, a complex number is a number
that can be expressed in the form , where and are real numbers
, and is a symbol
called the imaginary unit
, and satisfying the equation . Because no "real" number satisfies this equation, was called an imaginary number
by René Descartes
. For the complex number , is called the and is called the . The set of complex numbers is denoted by either of the symbols
or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.
Complex numbers allow solutions to all polynomial equation
s, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra
asserts that every polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation
has no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions and .
Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule combined with the associative
laws. Every nonzero complex number has a multiplicative inverse
. This makes the complex numbers a field
that has the real numbers as a subfield. The complex numbers form also a real vector space
of dimension two, with as a standard basis
This standard basis makes the complex numbers a Cartesian plane
, called the complex plane
. This allows a geometric interpretation of the complex numbers and their operations, and conversely expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the real line
which is identified to the horizontal axis of the complex plane. The complex numbers of absolute value
one form the unit circle
. The addition of a complex number is a translation
in the complex plane, and the multiplication by a complex number is a similarity
centered at the origin. The complex conjugation
is the reflection symmetry
with respect to the real axis. The complex absolute value is a Euclidean norm
In summary, the complex numbers form a rich structure that is simultaneously an algebraically closed field
, a commutative algebra
over the reals, and a Euclidean vector space
of dimension two.
A complex number is a number of the form , where and are real numbers
, and is an indeterminate satisfying . For example, is a complex number.
This way, a complex number is defined as a polynomial
with real coefficients in the single indeterminate , for which the relation is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials. The relation induces the equalities and which hold for all integers ; these allow the reduction of any polynomial that results from the addition and multiplication of complex numbers to a linear polynomial in , again of the form with real coefficients
The real number is called the ''real part'' of the complex number ; the real number is called its ''imaginary part''. To emphasize, the imaginary part does not include a factor ; that is, the imaginary part is , not .
Formally, the complex numbers are defined as the quotient ring
of the polynomial ring
in the indeterminate , by the ideal
generated by the polynomial (see below
A real number can be regarded as a complex number , whose imaginary part is 0. A purely imaginary number
is a complex number , whose real part is zero. As with polynomials, it is common to write for and for . Moreover, when the imaginary part is negative, that is, , it is common to write instead of ; for example, for , can be written instead of .
Since the multiplication of the indeterminate and a real is commutative in polynomials with real coefficients, the polynomial may be written as This is often expedient for imaginary parts denoted by expressions, for example, when is a radical.
The real part of a complex number is denoted by ,
; the imaginary part of a complex number is denoted by ,
of all complex numbers is denoted by
) or (upright bold).
In some disciplines, particularly in electromagnetism
and electrical engineering
, is used instead of as is frequently used to represent electric current
. In these cases, complex numbers are written as , or .
A complex number can thus be identified with an ordered pair
of real numbers, which in turn may be interpreted as coordinates of a point in a two-dimensional space. The most immediate space is the Euclidean plane with suitable coordinates, which is then called complex plane or Argand diagram
named after Jean-Robert Argand
. Another prominent space on which the coordinates may be projected is the two-dimensional surface of a sphere, which is then called Riemann sphere
Cartesian complex plane
The definition of the complex numbers involving two arbitrary real values immediately suggests the use of Cartesian coordinates in the complex plane. The horizontal (''real'') axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical (''imaginary'') axis, with increasing values upwards.
A charted number may be viewed either as the coordinatized
point or as a position vector
from the origin to this point. The coordinate values of a complex number can hence be expressed in its ''Cartesian'', ''rectangular'', or ''algebraic'' form.
Notably, 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 (see below
) corresponds to multiplying their magnitudes and adding the angles they make with the real axis. Viewed in this way, the multiplication of a complex number by corresponds to rotating the position vector counterclockwise
by a quarter turn
) about the origin—a fact which can be expressed algebraically as follows:
Polar complex plane
Modulus and argument
An alternative option for coordinates in the complex plane is the polar coordinate system
that uses the distance of the point from the origin
(), and the angle subtended between the positive real axis
and the line segment in a counterclockwise sense. This leads to the polar form of complex numbers.
The ''absolute value
'' (or ''modulus'' or ''magnitude'') of a complex number is
If is a real number (that is, if ), then . That is, the absolute value of a real number equals its absolute value as a complex number.
