A number of the form
,
where
and
are real numbers (cf.
Real number)
and
is the so-called
imaginary unit,
that is, a number whose square is equal to
(in engineering literature, the notation
is also used):
is called the
real part
of the complex number
and
its
imaginary part
(written
,
).
The real numbers can be regarded as special complex numbers, namely those with
.
Complex numbers that are not real, that is, for which
,
are sometimes called
imaginary numbers.
The complicated historical process of the development of the notion of a
complex number is reflected in the above
terminology which is mainly of traditional origin.
Algebraically speaking, a complex number is an element of the (algebraic)
extension
of the field of real numbers
obtained by the adjunction to the field
of a root
of the polynomial
.
The field
obtained in this way is called the
field of complex numbers
or the
complex number field.
The most important property of the field
is that it is
algebraically closed,
that is, any polynomial with coefficients in
splits into linear factors. The property of being algebraically closed
can be expressed in other words by saying that any polynomial of degree
with coefficients in
has at least one root in
(the
d'Alembert–Gauss theorem
or
fundamental theorem of algebra).
The field
can be constructed as follows. The elements
,
or
complex numbers,
are taken to be the points
,
of the plane
in Cartesian rectangular coordinates
and
,
and
.
Here the
sum of two complex numbers
and
is the complex number
,
that is,
and the
product
of those complex numbers is the complex number
,
that is,
The zero element
is the same as the origin of coordinates, and the complex number
is the identity of
.
The plane
whose points are identified with the elements of
is called the
complex plane.
The real numbers
are identified here with the points
,
of the
-axis
which, when referring to the complex plane, is called the
real axis.
The points
,
are situated on the
-axis,
called the
imaginary axis
of the complex plane
;
numbers of the form
are called
pure imaginary.
The representation of elements
of
,
or complex numbers, as points of the complex plane with
the rules
(1)
and
(2)
is equivalent to the
above more widely used form of notating complex numbers:
also called the
algebraic
or
Cartesian form
of writing complex numbers. With reference to the algebraic form, the
rules
(1)
and
(2)
reduce to the simple condition that
all operations with complex numbers are carried out as for
polynomials, taking into account the property of the imaginary unit:
.
The complex numbers
and
are called
conjugate
or
complex conjugates
in the plane
;
they are symmetrically situated with respect to the real axis. The sum
and the product of two conjugate complex numbers are the real numbers
where
is called the
modulus
or
absolute value
of
.
The following inequalities always hold:
A complex number
is different from 0 if and only if
.
The mapping
is an automorphism of the complex plane of order 2 (that is,
)
that leaves all points of the real axis fixed. Furthermore,
,
.
The operations of addition and multiplication
(1)
and
(2)
are
commutative and associative, they are related by the
distributive law, and they have the inverse operations
subtraction
and
division
(except for division by zero). The latter are expressed in algebraic form as:
Division of a complex number
by a complex number
thus reduces to multiplication of
by
It is an important question whether the extension
of the field of reals constructed above, with the rules of
operation indicated, is the only possible one or whether essentially
different variants are conceivable. The answer is given by the
uniqueness theorem:
Every (algebraic) extension of the field
obtained from
by adjoining a root
of the equation
is isomorphic to
,
that is, only the above rules of operation with complex
numbers are compatible with the requirement that the root
be algebraically adjoined. This fact, however, does not exclude the
existence of interpretations of complex numbers other than as
points of the complex plane. The following two
interpretations are most frequently employed in applications.
Vector interpretation.
A complex number
can be identified with the
vector
with coordinates
and
starting from the origin (see
Fig.).
Figure: c024140a
In this interpretation, addition and subtraction of complex numbers is
carried out according to the rules of addition and
subtraction of vectors. However, multiplication and division of complex
numbers, which must be performed according to
(2)
and
(3),
do not have immediate analogues in vector algebra (see
[4],
[5]).
The vector interpretation of complex numbers is immediately
applicable, for example, in electrical engineering in the
description of alternating sinusoidal currents and voltages.
Matrix interpretation.
The complex number
can be identified with a
-matrix
of special type
where the operations of addition, subtraction and multiplication are carried
out according to the usual rules of matrix algebra.
