Encyclopedia
In
mathematics, a
complex number is a number of the form
where
a and
b are real numbers, and
i is the imaginary unit, with the property
i 2 = −1. The real number
a is called the
real part of the complex number, and the real number
b is the
imaginary part. When the imaginary part
b is 0, the complex number is just the real number
a.
For example, 3 + 2
i is a
complex number, with real part
3 and imaginary part
2 .
Complex numbers can be added, subtracted, multiplied, and divided like real numbers, but they have additional elegant properties. For example, real numbers alone do not provide a solution for every
polynomial algebraic equation with real coefficients, while complex numbers do .
In some fields , complex numbers are written as
a +
bj.
Definitions
Equality
Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. That is, if and only if and
Notation and operations
The
set of all complex numbers is usually denoted by
C, or in
blackboard bold by . The real numbers,
R, may be regarded as "lying in"
C by considering every real number as a complex: .
Complex numbers are added, subtracted, and multiplied by formally applying the
associative, commutative and distributive laws of algebra, together with the equation
i 2 = −1:
Division of complex numbers can also be defined . Thus, the set of complex numbers forms a field which, in contrast to the real numbers, is algebraically closed.
In mathematics, the adjective "complex" means that the field of complex numbers is the underlying number field considered, for example complex analysis, complex matrix,
complex polynomial and
complex Lie algebra.
The complex number field
Formally, the complex numbers can be defined as ordered pairs of real numbers together with the operations:
So defined, the complex numbers form a field, the complex number field, denoted by
C.
Since a complex number
a +
bi is uniquely specified by an ordered pair of real numbers, the complex numbers are in
one-to-one correspondence with points on a plane, called the
complex plane.
We identify the real number
a with the complex number , and in this way the field of real numbers
R becomes a subfield of
C. The imaginary unit
i is the complex number .
In
C, we have:
- additive identity :
- multiplicative identity :
- additive inverse of :
- multiplicative inverse of non-zero :
C can also be defined as the topological closure of the algebraic numbers or as the algebraic closure of
R, both of which are described below.
The complex plane
A complex number can be viewed as a point or a position vector on a two-dimensional
Cartesian coordinate system called the
complex plane or
Argand diagram .
The Cartesian coordinates of the complex number are the real part
x and the imaginary part
y, while the
polar coordinates are
r = |
z|, called the
absolute value or modulus, and f = arg, called the
complex argument of
z . Together with
Euler's formula we have
The notation cis f is sometimes used for cos f +
i sin f.
The complex argument of 0 is not defined by the equations above. There are two possible approaches for this case. The first is to consider arg an undefined form, just like 0/0. The other is to choose some fixed value and to define arg to have that value. For this approach, a conventional choice is to set arg = 0.
Note that for a non-zero complex number the complex argument is unique modulo 2p, that is, if any two values of the complex argument exactly differ by an integer multiple of 2p, they are considered equivalent.
By simple
trigonometric identities,
we see that
and that
Now the addition of two complex numbers is just the vector addition of two vectors, and the multiplication with a fixed complex number can be seen as a simultaneous rotation and stretching.
Multiplication with i corresponds to a counter clockwise rotation by 90 degrees . The geometric content of the equation
i2 = −1 is that a sequence of two 90 degree rotations results in a 180 degree rotation. Even the fact · = +1 from arithmetic can be understood geometrically as the combination of two 180 degree turns.
Absolute value, conjugation and distance
The
absolute value of a complex number
z =
r eif is defined as |
z| =
r. Algebraically, if
z =
a +
ib, then
One can check readily that the absolute value has three important properties:
if and only if
for all complex numbers
z and
w. It then follows, for example, that and . By defining the distance function
d = |
z −
w| we turn the complex numbers into a metric space and we can therefore talk about limits and continuity. The addition, subtraction, multiplication and division of complex numbers are then continuous operations. Unless anything else is said, this is always the metric being used on the complex numbers.
The complex conjugate of the complex number
z =
a +
ib is defined to be
a -
ib, written as or . As seen in the figure, is the "reflection" of
z about the real axis. The following can be checked:
-
-
-
-
- if and only if z is real
-
-
- if z is non-zero.
The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.