By Pythagoras' theorem
, the absolute value of a complex number is the distance to the origin of the point representing the complex number in the complex plane
'' of (in many applications referred to as the "phase" )
is the angle of the radius
with the positive real axis, and is written as . As with the modulus, the argument can be found from the rectangular form —by applying the inverse tangent to the quotient of imaginary-by-real parts. By using a half-angle identity, a single branch of the arctan suffices to cover the range of the -function, , and avoids a more subtle case-by-case analysis
Normally, as given above, the principal value
in the interval is chosen. Values in the range are obtained by adding — if the value is negative. The value of is expressed in radian
s in this article. It can increase by any integer multiple of and still give the same angle, viewed as subtended by the rays of the positive real axis and from the origin through . Hence, the arg function is sometimes considered as multivalued
. The polar angle for the complex number 0 is indeterminate, but arbitrary choice of the polar angle 0 is common.
The value of equals the result of atan2
Together, and 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 formula
this can be written as
Using the function, this is sometimes abbreviated to
In angle notation
, often used in electronics
to represent a phasor
with amplitude and phase , it is written as
When visualizing complex functions
, both a complex input and output are needed. Because each complex number is represented in two dimensions, visually graphing a complex function would require the perception of a four dimensional space
, which is possible only in projections. Because of this, other ways of visualizing complex functions have been designed.
In domain coloring
the output dimensions are represented by color and brightness, respectively. Each point in the complex plane as domain is ''ornated'', typically with ''color'' representing the argument of the complex number, and ''brightness'' representing the magnitude. Dark spots mark moduli near zero, brighter spots are farther away from the origin, the gradation may be discontinuous, but is assumed as monotonous. The colors often vary in steps of for to from red, yellow, green, cyan, blue, to magenta. These plots are called color wheel graphs
. This provides a simple way to visualize the functions without losing information. The picture shows zeros for and poles at
s are another way to visualize complex functions. Riemann surfaces can be thought of as deformations
of the complex plane; while the horizontal axes represent the real and imaginary inputs, the single vertical axis only represents either the real or imaginary output. However, Riemann surfaces are built in such a way that rotating them 180 degrees shows the imaginary output, and vice versa. Unlike domain coloring, Riemann surfaces can represent multivalued function
s like .
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 irreducibilis
''). This conundrum led Italian mathematician Gerolamo Cardano
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 algebra
, which shows that with complex numbers, a solution exists to every polynomial equation
of degree one or higher. Complex numbers thus form an algebraically closed field
, where any polynomial equation has a root
Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician Rafael Bombelli
. A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton
, who extended this abstraction to the theory of quaternions
The earliest fleeting reference to square root
s of negative number
s can perhaps be said to occur in the work of the Greek mathematician Hero of Alexandria
in the 1st century AD
, where in his ''Stereometrica
'' he considers, apparently in error, the volume of an impossible frustum
of a pyramid
to arrive at the term in his calculations, although negative quantities were not conceived of in Hellenistic mathematics
and Hero merely replaced it by its positive .
The impetus to study complex numbers as a topic in itself first arose in the 16th century when algebraic solution
s for the roots of cubic
and quartic polynomial
s were discovered by Italian mathematicians (see Niccolò Fontana Tartaglia
, Gerolamo Cardano
). It was soon realized (but proved much later)
[ that these formulas, even if one was interested only in real solutions, sometimes required the manipulation of square roots of negative numbers. As an example, Tartaglia's formula for a cubic equation of the form gives the solution to the equation as
At first glance this looks like nonsense. However, formal calculations with complex numbers show that the equation has solutions , and . Substituting these in turn for in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of . 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 unavoidable. Rafael Bombelli was the first to address explicitly 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 Descartes in 1637, who was at pains to stress their unreal 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 and , and which was also used in complex number calculations with one of , positive and the other negative. The incorrect use of this identity (and the related identity ) in the case when both and are negative even bedeviled Euler. This difficulty eventually led to the convention of using the special symbol 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 Moivre 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 formula:
In 1748 Leonhard Euler went further and obtained Euler's formula of complex analysis:
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 Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's ''A Treatise of Algebra''.
Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra. Carl Friedrich Gauss had earlier published an essentially topological 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.
If one formerly contemplated this subject from a false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, −1, positive, negative, or imaginary (or even impossible) units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness. — Gauss (1831)
In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée, Mourey, Warren, Français and his brother, Bellavitis.
The English mathematician G.H. Hardy remarked that Gauss was the first mathematician to use complex numbers in 'a really confident and scientific way' although mathematicians such as Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.
Augustin Louis Cauchy and Bernhard Riemann 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 (1821) called the ''reduced form'' (l'expression réduite) and apparently introduced the term ''argument''; Gauss used for , introduced the term ''complex number'' for , and called 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 Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others.
