A
complex number is a
numberA number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
consisting of a
realIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
part and an
imaginaryAn imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...
part. Complex numbers extend the idea of the one-dimensional
number lineIn basic mathematics, a number line is a picture of a straight line on which every point is assumed to correspond to a real number and every real number to a point. Often the integers are shown as specially-marked points evenly spaced on the line...
to the two-dimensional
complex planeIn mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
by using the number line for the real part and adding a vertical axis to plot the imaginary part. In this way the complex numbers contain the ordinary real numbers while extending them in order to solve problems that would be impossible with only real numbers.
Complex numbers are used in many scientific fields, including
engineeringEngineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
,
electromagnetismElectromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
, quantum physics,
applied mathematicsApplied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
, and
chaos theoryChaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
. Italian mathematician
Gerolamo CardanoGerolamo Cardano was an Italian Renaissance mathematician, physician, astrologer and gambler...
is the first known to have introduced complex numbers; he called them "fictitious", during his attempts to find solutions to cubic equations in the 16th century.
Introduction and definition
Complex numbers have been introduced to allow for solutions of certain equations that have no real solution: the equation
has no real solution
x, since the square of
x is 0 or positive, so cannot be zero. Complex numbers are a solution to this problem. The idea is to enhance the real numbers by introducing a
non-real number iIn mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
whose square is −1, so that and are the two solutions to the preceding equation.
Definition
A
complex number is an expression of the form
where
a and
b are
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s and
i is the
imaginary unitIn mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
, satisfying
i2 = −1. For example, −3.5 + 2
i is a complex number.
The real number
a of the complex number
z =
a +
bi is called the
real part of
z and the real number
b is the
imaginary part. They are denoted Re(
z) or ℜ(
z) and Im(
z) or ℑ(
z), respectively. For example,
Some authors write
a+
ib instead of
a+
bi. In some disciplines (in particular,
electrical engineeringElectrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...
, where
i is a symbol for
currentElectric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
), in order to avoid notational conflict, the
imaginary unitIn mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
i is instead written as
j, so complex numbers are written as
a +
bj or
a +
jb.
A real number
a can usually be regarded as a complex number with an imaginary part of zero, that is to say,
a + 0
i. However the sets are defined differently and have slightly different operations defined, for instance comparison operations are not defined for complex numbers. Complex numbers whose real part is zero, that is to say, those of the form 0 +
bi, are called
imaginary numbers. It is common to write
a for
a + 0
i and
bi for . Moreover, when the imaginary part is negative, it is common to write
a − bi with
b > 0 instead of
a + (−
b)
i, for example 3 − 4
i instead of 3 + (−4)
i.
The set of all complex numbers is denoted by
C or
.
The complex plane
A complex number can be viewed as a point or position vector in a two-dimensional
Cartesian coordinate systemA Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
called the
complex planeIn mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
or Argand diagram (see and ), named after
Jean-Robert ArgandJean-Robert Argand was a gifted amateur mathematician. In 1806, while managing a bookstore in Paris, he published the idea of geometrical interpretation of complex numbers known as the Argand diagram.-Life:...
. The numbers are conventionally plotted using the real part as the horizontal component, and imaginary part as vertical (see Figure 1). These two values used to identify a given complex number are therefore called its
Cartesian,
rectangular, or
algebraic form.
The defining characteristic of a position vector is that it has magnitude and direction. These are emphasised in a complex number's
polar form and it turns out notably that the operations of addition and multiplication take on a very natural geometric character when complex numbers are viewed as position vectors: addition corresponds to vector addition while multiplication corresponds to multiplying their magnitudes and adding their arguments (i.e. the angles they make with the
x axis). Viewed in this way the multiplication of a complex number by
i corresponds to rotating a complex number counterclockwise through 90° about the origin:
.
History in brief
- Main section: History
The solution in radicals (without trigonometric functions) of a general cubic equation contains the square roots of negative numbers when all three roots are real numbers, a situation that cannot be rectified by factoring aided by the rational root test if the cubic is irreducible (the so-called
casus irreducibilisIn algebra, casus irreducibilis is one of the cases that may arise in attempting to solve a cubic equation with integer coefficients with roots that are expressed with radicals...
). This conundrum led Italian mathematician
Gerolamo CardanoGerolamo Cardano was an Italian Renaissance mathematician, physician, astrologer and gambler...
to conceive of complex numbers in around 1545, though his understanding was rudimentary.
Work on the problem of general polynomials ultimately led to the
fundamental theorem of algebraThe fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
, which shows that with complex numbers, a solution exists to every
polynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
equation of degree one or higher. Complex numbers thus form an
algebraically closedIn mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
fieldIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, where any
polynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
equation has a root.
Many mathematicians contributed to the full development of complex numbers. The rules for addition, subtraction, multiplication, and division of complex numbers were developed by the Italian mathematician
Rafael BombelliRafael Bombelli was an Italian mathematician.Born in Bologna, he is the author of a treatise on algebra and is a central figure in the understanding of imaginary numbers....
