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 fo ...
is the number with an equal
real 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 ...
. 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, 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 ...
.
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 th ...
).
If a root of a
univariate 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 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'' (franchi ...
, 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 ...
, 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 def ...
(†) 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 n ...
, 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 i ...
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, 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 multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
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 of ...
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 exampl ...
with
real 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''&nbs ...
).
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 deriv ...
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 isom ...
(where the topology on
is taken to be the standard topology) and
antilinear, 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 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
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 automorphis ...
. 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 po ...
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 dialecti ...
:
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's book ''Inversive Geometry'' (1933), written with his son Frank Vigor Morley.
Generalizations
The other planar real unital algebras,
dual numbers, 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 of
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 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 continuou ...
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 quater ...
s and
split-quaternions: 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 of ...
, this reversal is not needed there.
There is also an abstract notion of conjugation for
vector spaces 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 fo ...
s. In this context, any
antilinear map 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, unc ...
on
#
for all
and
#
for all
is called a , or a
real structure. As the involution
is
antilinear, 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
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
[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