That conjugation commutes with all the algebraic operations is rooted in the ambiguity in choice of
i . It is important to note, however, that the function is not differentiable .
Complex fractions
Given a complex number which is to be divided by another complex number whose magnitude is non-zero, there are two ways to do this; in either case it is the same as multiplying the first by the multiplicative inverse of the second. The first way has already been implied: to convert both complex numbers into exponential form, from which their quotient is easy to derive. The second way is to express the division as a fraction, then to multiply both numerator and denominator by the complex conjugate of the denominator. This causes the denominator to simplify into a real number:
Matrix representation of complex numbers
While usually not useful, alternative representations of complex fields can give some insight into their nature. One particularly elegant representation interprets every complex number as 2×2 matrix with real entries which stretches and rotates the points of the plane. Every such matrix has the form
with real numbers
a and
b. The sum and product of two such matrices is again of this form. Every non-zero such matrix is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a field. In fact, this is exactly the field of complex numbers. Every such matrix can be written as
which suggests that we should identify the real number 1 with the matrix
and the imaginary unit
i with
a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeed equal to −1.
The absolute value of a complex number expressed as a matrix is equal to the
square root of the determinant of that matrix. If the matrix is viewed as a transformation of a plane, then the transformation rotates points through an angle equal to the argument of the complex number and scales by a factor equal to the complex number's absolute value. The conjugate of the complex number
z corresponds to the transformation which rotates through the same angle as
z but in the opposite direction, and scales in the same manner as
z; this can be described by the transpose of the matrix corresponding to
z.
If the matrix elements are themselves complex numbers, then the resulting algebra is that of the
quaternions. In this way, the matrix representation can be seen as a way of expressing the Cayley-Dickson construction of algebras.
Geometric interpretation of the operations on complex numbers
Consider a plane. One point is the
origin, 0. Another point is the
unity, 1.
Addition
The
sum of two points
A and
B is the point
X =
A+
B such that the
triangles with vertices 0,
A,
B and
X,
B,
A are similar.
Multiplication
The
product of two points
A and
B is the point
X =
AB such that the triangles with vertices 0, 1,
A, and
0,
B,
X are similar.
Conjugation
The
complex conjugate of a point
A is a point
X =
A* such that the triangles with vertices 0, 1,
A and 0, 1,
X are mirror image of each other.
Some properties
Real vector space
C is a two-dimensional real vector space.
Unlike the reals, complex numbers cannot be ordered in any way that is compatible with its arithmetic operations:
C cannot be turned into an ordered field.
R-linear maps
C ?
C have the general form
with complex coefficients
a and
b. Only the first term is
C-linear; also only the first term is holomorphic; the second term is real-differentiable, but does not satisfy the Cauchy-Riemann equations.
The function
corresponds to rotations combined with scaling, while the function
corresponds to reflections combined with scaling.
Solutions of polynomial equations
A
root of the
polynomial p is a complex number
z such
that
p = 0.
A most striking result is that all polynomials of
degree
n with real or complex coefficients have exactly
ncomplex roots . This is known as the fundamental theorem of algebra, and shows that the complex numbers are an algebraically closed field.
Indeed, the complex number field is the algebraic closure of the real number field, and
Cauchy constructed complex numbers in this way. It can be identified as the quotient ring of the
polynomial ring
R[
X] by the ideal generated by the polynomial
X2 + 1:
This is indeed a field because
X2 + 1 is irreducible, hence generating a maximal ideal, in
R[
X]. The image of
X in this quotient ring becomes the imaginary unit
i.
Algebraic characterization
The field
C is characterized by the following three facts:
- its characteristic is 0
- its transcendence degree over the prime field is the cardinality of the continuum
- it is algebraically closed
Consequently,
C contains many proper subfields which are isomorphic to
C. Another consequence of this characterization is that the Galois group of
C over the rational numbers is enormous, with cardinality equal to that of the power set of the continuum.
Characterization as a topological field
As noted above, the algebraic characterization of
C fails to capture some of its most important properties. These properties, which underpin the foundations of complex analysis, arise from the
topology of
C. The following properties characterize
C as a topological field:
- C is a field.
- C contains a subset P of nonzero elements satisfying:
- P is closed under addition, multiplication and taking inverses.