Relations and operations
Complex numbers have a similar definition of equality to real numbers; two complex numbers and are equal if and only if both their real and imaginary parts are equal, that is, if and . Nonzero complex numbers written in polar form are equal if and only if they have the same magnitude and their arguments differ by an integer multiple of .
Unlike the real numbers, there is no natural ordering of the complex numbers. In particular, there is no linear ordering on the complex numbers that is compatible with addition and multiplication – the complex numbers cannot have the structure of an ordered field. This is e.g. because every non-trivial sum of squares in an ordered field is , and is a non-trivial sum of squares.
Thus, complex numbers are naturally thought of as existing on a two-dimensional plane.
The ''complex conjugate'' of the complex number is given by . It is denoted by either or . This unary operation on complex numbers cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division.
Geometrically, is the "reflection" of about the real axis. Conjugating twice gives the original complex number
which makes this operation an involution. The reflection leaves both the real part and the magnitude of unchanged, that is
The imaginary part and the argument of a complex number change their sign under conjugation
For details on argument and magnitude, see the section on Polar form.
The product of a complex number and its conjugate is known as the ''absolute square''. It is always a non-negative real number and equals the square of the magnitude of each:
This property can be used to convert a fraction with a complex denominator to an equivalent fraction with a real denominator by expanding both numerator and denominator of the fraction by the conjugate of the given denominator. This process is sometimes called "rationalization" of the denominator (although the denominator in the final expression might be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.
The real and imaginary parts of a complex number can be extracted using the conjugation:
Moreover, a complex number is real if and only if it equals its own conjugate.
Conjugation distributes over the basic complex arithmetic operations:
Conjugation is also employed in inversive geometry, a branch of geometry studying reflections more general than ones about a line. In the network analysis of electrical circuits, the complex conjugate is used in finding the equivalent impedance when the maximum power transfer theorem is looked for.
Addition and subtraction
Two complex numbers and are most easily added by separately adding their real and imaginary parts of the summands. That is to say:
Similarly, subtraction can be performed as
Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers and , interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices , and the points of the arrows labeled and (provided that they are not on a line). Equivalently, calling these points , , respectively and the fourth point of the parallelogram the triangles and are congruent. A visualization of the subtraction can be achieved by considering addition of the negative subtrahend.
Since the real part, the imaginary part, and the indeterminate in a complex number are all considered as numbers in themselves, two complex numbers, given as and are multiplied under the rules of the distributive property, the commutative properties and the defining property in the following way
Reciprocal and division
Using the conjugation, the reciprocal of a nonzero complex number can always be broken down to
since ''non-zero'' implies that is greater than zero.
This can be used to express a division of an arbitrary complex number by a non-zero complex number as
Multiplication and division in polar form
Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates. Given two complex numbers and , because of the trigonometric identities
we may derive
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 corresponds to a quarter-turn counter-clockwise, which gives back . The picture at the right illustrates the multiplication of
Since the real and imaginary part of are equal, the argument of that number is 45 degrees, or (in radian). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles 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 .
Similarly, division is given by
The square roots of (with ) are , where
where is the signum function. This can be seen by squaring to obtain . Here is called the modulus of , and the square root sign indicates the square root with non-negative real part, called the principal square root; also where .
The exponential function can be defined for every complex number by the power series
which has an infinite radius of convergence.
The value at of the exponential function is Euler's number
If is real, one has
Analytic continuation allows extending this equality for every complex value of , and thus to define the complex exponentiation with base as
The exponential function satisfies the functional equation
This can be proved either by comparing the power series expansion of both members or by applying analytic continuation from the restriction of the equation to real arguments.
Euler's formula states that, for any real number ,
The functional equation implies thus that, if and are real, one has
which is the decomposition of the exponential function into its real and imaginary parts.
In the real case, the natural logarithm can be defined as the inverse
of the exponential function. For extending this to the complex domain, one can start from Euler's formula. It implies that, if a complex number is written in polar form
with then with
as complex logarithm one has a proper inverse:
However, because cosine and sine are periodic functions, the addition of an integer multiple of to does not change . For example, , so both and are possible values for the natural logarithm of .
Therefore, if the complex logarithm is not to be defined as a multivalued function
one has to use a branch cut and to restrict the codomain, resulting in the bijective function
If is not a non-positive real number (a positive or a non-real number), the resulting principal value of the complex logarithm is obtained with . It is an analytic function outside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number , where the principal value is .