. A more abstract formalism for the complex numbers was further developed by the Irish mathematician
William Rowan HamiltonSir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...
, who extended this abstraction to the theory of quaternions.
Conjugation
The
complex conjugateIn mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
of the complex number
z =
x +
yi is defined to be
x −
yi. It is denoted
or
. Geometrically,
is the "reflection" of
z about the real axis. In particular, conjugating twice gives the original complex number:
.
The real and imaginary parts of a complex number can be extracted using the conjugate:
-
-
Moreover, a complex number is real if and only if it equals its conjugate.
Conjugation distributes over the standard arithmetic operations:
-
-
-
The
reciprocalIn 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/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
of a nonzero complex number
z =
x +
yi is given by
-
This formula can be used to compute the multiplicative inverse of a complex number if it is given in rectangular coordinates. Inversive geometry, a branch of geometry studying more general reflections than ones about a line, can also be expressed in terms of complex numbers.
Addition and subtraction
Complex numbers are
addedAddition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
by adding the real and imaginary parts of the summands. That is to say:
Similarly,
subtractionIn arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
is defined by
Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers
A and
B, interpreted as points of the complex plane, is the point
X obtained by building a
parallelogramIn Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure...
three of whose vertices are 0,
A and
B. Equivalently,
X is the point such that the
triangleA triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
s with vertices 0,
A,
B, and
X,
B,
A, are
congruentIn geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
.
Multiplication and division
The multiplication of two complex numbers is defined by the following formula:
In particular, the square of the imaginary unit is −1:
The preceding definition of multiplication of general complex numbers is the natural way of extending this fundamental property of the imaginary unit. Indeed, treating
i as a variable, the formula follows from this
(distributive law)
-
-
(commutative law of addition—the order of the summands can be changed)
(commutative law of multiplication—the order of the factors can be changed)
(fundamental property of the imaginary unit).
The division of two complex numbers is defined in terms of complex multiplication, which is described above, and real division:
Division can be defined in this way because of the following observation:
As shown earlier,
is the complex conjugate of the denominator
. The real part
c and the imaginary part
d of the denominator must not both be zero for division to be defined.
Square root
The square roots of
a +
bi (with
b ≠ 0) are
, where
and
where sgn is the
signumIn mathematics, the sign function is an odd mathematical function that extracts the sign of a real number. To avoid confusion with the sine function, this function is often called the signum function ....
function. This can be seen by squaring
to obtain
a +
bi. Here
is called the
modulusIn mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
of
a + bi, and the square root with non-negative real part is called the
principal square root.
Polar form
Absolute value and argument
Another way of encoding points in the complex plane other than using the
x- and
y-coordinates is to use the distance of a point
P to
O, the point whose coordinates are (0, 0) (
originIn mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...
), and the angle of the line through
P and
O. This idea leads to the polar form of complex numbers.
The
absolute valueIn mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
(or
modulus or
magnitude) of a complex number is
If
z is a real number (i.e.,
y = 0), then
r = |
x|. In general, by Pythagoras' theorem,
r is the distance of the point
P representing the complex number
z to the origin.
The
argumentIn mathematics, arg is a function operating on complex numbers . It gives the angle between the line joining the point to the origin and the positive real axis, shown as in figure 1 opposite, known as an argument of the point In mathematics, arg is a function operating on complex numbers...
or
phase of
z is the angle of the
radiusIn classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
OP with the positive real axis, and is written as
. As with the modulus, the argument can be found from the rectangular form
:
The value of
φ must always be expressed in
radianRadian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s. It can change by any multiple of 2
π and still give the same angle. Hence, the arg function is sometimes considered as
multivaluedIn mathematics, a multivalued function is a left-total relation; i.e. every input is associated with one or more outputs...
. Normally, as given above, the
principal valueIn considering complex multiple-valued functions in complex analysis, the principal values of a function are the values along one chosen branch of that function, so it is single-valued.-Motivation:...
in the interval
is chosen. Values in the range
are obtained by adding
if the value is negative. The polar angle for the complex number 0 is undefined, but arbitrary choice of the angle 0 is common.
The value of
φ equals the result of
atan2In trigonometry, the two-argument function atan2 is a variation of the arctangent function. For any real arguments and not both equal to zero, is the angle in radians between the positive -axis of a plane and the point given by the coordinates on it...
:
.
