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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a complex number is an element of a number system that extends the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s with a specific element denoted , called the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...
. For the complex number a+bi, is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C 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 equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation (x+1)^2 = -9 has no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions -1+3i and -1-3i. Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule i^=-1 combined with the associative, commutative, and distributive laws. Every nonzero complex number has a
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
. This makes the complex numbers a field that has the real numbers as a subfield. The complex numbers also form a real vector space of dimension two, with as a
standard basis In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the ...
. 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 Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
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 In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
, a
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
over the reals, and a Euclidean vector space of dimension two.


Definition

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 In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
with real coefficients 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 mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
in the indeterminate , by the
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
generated by the polynomial (see
below Below may refer to: *Earth * Ground (disambiguation) *Soil *Floor * Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Ernst von Below (1863–1955), German World War I general *Fred Below ...
).


Notation

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 , \mathcal(z), or \mathfrak(z); the imaginary part of a complex number is denoted by , \mathcal(z), or \mathfrak(z). For example, \operatorname(2 + 3i) = 2 \quad \text \quad \operatorname(2 + 3i) = 3~. The
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of all complex numbers is denoted by \Complex ( blackboard bold) or (upright bold). In some disciplines, particularly in
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
and
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, is used instead of as is frequently used to represent
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The movi ...
. In these cases, complex numbers are written as , or .


Visualization

A complex number can thus be identified with an
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
(\Re (z),\Im (z)) 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 In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
.


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 Below may refer to: *Earth * Ground (disambiguation) *Soil *Floor * Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Ernst von Below (1863–1955), German World War I general *Fred 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 Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...
by a quarter turn ( 90°) about the origin—a fact which can be expressed algebraically as follows: (a + bi)\cdot i = ai + b(i)^2 = -b + ai .


Polar complex plane


Modulus and argument

An alternative option for coordinates in the complex plane is the
polar coordinate system In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
that uses the distance of the point from the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
(), and the angle subtended between the positive real axis and the line segment in a counterclockwise sense. This leads to the polar form :z=re^=r(\cos\varphi +i\sin\varphi) of a complex number, where is the absolute value of , and \varphi is the argument of . The ''absolute value'' (or ''modulus'' or ''magnitude'') of a complex number is r=, z, =\sqrt. 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. The ''argument'' of (in many applications referred to as the "phase" ) is the angle of the
radius In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''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 \varphi = \arg (x+yi) = \begin 2 \arctan\left(\dfrac\right) &\text y \neq 0 \text x > 0, \\ \pi &\text x < 0 \text y = 0, \\ \text &\text x = 0 \text y = 0. \end Normally, as given above, the principal value in the interval is chosen. If the arg value is negative, values in the range or can be obtained by adding . The value of is expressed in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
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 In computing and mathematics, the function atan2 is the 2-argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi < \theta \leq \pi) between the positive
: \varphi = \operatorname\left(\operatorname(z),\operatorname(z) \right). 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'' z = r(\cos \varphi + i\sin \varphi ). Using
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
this can be written as z = r e^ \text z = r \exp i \varphi. Using the function, this is sometimes abbreviated to z = r \operatorname\mathrm \varphi. In angle notation, often used in
electronics The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
to represent a phasor with amplitude and phase , it is written as z = r \angle \varphi .


Complex graphs

When visualizing
complex functions Complex 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 helpful in many branches of mathematics, including algebraic ...
, 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 \pm \sqrt.