If is real and complex, the exponentiation is defined as
where denotes the natural logarithm.
It seems natural to extend this formula to complex values of , but there are some difficulties resulting from the fact that the complex logarithm is not really a function, but a multivalued function.
It follows that if is as above, and if is another complex number, then the ''exponentiation'' is the multivalued function
Integer and fractional exponents
If, in the preceding formula, is an integer, then the sine and the cosine are independent of . Thus, if the exponent is an integer, then is well defined, and the exponentiation formula simplifies to de Moivre's formula:
The th roots of a complex number are given by
for . (Here is the usual (positive) th root of the positive real number .) Because sine and cosine are periodic, other integer values of do not give other values.
While the th root of a positive real number is chosen to be the ''positive'' real number satisfying , there is no natural way of distinguishing one particular complex th root of a complex number. Therefore, the th root is a -valued function of . This implies that, contrary to the case of positive real numbers, one has
since the left-hand side consists of values, and the right-hand side is a single value.
The set of complex numbers is a field.
[See , pages 15–16.] 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 , its additive inverse is also a complex number; and third, every nonzero complex number has a reciprocal complex number. Moreover, these operations satisfy a number of laws, for example the law of commutativity of addition and multiplication for any two complex numbers and :
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, is not an ordered field, that is to say, it is not possible to define a relation that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so precludes the existence of an ordering on
When the underlying field for a mathematical topic or construct is the field of complex numbers, the topic's name is usually modified to reflect that fact. For example: complex analysis, complex matrix, complex polynomial, and complex Lie algebra.
Solutions of polynomial equations
Given any complex numbers (called coefficients) , the equation
has at least one complex solution ''z'', provided that at least one of the higher coefficients is nonzero. This is the statement of the ''fundamental theorem of algebra'', of Carl Friedrich Gauss and Jean le Rond d'Alembert. Because of this fact, is called an algebraically closed field. This property does not hold for the field of rational numbers (the polynomial does not have a rational root, since is not a rational number) nor the real numbers (the polynomial does not have a real root for , since the square of is positive for any real number ).
There are various proofs of this theorem, by either analytic methods such as Liouville's theorem, or topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of ''odd'' degree has at least one real root.
Because of this fact, theorems that hold ''for any algebraically closed field'' apply to For example, any non-empty complex square matrix has at least one (complex) eigenvalue.
The field has the following three properties:
* First, it has characteristic 0. This means that for any number of summands (all of which equal one).
* Second, its transcendence degree over , the prime field of is the cardinality of the continuum.
* Third, it is algebraically closed (see above).
It can be shown that any field having these properties is isomorphic (as a field) to For example, the algebraic closure of the field of the -adic number also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields). Also, is isomorphic to the field of complex Puiseux series. However, specifying an isomorphism requires the axiom of choice. Another consequence of this algebraic characterization is that contains many proper subfields that are isomorphic to .
Characterization as a topological field
The preceding characterization of describes only the algebraic aspects of That is to say, the properties of nearness and continuity, which matter in areas such as analysis and topology, are not dealt with. The following description of as a topological field (that is, a field that is equipped with a topology, which allows the notion of convergence) does take into account the topological properties. contains a subset (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:
* is closed under addition, multiplication and taking inverses.
* If and are distinct elements of , then either or is in .
* If is any nonempty subset of , then for some in
Moreover, has a nontrivial involutive automorphism (namely the complex conjugation), such that is in for any nonzero in
Any field with these properties can be endowed with a topology by taking the sets as a base, where ranges over the field and ranges over . With this topology is isomorphic as a ''topological'' field to
The only connected locally compact topological fields are and This gives another characterization of as a topological field, since can be distinguished from because the nonzero complex numbers are connected, while the nonzero real numbers are not.
Construction as ordered pairs
William Rowan Hamilton introduced the approach to define the set of complex numbers as the set of of real numbers, in which the following rules for addition and multiplication are imposed:
It is then just a matter of notation to express as .
Construction as a quotient field
Though this low-level construction does accurately describe the structure of the complex numbers, the following equivalent definition reveals the algebraic nature of more immediately. This characterization relies on the notion of fields and polynomials. A field is a set endowed with addition, subtraction, multiplication and division operations that behave as is familiar from, say, rational numbers. For example, the distributive law
must hold for any three elements , and of a field. The set of real numbers does form a field. A polynomial with real coefficients is an expression of the form
where the are real numbers. The usual addition and multiplication of polynomials endows the set