Together,
r and
φ give another way of representing complex numbers, the
polar form, as the combination of modulus and argument fully specify the position of a point on the plane. Recovering the original rectangular co-ordinates from the polar form is done by the formula called
trigonometric form
Using
Euler's formulaEuler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
this can be written as
Using the cis function, this is sometimes abbreviated to
In
angle notation, often used in
electronicsElectronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...
to represent a phasor with amplitude
r and phase
φ it is written as
Multiplication, division and exponentiation in polar form
The relevance of representing complex numbers in polar form stems from the fact that the formulas for multiplication, division and exponentiation are simpler than the ones using Cartesian coordinates. Given two complex numbers
z1 =
r1(cos φ
1 +
isin φ
1) and
z2 =
r2(cos φ
2 +
isin φ
2) the formula for multiplication is
In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. For example, multiplying by
i corresponds to a quarter-rotation counter-clockwise, which gives back
i 2 = −1. The picture at the right illustrates the multiplication of
Since the real and imaginary part of 5+5
i are equal, the argument of that number is 45 degrees, or π/4 (in
radianRadian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
). On the other hand, it is also the sum of the angles at the origin of the red and blue triangle are arctan(1/3) and arctan(1/2), respectively. Thus, the formula
holds. As the arctan function can be approximated highly efficiently, formulas like this—known as Machin-like formulas—are used for high-precision approximations of π.
Similarly, division is given by
This also implies
de Moivre's formulaIn mathematics, de Moivre's formula , named after Abraham de Moivre, states that for any complex number x and integer n it holds that...
for exponentiation of complex numbers with integer exponents:
The
n-th
rootsIn mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...
of
z are given by
for any integer . Here
is the usual (positive)
nth root of the positive real number
r. While the
nth root of a positive real number
r is chosen to be the
positive real number
c satisfying
cn =
x there is no natural way of distinguishing one particular complex
nth root of a complex number. Therefore, the
nth root of
z is considered as a
multivalued functionIn mathematics, a multivalued function is a left-total relation; i.e. every input is associated with one or more outputs...
(in
z), as opposed to a usual function
f, for which
f(
z) is a uniquely defined number. Formulas such as
(which holds for positive real numbers), do in general not hold for complex numbers.
Field structure
The set
C of complex numbers is a
fieldIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
. Briefly, this means that the following facts hold: first, any two complex numbers can be added and multiplied to yield another complex number. Second, for any complex number
a, its negative −
a is also a complex number; and third, every nonzero complex number has a
reciprocalIn 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/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
complex number. Moreover, these operations satisfy a number of laws, for example the law of
commutativityIn mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
of addition and multiplication for any two complex numbers
z1 and
z2:
These two laws and the other requirements on a field can be proven by the formulas given above, using the fact that the real numbers themselves form a field.
Unlike the reals,
C is not an
ordered fieldIn mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and...
, that is to say, it is not possible to define a relation
z1 <
z2 that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so
i2 = −1 precludes the existence of an
orderingIn set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
on
C.
When the underlying field for a mathematical topic or construct is the field of complex numbers, the thing's name is usually modified to reflect that fact. For example:
complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, complex
matrixIn mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
, complex
polynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
, and complex
Lie algebraIn mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
.
Solutions of polynomial equations
Given any complex numbers (called
coefficientIn mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
s)
a0, ...,
an, the equation
has at least one complex solution
z, provided that at least one of the higher coefficients,
a1, ...,
an, is nonzero. This is the statement of the
fundamental theorem of algebraThe fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
. Because of this fact,
C is called an
algebraically closed fieldIn mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
. This property does not hold for the field of rational numbers
Q (the polynomial
x2 − 2 does not have a rational root, since √ is not a rational number) nor the
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s
R (the polynomial
x2 +
a does not have a real solution for
a > 0, since the square of
x is positive for any real number
x).
There are various proofs of this theorem, either by analytic methods such as
Liouville's theoremIn complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that |f| ≤ M for all z in C is constant.The theorem is considerably improved by...
, or
topologicalTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
ones such as the
winding numberIn mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point...
, or a proof combining
Galois theoryIn mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
and the fact that any real polynomial of
odd degree has at least one root.
Because of this fact, theorems that hold "
for any algebraically closed field", apply to
C. For example, any complex
matrixIn mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
has at least one (complex) eigenvalue.
Algebraic characterization
The field
C has the following three properties: first, it has
characteristicIn mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
0. This means that 1 + 1 + ... + 1 ≠ 0 for any number of summands (all of which equal one). Second, its
transcendence degreeIn abstract algebra, the transcendence degree of a field extension L /K is a certain rather coarse measure of the "size" of the extension...
over
Q, the prime field of
C is the
cardinality of the continuumIn set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by |\mathbb R| or \mathfrak c ....
. Third, it is algebraically closed (see above). It can be shown that any field having these properties is isomorphic (as a field) to
C. For example, the algebraic closure of
Qp also satisfies these three properties, so these two fields are isomorphic. Also,
C is isomorphic to the field of complex
Puiseux seriesIn mathematics, Puiseux series are a generalization of formal power series, first introduced by Isaac Newton in 1676 and rediscovered by Victor Puiseux in 1850, that allows for negative and fractional exponents of the indeterminate...
. However, specifying an isomorphism requires the
axiom of choice. Another consequence of this algebraic characterization is that
C contains many proper subfields which are isomorphic to
C (the same is true of
R, which contains many sub fields isomorphic to itself).