By using polar coordinates in the complex plane
,
that is, the radius vector
and polar angle
,
called here the
argument
of
(sometimes also called the
phase
of
),
one obtains the
trigonometric
or
polar form
of a complex number:
The argument
is a many-valued real-valued function of the complex number
,
whose values for a given
differ by integral multiples of
;
the argument of the complex number
is not defined. One usually takes the
principal value of the argument
,
defined by the additional condition
.
The
Euler formulas
transform the trigonometric form
(4)
into the
exponential form
of a complex number:
The forms
(4)
and
(5)
are particularly suitable for
carrying out multiplication and division of complex numbers:
Under multiplication (or division) of complex numbers the moduli are
multiplied (or divided) and the arguments are added (or subtracted).
Raising to a power or extracting a root is carried out according to the so-called
de Moivre formulas:
where the first of these is also applicable for negative integer exponents
.
Geometrically, multiplication of a complex number
by a complex number
reduces to rotating the vector
over the angle
(anti-clockwise if
)
and subsequently multiplying its length by
;
in particular, multiplication by a complex number
,
which has modulus one, is merely rotation over the angle
.
Thus, complex numbers can be interpreted as operators of a special type (affinors, cf.
Affinor).
In this connection, the mixed vector-matrix interpretation of
multiplication of complex numbers is sometimes useful:
in which the multiplicand is treated as a
matrix-vector and the multiplier as a matrix-operator.
The bijection
induces on the field
the topology of the
-dimensional real vector space
;
this topology is compatible with the field structure of
and thus
is a topological field. The modulus
is the Euclidean norm of the complex number
,
and
endowed with this norm is a complex one-dimensional Euclidean space, also called the
complex
-plane.
The topological product
(
times,
)
is a complex
-dimensional
Euclidean space. For a satisfactory analysis of functions
it is usually necessary to consider their behaviour in the
complex domain. This is due to the fact that
is algebraically closed. Even the behaviour of such elementary functions as
,
,
,
can be properly understood only when they are
regarded as functions of a complex variable (see
Analytic function).
Apparently, imaginary quantities first occurred in the celebrated work
The great art, or the rules of algebra
by
G. Cardano,
1545,
who regarded them as useless and unsuitable
for applications.
R. Bombelli
(1572)
was the first to realize the
value of the use of imaginary quantities, in particular for the solution of the
cubic equation
in the so-called irreducible case (when the real roots are
expressed in terms of cube roots of imaginary quantities, cf.
Cardano formula).
He gave some of the simplest rules of operation
with complex numbers. In general, expressions of the form
,
,
appearing in the solution of
quadratic
and cubic equations were called
"imaginary"
in the
16th century
and
17th century.
However, even for many of the great
scholars of the
17th century,
the algebraic and geometric nature
of imaginary quantities was unclear and even mystical. It is
known, for example, that
I. Newton
did not include imaginary quantities
within the notion of number, and that
G. Leibniz
said that
"complex numbers are a fine and wonderful refuge of the divine spirit, as if it were an amphibian of existence and non-existence" .
The problem of expressing the
-th
roots of a given number was mainly solved in the papers of
A. de Moivre
(1707,
1724)
and
R. Cotes
(1722).
The symbol
was proposed by
L. Euler
(1777,
published
1794).
It
was he who in
1751
asserted that the field
is algebraically closed;
J. d'Alembert
(1747)
came to the
same conclusion. The first rigorous proof of this fact is
due to
C.F. Gauss
(1799),
who introduced the term
"complex number"
in
1831.
The complete geometric interpretation of complex
numbers and operations on them appeared first in the work
of
C. Wessel
(1799).
The geometric representation of
complex numbers, sometimes called the
"Argand diagramArgand diagram" ,
came into use after
the publication in
1806
and
1814
of papers by
J.R. Argand,
who rediscovered, largely independently, the findings of Wessel.
The purely arithmetic theory of complex numbers as pairs of real
numbers was introduced by
W. Hamilton
(1837).
He found
a generalization of complex numbers, namely the quaternions (cf.
Quaternion),
which form a non-commutative algebra. More generally, it was
proved at the end of the
19th century
that any
extension of the notion of number beyond the complex
numbers requires sacrificing some property of the
usual operations (primarily commutativity). See also
Hypercomplex number;
Double and dual numbers;
Cayley numbers.