- If x and y are distinct elements of P, then either x-y or y-x is in P
- If S is any nonempty subset of P, then S+P=x+P for some x in C.
- C has a nontrivial involutive automorphism x->x*, fixing P and such that xx* is in P for any nonzero x in C.
Given these properties, one can then define a topology on
C by taking the setsas a base, where
x ranges over
C, and
p ranges over
P.
To see that these properties characterize
C as a topological field, one notes that
P ? ?
-P is an ordered Dedekind-complete field and thus can be identified with the real numbers
R by a unique field isomorphism. The last property is easily seen to imply that the Galois group over the real numbers is of order two, completing the characterization.
Pontryagin has shown that the only
connected locally compact topological fields are
R and
C. This gives another characterization of
C as a topological field, since
C can be distinguished from
R by noting the nonzero complex numbers are
connected whereas the nonzero real numbers are not.
Complex analysis
The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis . Unlike real functions which are commonly represented as two dimensional graphs, complex functions have four dimensional graphs
and may usefully be illustrated by color coding a three dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.
Applications
The words "real" and "imaginary" were meaningful when complex numbers were used mainly as an aid in manipulating "real" numbers, with only the "real" part directly describing the world. Later applications, and especially the discovery of quantum mechanics, showed that nature has no preference for "real" numbers and its most
real descriptions often require complex numbers, the "imaginary" part being just as physical as the "real" part.
Control theory
In
control theory, systems are often transformed from the time domain to the frequency domain using the
Laplace transform. The system's poles and zeros are then analyzed in the
complex plane. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane.
In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes, i.e. have real part greater than or less than zero. If a system has poles that are
- in the right half plane, it will be unstable,
- all in the left half plane, it will be stable,
- on the imaginary axis, it will have marginal stability.
If a system has zeros in the right half plane, it is a
nonminimum phase system.
Signal analysis
Complex numbers are used in signal analysis and other fields as a convenient description for periodically varying signals. The absolute value |
z| is interpreted as the
amplitude and the argument arg as the phase of a
sine wave of given
frequency.
If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as complex valued functions of the form
where ? represents the
angular frequency and the complex number
z encodes the phase and amplitude as explained above.
In
electrical engineering, the Fourier transform is used to analyze varying
voltages and
currents. The treatment of
resistors,
capacitors, and
inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis to transmit, compress, restore, and otherwise process digital
audio signals, still images, and
video signals.
Frequency domain electromagnetism
Maxwell's equations are written in terms of real vector function's of space and time. When fourier transformed to functions of space and frequency the fields become complex, as in signal analysis. The majority of electromagnetic calculations are done in the frequency domain.
Sign convention
In such calculations, engineers tend to use to describe a plane wave, for compatibility with signal analysis, while physicists tend to use for compatibility with quantum mechanics and other scattering calculations. It is therefore sometimes said that j = –i.
Improper integrals
In applied fields, the use of complex analysis is often used to compute certain real-valued
improper integrals, by means of complex-valued functions. Several methods exist to do this, see
methods of contour integration.
Quantum mechanics
The complex number field is also of utmost importance in
quantum mechanicssince the underlying theory is built on Hilbert spaces over
C. The more limited original formulations of
Schrödinger and
Heisenberg are also in terms of complex numbers.
Relativity
In
special and
general relativity, some formulae for the metric on spacetime become simpler if one takes the time variable to be imaginary.
Applied mathematics
In
differential equations, it is common to
first find all complex roots
r of the characteristic equation of a
linear differential equation and then attempt to solve the system
in terms of base functions of the form
f =
ert.
Fluid dynamics
In fluid dynamics, complex functions are used to describe potential flow in 2d.
Fractals
Certain
fractals are plotted in the complex plane e.g.
Mandelbrot set and
Julia set.
History
The earliest fleeting reference to
square roots of negative numbers perhaps occurred in the work of the
Greek mathematician and inventor
Heron of Alexandria in the
1st century CE, when he considered the volume of an impossible
frustum of a
pyramid , though negative numbers were not conceived in the
Hellenistic world.