Characterization as a topological field
The preceding characterization of
C describes the algebraic aspects of
C, only. That is to say, the properties of nearness and continuity, which matter in areas such as
analysisAnalysis is the process of breaking a complex topic or substance into smaller parts to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle , though analysis as a formal concept is a relatively recent development.The word is...
and
topologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, are not dealt with. The following description of
C as a
topological fieldIn mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as mapswhere R × R carries the product topology.- General comments :...
(that is, a field that is equipped with a
topologyTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, which allows the notion of convergence) does take into account the topological properties.
C contains a subset
P (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:
- 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.
Moreover,
C has a nontrivial involutive
automorphismIn 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 of an object forms a group, called the automorphism...
(namely the complex conjugation), such that
xx∗ is in
P for any nonzero
x in
C.
Any field
F with these properties can be endowed with a topology by taking the sets
B(
x,
p) = {
y |
p − (
y −
x)(
y −
x)
∗ ∈
P} as a
baseIn mathematics, a base B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T...
, where
x ranges over the field and
p ranges over
P. With this topology
F is isomorphic as a
topological field to
C.
The only
connectedIn topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
locally compact
topological fieldsIn mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as mapswhere R × R carries the product topology.- General comments :...
are
R and
C. This gives another characterization of
C as a topological field, since
C can be distinguished from
R because the nonzero complex numbers are
connectedIn topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
, while the nonzero real numbers are not.
Formal development
Above, complex numbers have been defined by introducing
i, the imaginary unit, as a symbol. More rigorously, the set
C of complex numbers can be defined as the set
R2 of ordered pairs (
a,
b) of real numbers. In this notation, the above formulas for addition and multiplication read
-
-
It is then just a matter of notation to express (a, b) as a + ib.
Though this low-level construction does accurately describe the structure of the complex numbers, the following equivalent definition reveals the algebraic nature of C more immediately. This characterization relies on the notion of fields and polynomials. A field is a set endowed with an addition, subtraction, multiplication and division operations which behave as is familiar from, say, rational numbers. For example, the distributive law
is required to hold for any three elements
x,
y and
z of a field. The set
R of real numbers does form a field. A polynomial p(
X) with real
coefficientIn mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
s is an expression of the form
where the
a0, ...,
an are real numbers. The usual addition and multiplication of polynomials endows the set
R[
X] of all such polynomials with a
ringIn mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
structure. This ring is called
polynomial ringIn mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
. The
quotient ringIn ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...
R[
X]/(
X2+1) can be shown to be a field.
This extension field contains two square roots of −1, namely (the
cosetIn mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
s of)
X and −
X, respectively. (The cosets of) 1 and
X form a basis of
R[
X]/(
X2+1) as a real
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
, which means that each element of the extension field can be uniquely written as a
linear combinationIn mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
in these two elements. Equivalently, elements of the extension field can be written as ordered pairs (
a,
b) of real numbers. Moreover, the above formulas for addition etc. correspond to the ones yielded by this
abstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
ic approach – the two definitions of the field
C are said to be
isomorphicIn abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
(as fields). Together with the above-mentioned fact that
C is algebraically closed, this also shows that
C is an
algebraic closureIn mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
of
R.
Matrix representation of complex numbers
Complex numbers can also be represented by 2×2
matricesIn mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
that have the following form:
Here the entries
a and
b are real numbers. The sum and product of two such matrices is again of this form, and the sum and product of complex numbers corresponds to the sum and
productIn mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
of such matrices. The geometric description of the multiplication of complex numbers can also be phrased in terms of
rotation matrices by using this correspondence between complex numbers and such matrices. Moreover, the square of the absolute value of a complex number expressed as a matrix is equal to the
determinantIn linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of that matrix:
The conjugate
corresponds to the
transposeIn linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
of the matrix.
Though this representation of complex numbers with matricies is the most common, many other representations arise from matrices
other than 
that square to the negative of the
identity matrixIn linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
. See the article on 2 × 2 real matrices for other representations of complex numbers.
Complex analysis
The study of functions of a complex variable is known as
complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
and has enormous practical use in
applied mathematicsApplied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
as well as in other branches of mathematics. Often, the most natural proofs for statements in
real analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
or even
number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
employ techniques from complex analysis (see
prime number theoremIn number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....
for an example). 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.
Complex exponential and related functions
The notions of
convergent series and
continuous functionIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s in (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said to converge if and only if its real and imaginary parts do. This is equivalent to the (ε, δ)-definition of limits, where the absolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view,
C, endowed with the
metricIn mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
is a complete
metric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
, which notably includes the
triangle inequalityIn mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side ....
for any two complex numbers
z1 and
z2.
Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the
exponential functionIn mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
exp(
z), also written
ez, is defined as the infinite series
and the series defining the real trigonometric functions
sineIn mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
and cosine, as well as hyperbolic functions such as sinh also carry over to complex arguments without change.
