In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the complex conjugate of a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
is the number with an equal
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
part and an
imaginary 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 me ...
. That is, (if
and
are real, then) the complex conjugate of
is equal to
The complex conjugate of
is often denoted as
or
.
In
polar form
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
, the conjugate of
is
This can be shown using
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for an ...
.
The product of a complex number and its conjugate is a real number:
(or
in
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
).
If a root of a
univariate
In mathematics, a univariate object is an expression, equation, function or polynomial involving only one variable. Objects involving more than one variable are multivariate. In some cases the distinction between the univariate and multivariate cas ...
polynomial with real coefficients is complex, then its
complex conjugate is also a root.
Notation
The complex conjugate of a complex number
is written as
or
The first notation, a
vinculum, avoids confusion with the notation for the
conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex con ...
of a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
, 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 fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, where
dagger
A dagger is a fighting knife with a very sharp point and usually two sharp edges, typically designed or capable of being used as a thrusting or stabbing weapon.State v. Martin, 633 S.W.2d 80 (Mo. 1982): This is the dictionary or popular-use de ...
(†) is used for the conjugate transpose, as well as electrical engineering and
computer engineering
Computer engineering (CoE or CpE) is a branch of electrical engineering and computer science that integrates several fields of computer science and electronic engineering required to develop computer hardware and software. Computer engineers ...
, where bar notation can be confused for the
logical negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
("NOT")
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
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. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
. If a complex number is
represented as a matrix, the notations are identical.
Properties
The following properties apply for all complex numbers
and
unless stated otherwise, and can be proved by writing
and
in the form
For any two complex numbers, conjugation is
distributive over addition, subtraction, multiplication and division:
[, Appendix D]
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 points of conjugation.
Conjugation does not change the modulus of a complex number:
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 inpu ...
, that is, the conjugate of the conjugate of a complex number
is
In symbols,
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
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...
of a complex number given in rectangular coordinates:
Conjugation is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
under composition with exponentiation to integer powers, with the exponential function, and with the natural logarithm for nonzero arguments:
[See Exponentiation#Non-integer powers of complex numbers.]
If
is a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
with
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
coefficients and
then
as well. Thus, non-real roots of real polynomials occur in complex conjugate pairs (''see''
Complex conjugate root theorem
In mathematics, the complex conjugate root theorem states that if ''P'' is a polynomial in one variable with real coefficients, and ''a'' + ''bi'' is a root of ''P'' with ''a'' and ''b'' real numbers, then its complex conjugate ''a''&nb ...
).
In general, if
is a
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
whose restriction to the real numbers is real-valued, and
and
are defined, then
The map
from
to
is a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
(where the topology on
is taken to be the standard topology) and
antilinear
In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if
\begin
f(x + y) &= f(x) + f(y) && \qquad \text \\
f(s x) &= \overline f(x) && \qquad \text \\
\end
hold for all vectors x, y \ ...
, if one considers
as a complex
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over itself. Even though it appears to be a
well-behaved
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition,
it is sometimes called well-behaved. Th ...
function, it is not
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
; it reverses orientation whereas holomorphic functions locally preserve orientation. It is
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
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 grass ...
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
. As it keeps the real numbers fixed, it is an element of the
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
of the
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
This Galois group has only two elements:
and the identity on
Thus the only two field automorphisms of
that leave the real numbers fixed are the identity map and complex conjugation.
Use as a variable
Once a complex number
or
is given, its conjugate is sufficient to reproduce the parts of the
-variable:
* Real part:
* Imaginary part:
*
Modulus (or absolute value):
*
Argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
:
so
Furthermore,
can be used to specify lines in the plane: the set
is a line through the origin and perpendicular to
since the real part of
is zero only when the cosine of the angle between
and
is zero. Similarly, for a fixed complex unit
the equation
determines the line through
parallel to the line through 0 and
These uses of the conjugate of
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 and geometry. Among his mathematical accomplishments was the discovery and proof of the celebr ...
's book ''Inversive Geometry'' (1933), written with his son Frank Vigor Morley.
Generalizations
The other planar real unital algebras,
dual numbers
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0.
Du ...
, and
split-complex number
In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components and , and is written z=x+yj, where j^2=1. The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number wi ...
s are also analyzed using complex conjugation.
For matrices of complex numbers,
where
represents the element-by-element conjugation of
[Arfken, ''Mathematical Methods for Physicists'', 1985, pg. 201] Contrast this to the property
where
represents the
conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex con ...
of
Taking the
conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex con ...
(or adjoint) of complex
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
generalizes complex conjugation. Even more general is the concept of
adjoint operator
In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule
:\langle Ax,y \rangle = \langle x,A^*y \rangle,
where ...
for operators on (possibly infinite-dimensional) complex
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s. All this is subsumed by the *-operations of
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous ...
s.
One may also define a conjugation for
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s and
split-quaternion
In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers.
After introduction in th ...
s: the conjugate of
is
All these generalizations are multiplicative only if the factors are reversed:
Since the multiplication of planar real algebras is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
, this reversal is not needed there.
There is also an abstract notion of conjugation for
vector spaces
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s. In this context, any
antilinear map
In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if
\begin
f(x + y) &= f(x) + f(y) && \qquad \text \\
f(s x) &= \overline f(x) && \qquad \text \\
\end
hold for all vectors x, y ...
that satisfies
#
where
and
is the
identity map
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unch ...
on
#
for all
and
#
for all
is called a , or a
real structure
In mathematics, a real structure on a complex vector space is a way to decompose the complex vector space in the direct sum of two real vector spaces. The prototype of such a structure is the field of complex numbers itself, considered as a compl ...
. As the involution
is
antilinear
In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if
\begin
f(x + y) &= f(x) + f(y) && \qquad \text \\
f(s x) &= \overline f(x) && \qquad \text \\
\end
hold for all vectors x, y \ ...
, it cannot be the identity map on
Of course,
is a
-linear transformation of
if one notes that every complex space
has a real form obtained by taking the same
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
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
[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.
See also
*
*
*
*
*
*
*
*
References
note
Bibliography
* Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Springer-Verlag, 1988. . (antilinear maps are discussed in section 3.3).
{{DEFAULTSORT:Complex Conjugate
Complex numbers