Complex numbers became more prominent in the
16th century, when closed formulas for the roots of
cubic and quartic
polynomials were discovered by Italian mathematicians . It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. For example, Tartaglia's cubic formula gives the following solution to the equation :
At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation has solutions −
i, and . Substituting these in turn for into the cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of
This was doubly unsettling since not even negative numbers were considered to be on firm ground at the time. The term "imaginary" for these quantities was coined by
René Descartes in 1637 and was meant to be derogatory . A further source of confusion was that the equation seemed to be capriciously inconsistent with the algebraic identity , which is valid for positive real numbers
a and
b, and which was also used in complex number calculations with one of
a,
b positive and the other negative. The incorrect use of this identity in the case when both
a and
b are negative even bedeviled
Euler. This difficulty eventually led to the convention of using the special symbol
i in place of to guard against this mistake.
The
18th century saw the labors of
Abraham de Moivre and
Leonhard Euler. To De Moivre is due the well-known formula which bears his name, de Moivre's formula:
and to Euler
Euler's formula of complex analysis:
The existence of complex numbers was not completely accepted until the geometrical interpretation had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by
Carl Friedrich Gauss, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in
Wallis's De Algebra tractatus.
Wessel's memoir appeared in the Proceedings of the Copenhagen Academy for 1799, and is exceedingly clear and complete, even in comparison with modern works. He also considers the sphere, and gives a
quaternion theory from which he develops a complete spherical trigonometry. In 1804 the Abbé Buée independently came upon the same idea which Wallis had suggested, that should represent a unit line, and its negative, perpendicular to the real axis. Buée's paper was not published until 1806, in which year Jean-Robert Argand also issued a pamphlet on the same subject. It is to Argand's essay that the scientific foundation for the graphic representation of complex numbers is now generally referred. Nevertheless, in 1831 Gauss found the theory quite unknown, and in 1832 published his chief memoir on the subject, thus bringing it prominently before the mathematical world. Mention should also be made of an excellent little treatise by Mourey , in which the foundations for the theory of directional numbers are scientifically laid. The general acceptance of the theory is not a little due to the labors of Augustin Louis Cauchy and
Niels Henrik Abel, and especially the latter, who was the first to boldly use complex numbers with a success that is well known.
The common terms used in the theory are chiefly due to the founders. Argand called the
direction factor, and the
modulus; Cauchy called the
reduced form ; Gauss used
i for , introduced the term
complex number for , and called the
norm.
The expression
direction coefficient, often used for , is due to Hankel , and
absolute value, for
modulus, is due to Weierstrass.
Following Cauchy and Gauss have come a number of contributors of high rank, of whom the following may be especially mentioned: Kummer , Leopold Kronecker , Scheffler , Bellavitis , Peacock , and De Morgan . Möbius must also be mentioned for his numerous memoirs on the geometric applications of complex numbers, and
Dirichlet for the expansion of the theory to include primes, congruences, reciprocity, etc., as in the case of real numbers.
A complex ring or field is a set of complex numbers which is closed under addition, subtraction, and multiplication.
Gauss studied complex numbers of the form , where
a and
b are integral, or rational . His student,
Ferdinand Eisenstein, studied the type , where is a complex root of . Other such classes of complex numbers are derived from the roots of unity for higher values of . This generalization is largely due to Kummer, who also invented ideal numbers, which were expressed as geometrical entities by Felix Klein in 1893. The general theory of fields was created by Évariste Galois, who studied the fields generated by the roots of any polynomial equation
The late writers on the general theory include
Weierstrass, Schwarz,
Richard Dedekind, Otto Hölder, Berloty,
Henri Poincaré, Eduard Study, and
Alexander MacFarlane.
The formally correct definition using pairs of real numbers was given in the
19th century.
See also
- Circular motion#Using complex numbers
- Complex geometry
- De Moivre's formula
- Euler's identity
...
Further reading
- An Imaginary Tale: The Story of , by Paul J. Nahin; Princeton University Press; ISBN 0-691-02795-1 . A gentle introduction to the history of complex numbers and the beginnings of complex analysis.
- Numbers, by H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, R. Remmert; Springer; ISBN 0-387-97497-0 . An advanced perspective on the historical development of the concept of number.
- The Road to Reality: A Complete Guide to the Laws of the Universe, by Roger Penrose; Alfred A. Knopf, 2005; ISBN 0-679-45443-8. Chapters 4-7 in particular deal extensively with complex numbers.
References
External links