Euler's identity states:
for any real number
φ, in particular
Unlike in the situation of real numbers, there is an
infinitude of complex solutions
z of the equation
for any complex number . It can be shown that any such solution
z—called
complex logarithmIn complex analysis, a complex logarithm function is an "inverse" of the complex exponential function, just as the natural logarithm ln x is the inverse of the real exponential function ex. Thus, a logarithm of z is a complex number w such that ew = z. The notation for such a w is log z...
of
a—satisfies
where arg is the
argumentIn mathematics, arg is a function operating on complex numbers . It gives the angle between the line joining the point to the origin and the positive real axis, shown as in figure 1 opposite, known as an argument of the point In mathematics, arg is a function operating on complex numbers...
defined above, and ln the (real)
natural logarithmThe natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...
. As arg is a
multivalued functionIn mathematics, a multivalued function is a left-total relation; i.e. every input is associated with one or more outputs...
, unique only up to a multiple of 2
π, log is also multivalued. The
principal valueIn considering complex multiple-valued functions in complex analysis, the principal values of a function are the values along one chosen branch of that function, so it is single-valued.-Motivation:...
of log is often taken by restricting the imaginary part to the
intervalIn mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
(−π,π].
Complex
exponentiationExponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
zω is defined as
Consequently, they are in general multi-valued. For
ω = 1 /
n, for some natural number
n, this recovers the non-unicity of
n-th roots mentioned above.
Complex numbers, unlike real numbers, do not in general satisfy the unmodified power and logarithm identities, particularly when naïvely treated as single-valued functions; see failure of power and logarithm identities. For example they do not satisfy
Both sides of the equation are multivalued by the definition of complex exponentiation given here, and the values on the left are a subset of those on the right.
Holomorphic functions
A function
f :
C →
C is called holomorphic if it satisfies the
Cauchy-Riemann equationsIn mathematics, the Cauchy–Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which must be satisfied if we know that a complex function is complex differentiable...
. For example, any
R-linear map
C →
C can be written in the form
with complex coefficients
a and
b. This map is holomorphic
if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
b = 0. The second summand
is real-differentiable, but does not satisfy the
Cauchy-Riemann equationsIn mathematics, the Cauchy–Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which must be satisfied if we know that a complex function is complex differentiable...
.
Complex analysis shows some features not apparent in real analysis. For example, any two holomorphic functions
f and
g that agree on an arbitrarily small open subset of
C necessarily agree everywhere.
Meromorphic functionIn complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
s, functions that can locally be written as
f(
z)/(
z −
z0)
n with a holomorphic function
f(
z), still share some of the features of holomorphic functions. Other functions have
essential singularitiesIn complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits extreme behavior.The category essential singularity is a "left-over" or default group of singularities that are especially unmanageable: by definition they fit into neither of the...
, such as sin(1/
z) at
z = 0.
Control theory
In
control theoryControl theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
, systems are often transformed from the
time domainTime domain is a term used to describe the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the case of continuous time, or at various...
to the
frequency domainIn electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....
using the
Laplace transform. The system's
poles and
zerosIn complex analysis, a zero of a holomorphic function f is a complex number a such that f = 0.-Multiplicity of a zero:A complex number a is a simple zero of f, or a zero of multiplicity 1 of f, if f can be written asf=g\,where g is a holomorphic function g such that g is not zero.Generally, the...
are then analyzed in the
complex plane. The
root locusRoot locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly the gain of a feedback system. This is a technique used in the field of control systems developed by Walter R...
,
Nyquist plotA Nyquist plot is a parametric plot of a transfer function used in automatic control and signal processing. The most common use of Nyquist plots is for assessing the stability of a system with feedback. In Cartesian coordinates, the real part of the transfer function is plotted on the X axis. The...
, and
Nichols plot techniques all make use of the complex plane.
In the root locus method, it is especially important whether the
poles and
zerosIn complex analysis, a zero of a holomorphic function f is a complex number a such that f = 0.-Multiplicity of a zero:A complex number a is a simple zero of f, or a zero of multiplicity 1 of f, if f can be written asf=g\,where g is a holomorphic function g such that g is not zero.Generally, the...
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
In electrical engineering, specifically signal processing and control theory, BIBO stability is a form of stability for linear signals and systems that take inputs. BIBO stands for Bounded-Input Bounded-Output...
,
- on the imaginary axis, it will have marginal stability
In the theory of dynamical systems, and control theory, a continuous linear time-invariant system is marginally stable if and only if the real part of every eigenvalue in the system's transfer-function is non-positive, and all eigenvalues with zero real value are simple roots...
.
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 for a convenient description for periodically varying signals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a
sine waveThe sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...
of a given
frequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
, the absolute value |
z| of the corresponding
z is the
amplitudeAmplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
and the argument arg(
z) the
phasePhase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...
.
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 ω represents the
angular frequencyIn physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
and the complex number
z encodes the phase and amplitude as explained above.
In
electrical engineeringElectrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...
, the
Fourier transformIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
is used to analyze varying
voltageVoltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
s and
currentsElectric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
. The treatment of
resistorA linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
s,
capacitorA capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
s, and
inductorAn inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
s 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
impedanceElectrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
. This approach is called phasor calculus. This use is also extended into
digital signal processingDigital signal processing is concerned with the representation of discrete time signals by a sequence of numbers or symbols and the processing of these signals. Digital signal processing and analog signal processing are subfields of signal processing...
and
digital image processingDigital image processing is the use of computer algorithms to perform image processing on digital images. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing...
