0,$ then $p\left\left(\overline\right\right) = 0$ as well. Thus, non-real roots of real polynomials occur in complex conjugate pairs (''see''
Complex conjugate root theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
). In general, if $\varphi$ is a
holomorphic function A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
whose restriction to the real numbers is real-valued, and $\varphi\left(z\right)$ and $\varphi\left(\overline\right)$ are defined, then $\varphi\left(\overline\right) = \overline.\,\!$ The map $\sigma\left(z\right) = \overline$ from $\Complex$ to $\Complex$ is a
homeomorphism and a donut (torus In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerne ...
(where the topology on $\Complex$ is taken to be the standard topology) and
antilinear In mathematics, a map (mathematics), mapping f:V\to W from a complex vector space to another is said to be antilinear (or conjugate-linear) if :f(ax + by) = \barf(x) + \barf(y) for all a, \, b \, \in \mathbb and all x, \, y \, \in V, where \bar an ...
, if one considers $\Complex$ as a complex
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
over itself. Even though it appears to be a
well-behaved In mathematics, a pathological object is one which possesses deviant, irregular or counterintuitive property, in such a way that distinguishes it from what is conceived as a typical object in the same category. The opposite of pathological is ...
function, it is not
holomorphic Image:Conformal map.svg, A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics, a holomorphic function is a complex-valued function of one or more complex number, complex variables that is, at every point of ...
; it reverses orientation whereas holomorphic functions locally preserve orientation. It is
bijective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
and compatible with the arithmetical operations, and hence is a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
automorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. As it keeps the real numbers fixed, it is an element of the
Galois group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of the
field extension In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
$\Complex/\R.$ This Galois group has only two elements: $\sigma$ and the identity on $\Complex.$ Thus the only two field automorphisms of $\Complex$ 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 $z = re^$ is given, its conjugate is sufficient to reproduce the parts of the $z$-variable: * Real part: $x = \operatorname\left(z\right) = \dfrac$ * Imaginary part: $y = \operatorname\left(z\right) = \dfrac$ *
Modulus (or absolute value) : $r= \left, z \ = \sqrt$ *
Argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ...
: $e^ = e^ = \sqrt,$ so $\theta = \arg z = \dfrac \ln\sqrt = \dfrac$ Furthermore, $\overline$ can be used to specify lines in the plane: the set $\left\$ is a line through the origin and perpendicular to $,$ since the real part of $z\cdot\overline$ is zero only when the cosine of the angle between $z$ and  is zero. Similarly, for a fixed complex unit $u = e^,$ the equation $\frac = u^2$ 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 Frank Morley (September 9, 1860 – October 17, 1937) was a leading mathematician, known mostly for his teaching and research in the fields of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) i ... 's book ''Inversive Geometry'' (1933), written with his son Frank Vigor Morley.

Generalizations

The other planar real algebras,
dual numbers In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
, and
split-complex number In abstract algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Cu ...
s are also analyzed using complex conjugation. For matrices of complex numbers, $\overline = \left(\overline\right) \left(\overline\right),$ where $\overline$ represents the element-by-element conjugation of $\mathbf.$ Contrast this to the property $\left(\mathbf\right)^*=\mathbf^* \mathbf^*,$ where $\mathbf^*$ represents the
conjugate transpose In mathematics, the conjugate transpose (or Hermitian transpose) of an ''m''-by-''n'' matrix (mathematics), matrix \boldsymbol with complex number, complex entries is the ''n''-by-''m'' matrix obtained from \boldsymbol by taking the transpose and ...
of $\mathbf.$ Taking the
conjugate transpose In mathematics, the conjugate transpose (or Hermitian transpose) of an ''m''-by-''n'' matrix (mathematics), matrix \boldsymbol with complex number, complex entries is the ''n''-by-''m'' matrix obtained from \boldsymbol by taking the transpose and ...
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
generalizes complex conjugation. Even more general is the concept of
adjoint operatorIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
for operators on (possibly infinite-dimensional) complex
Hilbert space The mathematical concept of a Hilbert space, named after David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries ...
s. All this is subsumed by the *-operations of
C*-algebra In mathematics, specifically in functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a commo ...
s. One may also define a conjugation for
quaternion In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ... s and split-quaternions: the conjugate of $a + bi + cj + dk$ is $a - bi - cj - dk.$ All these generalizations are multiplicative only if the factors are reversed: $^* = w^* z^*.$ Since the multiplication of planar real algebras is
commutative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, this reversal is not needed there. There is also an abstract notion of conjugation for vector spaces $V$ over the
complex number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s. In this context, any
antilinear map In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
$\varphi: V \to V$ that satisfies # $\varphi^2 = \operatorname_V\,,$ where $\varphi^2 = \varphi \circ \varphi$ and $\operatorname_V$ is the
identity map on $V,$ # $\varphi\left(zv\right) = \overline \varphi\left(v\right)$ for all $v \in V, z \in \Complex,$ and # $\varphi\left\left(v_1 + v_2\right\right) = \varphi\left\left(v_1\right\right) + \varphi\left\left(v_2\right\right)\,$ for all $v_1 v_2, \in V,$ is called a , or a
real structure In mathematics, a real structure on a complex number, complex vector space is a way to decompose the complex vector space in the direct sum of vector spaces, direct sum of two real number, real vector spaces. The prototype of such a structure is the ...
. As the involution $\varphi$ is
antilinear In mathematics, a map (mathematics), mapping f:V\to W from a complex vector space to another is said to be antilinear (or conjugate-linear) if :f(ax + by) = \barf(x) + \barf(y) for all a, \, b \, \in \mathbb and all x, \, y \, \in V, where \bar an ...
, it cannot be the identity map on $V.$ Of course, $\varphi$ is a $\R$-linear transformation of $V,$ if one notes that every complex space $V$ has a real form obtained by taking the same
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
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 $V.$Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Springer-Verlag, 1988, p. 29 One example of this notion is the conjugate transpose operation of complex matrices defined above. However, on generic complex vector spaces, there is no notion of complex conjugation.