, which utilize digital versions of Fourier analysis (and
waveletA wavelet is a wave-like oscillation with an amplitude that starts out at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one might see recorded by a seismograph or heart monitor. Generally, wavelets are purposefully crafted to have...
analysis) to transmit,
compressIn computer science and information theory, data compression, source coding or bit-rate reduction is the process of encoding information using fewer bits than the original representation would use....
, restore, and otherwise process
digitalA digital system is a data technology that uses discrete values. By contrast, non-digital systems use a continuous range of values to represent information...
audioSound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.-Propagation of...
signals, still images, and
videoVideo is the technology of electronically capturing, recording, processing, storing, transmitting, and reconstructing a sequence of still images representing scenes in motion.- History :...
signals.
Improper integrals
In applied fields, complex numbers are often used to compute certain real-valued
improper integralIn calculus, an improper integral is the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits....
s, by means of complex-valued functions. Several methods exist to do this; see
methods of contour integrationIn the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.Contour integration is closely related to the calculus of residues, a methodology of complex analysis....
.
Quantum mechanics
The complex number field is relevant in the mathematical formulations of quantum mechanics, where complex
Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
s provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the
SchrödingerErwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...
equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
and
HeisenbergWerner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
's
matrix mechanicsMatrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.Matrix mechanics was the first conceptually autonomous and logically consistent formulation of quantum mechanics. It extended the Bohr Model by describing how the quantum jumps...
– make use of complex numbers.
Relativity
In
specialSpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
and
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, some formulas for the metric on
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
become simpler if one takes the time variable to be imaginary. (This is no longer standard in classical relativity, but is
used in an essential wayIn physics, Wick rotation, named after Gian-Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable...
in
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
.) Complex numbers are essential to
spinorIn mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
s, which are a generalization of the
tensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
s used in relativity.
Dynamic equations
In differential equations, it is common to first find all complex roots
r of the characteristic equation of a
linear differential equationLinear differential equations are of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...
or equation system and then attempt to solve the system in terms of base functions of the form
f(
t) =
ert. Likewise, in difference equations, the complex roots
r of the characteristic equation of the difference equation system are used, to attempt to solve the system in terms of base functions of the form
f(
t) =
r t.
Fluid dynamics
In
fluid dynamicsIn physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, complex functions are used to describe potential flow in two dimensions.
Fractals
Certain
fractalA fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
s are plotted in the complex plane, e.g. the
Mandelbrot setThe Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape...
and
Julia setIn the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function...
s.
Algebraic number theory
As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in
C. A fortiori, the same is true if the equation has rational coefficients. The roots of such equations are called algebraic numbers – they are a principal object of study in
algebraic number theoryAlgebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
. Compared to , the algebraic closure of
Q, which also contains all algebraic numbers,
C has the advantage of being easily understandable in geometric terms. In this way, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, more specifically applying the machinery of
field theoryField theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined....
to the number field containing
roots of unityIn mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
, it can be shown that it is not possible to construct a regular nonagon using only compass and straightedge – a purely geometric problem.
Another example are Pythagorean triples (
a,
b,
c), that is to say integers satisfying
(which implies that the triangle having sidelengths
a,
b, and
c is a
right triangleA right triangle or right-angled triangle is a triangle in which one angle is a right angle . The relation between the sides and angles of a right triangle is the basis for trigonometry.-Terminology:The side opposite the right angle is called the hypotenuse...
). They can be studied by considering
Gaussian integerIn number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. The Gaussian integers are a special case of the quadratic...
s, that is, numbers of the form
x +
iy, where
x and
y are integers.
Analytic number theory
Analytic number theoryIn mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the Riemann zeta-function ζ(
s) is related to the distribution of
prime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s.
History
The earliest fleeting reference to
square rootIn mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
s of negative numbers can perhaps be said to occur in the work of the Greek mathematician
Heron of AlexandriaHero of Alexandria was an ancient Greek mathematician and engineerEnc. Britannica 2007, "Heron of Alexandria" who was active in his native city of Alexandria, Roman Egypt...
in the 1st century AD, where in his
Stereometrica he considers, apparently in error, the volume of an impossible
frustumIn geometry, a frustum is the portion of a solid that lies between two parallel planes cutting it....
of a
pyramidA pyramid is a structure whose outer surfaces are triangular and converge at a single point. The base of a pyramid can be trilateral, quadrilateral, or any polygon shape, meaning that a pyramid has at least three triangular surfaces...
to arrive at the term
in his calculations, although negative quantities were not conceived of in
Hellenistic mathematicsGreek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...
and Heron merely replaced it by its positive.
The impetus to study complex numbers proper first arose in the 16th century when
algebraic solutionsAn algebraic solution is a closed form expression that is the solution of an algebraic equation in terms of the coefficients, relying only on addition, subtraction, multiplication, division, and the extraction of roots ....
for the roots of cubic and quartic
polynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s were discovered by Italian mathematicians (see
Niccolo Fontana TartagliaNiccolò Fontana Tartaglia was a mathematician, an engineer , a surveyor and a bookkeeper from the then-Republic of Venice...
,
Gerolamo CardanoGerolamo Cardano was an Italian Renaissance mathematician, physician, astrologer and gambler...
). 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. As an example, Tartaglia's cubic formula gives the solution to the equation
x3 −
x = 0 as
At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation
z3 =
i has solutions
–i,
and
. Substituting these in turn for
in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of
x3 –
x = 0. Of course this particular equation can be solved at sight but it does illustrate that when general formulas are used to solve cubic equations with real roots then, as later mathematicians showed rigorously, the use of complex numbers
is unavoidableIn algebra, casus irreducibilis is one of the cases that may arise in attempting to solve a cubic equation with integer coefficients with roots that are expressed with radicals...
.
Rafael BombelliRafael Bombelli was an Italian mathematician.Born in Bologna, he is the author of a treatise on algebra and is a central figure in the understanding of imaginary numbers....
was the first to explicitly address these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic trying to resolve these issues.
The term "imaginary" for these quantities was coined by
René DescartesRené Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...
in 1637, although he was at pains to stress their imaginary nature A further source of confusion was that the equation
seemed to be capriciously inconsistent with the algebraic identity
, which is valid for non-negative 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 (and the related 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 . Even so Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book,
Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.
In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730
Abraham de MoivreAbraham de Moivre was a French mathematician famous for de Moivre's formula, which links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He was a friend of Isaac Newton, Edmund Halley, and James Stirling...
noted that the complicated identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be simply re-expressed by the following well-known formula which bears his name,
de Moivre's formulaIn mathematics, de Moivre's formula , named after Abraham de Moivre, states that for any complex number x and integer n it holds that...
:
In 1748
Leonhard EulerLeonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
went further and obtained
Euler's formulaEuler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
of
complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
:
by formally manipulating complex
power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.
The idea of a complex number as a point in the complex plane (above) was first described by
Caspar WesselCaspar Wessel was a Norwegian-Danish mathematician and cartographer. In 1799, Wessel was the first person to describe the complex numbers. He was the younger brother of poet and playwright Johan Herman Wessel....
in 1799, although it had been anticipated as early as 1685 in
Wallis's De Algebra tractatus.
Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806
Jean-Robert ArgandJean-Robert Argand was a gifted amateur mathematician. In 1806, while managing a bookstore in Paris, he published the idea of geometrical interpretation of complex numbers known as the Argand diagram.-Life:...
independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra. Gauss had earlier published an essentially
topologicalTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1". It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology. The English mathematician
G. H. HardyGodfrey Harold “G. H.” Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....
remarked that Gauss was the first mathematician to use complex numbers in 'a really confident and scientific way' although mathematicians such as
Niels Henrik AbelNiels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...
and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.
Augustin Louis CauchyBaron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...
and
Bernhard RiemannGeorg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
together brought the fundamental ideas of complex analysis to a high state of completion, commencing around 1825 in Cauchy's case.
The common terms used in the theory are chiefly due to the founders. Argand called
the
direction factor, and
the
modulus; Cauchy (1828) called
the
reduced form (l'expression réduite) and apparently introduced the term
argument; Gauss used
i for
, introduced the term
complex number for
a +
bi, and called
a2 +
b2 the
norm. The expression
direction coefficient, often used for
, is due to Hankel (1867), and
absolute value, for
modulus, is due to Weierstrass.
Later classical writers on the general theory include
Richard DedekindJulius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
,
Otto HölderOtto Ludwig Hölder was a German mathematician born in Stuttgart.Hölder first studied at the Polytechnikum and then in 1877 went to Berlin where he was a student of Leopold Kronecker, Karl Weierstraß, and Ernst Kummer.He is famous for many things including: Hölder's inequality, the Jordan–Hölder...
,
Felix KleinChristian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
,
Henri PoincaréJules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
,
Hermann SchwarzKarl Hermann Amandus Schwarz was a German mathematician, known for his work in complex analysis. He was born in Hermsdorf, Silesia and died in Berlin...
,
Karl WeierstrassKarl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....
and many others.
Generalizations and related notions
The process of extending the field
R of reals to
C is known as
Cayley-Dickson constructionIn mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one...
. It can be carried further to higher dimensions, yielding the
quaternionIn mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s
H and
octonionIn mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H...
s
O which (as a real vector space) are of dimension 4 and 8, respectively. However, with increasing dimension, the algebraic properties familiar from real and complex numbers vanish: the quaternions are only a skew field, i.e.
x·y ≠
y·x for two quaternions, the multiplication of octonions fails (in addition to not being commutative) to be associative: (
x·y)·
z ≠
x·(
y·z). However, all of these are
normed division algebraIn mathematics, a normed division algebra A is a division algebra over the real or complex numbers which is also a normed vector space, with norm || · || satisfying the following property:\|xy\| = \|x\| \|y\| for all x and y in A....
s over
R. By Hurwitz's theorem they are the only ones. The next step in the Cayley-Dickson construction, the
sedenionIn abstract algebra, sedenions form a 16-dimensional non-associative algebra over the reals obtained by applying the Cayley–Dickson construction to the octonions...
s fail to have this structure.
The Cayley-Dickson construction is closely related to the
regular representationIn mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation....
of
C, thought of as an
R-
algebraIn mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R....
(an
R-vector space with a multiplication), with respect to the basis 1,
i. This means the following: the
R-linear map
for some fixed complex number
w can be represented by a 2×2 matrix (once a basis has been chosen). With respect to the basis 1,
i, this matrix is
i.e., the one mentioned in the section on matrix representation of complex numbers above. While this is a linear representation of
C in the 2 × 2 real matrices, it is not the only one. Any matrix
has the property that its square is the negative of the identity matrix:
J2 = −
I. Then
is also isomorphic to the field
C, and gives an alternative complex structure on
R2. This is generalized by the notion of a
linear complex structureIn mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, −I. Such a structure on V allows one to define multiplication by complex scalars in a canonical fashion so as to regard V as a complex vector space.Complex structures have...
.
Hypercomplex numberIn mathematics, a hypercomplex number is a traditional term for an element of an algebra over a field where the field is the real numbers or the complex numbers. In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established...
s also generalize
R,
C,
H, and
O. For example this notion contains the
split-complex numberIn abstract algebra, the split-complex numbers are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the formwhere x and y are real numbers...
s, which are elements of the ring
R[
x]/(
x2 − 1) (as opposed to
R[
x]/(
x2 + 1)). In this ring, the equation
a2 = 1 has four solutions.
The field
R is the completion of
Q, the field of
rational numberIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s, with respect to the usual
absolute valueIn mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
metricIn mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
. Other choices of
metricIn mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
s on
Q lead to the fields
Qp of
p-adic numbersIn mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
(for any
prime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
p), which are thereby analogous to
R. There are no other nontrivial ways of completing
Q than
R and
Qp, by
Ostrowski's theoremOstrowski's theorem, due to Alexander Ostrowski , states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.- Definitions :...
. The algebraic closure
of
Qp still carry a norm, but (unlike
C) are not complete with respect to it. The completion
of
turns out to be algebraically closed. This field is called
p-adic complex numbers by analogy.
The fields
R and
Qp and their finite field extensions, including
C, are
local fieldIn mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
s.
See also
- Circular motion using complex numbers
- Complex base systems
In arithmetic, a complex base system is a positional numeral system whose radix is an imaginary or complex number In arithmetic, a complex base system is a positional numeral system whose radix is an imaginary (proposed by Donald Knuth in 1955) or complex number In arithmetic, a complex base...
- Complex geometry
In mathematics, complex geometry is the study of complex manifolds and functions of many complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric chapters of complex analysis....
- Complex square root
- Domain coloring
Domain coloring is a technique for visualizing functions of a complex variable. The term "domain coloring" was coined by Frank Farris possibly around 1998...
- Eisenstein integer
In mathematics, Eisenstein integers , also known as Eulerian integers , are complex numbers of the formz = a + b\omega \,\!where a and b are integers and...
- Euler's identity
- Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. The Gaussian integers are a special case of the quadratic...
- Mandelbrot set
The Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape...
- Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
- Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
(extended complex plane)
- Root of unity
In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
Historical references
-
- A gentle introduction to the history of complex numbers and the beginnings of complex analysis.
- An advanced perspective on the historical development of the concept of number.
Further reading
- The Road to Reality: A Complete Guide to the Laws of the Universe, by Roger Penrose
Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...
; Alfred A. Knopf, 2005; ISBN 0-679-45443-8. Chapters 4-7 in particular deal extensively (and enthusiastically) with complex numbers.
- Unknown Quantity: A Real and Imaginary History of Algebra, by John Derbyshire; Joseph Henry Press; ISBN 0-309-09657-X (hardcover 2006). A very readable history with emphasis on solving polynomial equations and the structures of modern algebra.
- Visual Complex Analysis, by Tristan Needham
Tristan Needham is a mathematician and author of thebook Visual Complex Analysis in which he uses a geometric approach to develop complex analysis ....
; Clarendon Press; ISBN 0-19-853447-7 (hardcover, 1997). History of complex numbers and complex analysis with compelling and useful visual interpretations.
External links
- Euler's work on Complex Roots of Polynomials at Convergence. MAA Mathematical Sciences Digital Library.
- John and Betty's Journey Through Complex Numbers
- Dimensions: a math film. Chapter 5 presents an introduction to complex arithmetic and stereographic projection
The stereographic projection, in geometry, is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it...
. Chapter 6 discusses transformations of the complex plane, Julia setIn the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function...
s, and the Mandelbrot setThe Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape...
.