History

The solution in
radicals Radical may refer to: Politics and ideology Politics *Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...
(without
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
) of a general cubic equation, when all three of its roots are real numbers, contains the square roots of negative numbers, a situation that cannot be rectified by factoring aided by the rational root test, if the cubic is irreducible; this is the so-called '' casus irreducibilis'' ("irreducible case"). This conundrum led Italian mathematician
Gerolamo Cardano Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
to conceive of complex numbers in around 1545 in his ''Ars Magna'', though his understanding was rudimentary; moreover he later dismissed complex numbers as "subtle as they are useless". Cardano did use imaginary numbers, but described using them as “mental torture.” This was prior to the use of the graphical complex plane. Cardano and other Italian mathematicians, notably Scipione del Ferro, in the 1500s created an algorithm for solving cubic equations which generally had one real solution and two solutions containing an imaginary number. Since they ignored the answers with the imaginary numbers, Cardano found them useless. 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 In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
, where any polynomial equation has a
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
. 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 Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
, who extended this abstraction to the theory of
quaternions In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
. The earliest fleeting reference to
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
s of negative numbers can perhaps be said to occur in the work of the
Greek mathematician Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathe ...
Hero of Alexandria Hero of Alexandria (; grc-gre, Ἥρων ὁ Ἀλεξανδρεύς, ''Heron ho Alexandreus'', also known as Heron of Alexandria ; 60 AD) was a Greek mathematician and engineer who was active in his native city of Alexandria, Roman Egypt. H ...
in the 1st century AD, where in his '' Stereometrica'' he considered, apparently in error, the volume of an impossible frustum of a
pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...
to arrive at the term \sqrt in his calculations, which today would simplify to \sqrt = 3i\sqrt. Negative quantities were not conceived of in
Hellenistic mathematics Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathe ...
and Hero merely replaced it by its positive \sqrt = 3\sqrt. The impetus to study complex numbers as a topic in itself first arose in the 16th century when algebraic solutions for the roots of cubic and quartic
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s were discovered by Italian mathematicians (see Niccolò Fontana Tartaglia,
Gerolamo Cardano Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
). It was soon realized (but proved much later) that these formulas, even if one were 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 \tfrac\left(\left(\sqrt\right)^+\left(\sqrt\right)^\right). At first glance this looks like nonsense. However, formal calculations with complex numbers show that the equation has three solutions: -i, \frac, \frac. Substituting these in turn for \sqrt^ 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 René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...
in 1637, who was at pains to stress their unreal nature A further source of confusion was that the equation \sqrt^2 = \sqrt\sqrt = -1 seemed to be capriciously inconsistent with the algebraic identity \sqrt\sqrt = \sqrt, 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 \frac = \sqrt) in the case when both and are negative even bedeviled
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
. This difficulty eventually led to the convention of using the special symbol in place of \sqrt 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 Abraham de Moivre FRS (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He move ...
noted that the identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be re-expressed by the following de Moivre's formula: (\cos \theta + i\sin \theta)^ = \cos n \theta + i\sin n \theta. In 1748, Euler went further and obtained
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
of complex analysis: \cos \theta + i\sin \theta = e ^ by formally manipulating complex
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
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 DanishNorwegian
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
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, \sqrt 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.
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 Norwegian Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.
Augustin-Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. H ...
and
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
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 r = \sqrt the ''modulus''; Cauchy (1821) called the ''reduced form'' (l'expression réduite) and apparently introduced the term ''argument''; Gauss used for \sqrt, 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 Ludwig Otto Hölder (December 22, 1859 – August 29, 1937) was a German mathematician born in Stuttgart. Early life and education Hölder was the youngest of three sons of professor Otto Hölder (1811–1890), and a grandson of professor Chri ...
, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others. Important work (including a systematization) in complex multivariate calculus has been started at beginning of the 20th century. Important results have been achieved by Wilhelm Wirtinger in 1927.


Relations and operations


Equality

Complex numbers have a similar definition of equality to real numbers; two complex numbers and are equal
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
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 .


Ordering

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. Hence, the complex numbers do not have the structure of an ordered field. One explanation for this is that every non-trivial sum of squares in an ordered field is nonzero, and is a non-trivial sum of squares. Thus, complex numbers are naturally thought of as existing on a two-dimensional plane.


Conjugate

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 \overline=z, which makes this operation an involution. The reflection leaves both the real part and the magnitude of unchanged, that is \operatorname(\overline) = \operatorname(z)\quad and \quad , \overline, = , z, . The imaginary part and the argument of a complex number change their sign under conjugation \operatorname(\overline) = -\operatorname(z)\quad \text \quad \operatorname \overline \equiv -\operatorname z \pmod . 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: z\cdot \overline = x^2 + y^2 = , z, ^2 = , \overline, ^2. 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: \operatorname(z) = \dfrac,\quad \text \quad \operatorname(z) = \dfrac. Moreover, a complex number is real if and only if it equals its own conjugate. Conjugation distributes over the basic complex arithmetic operations: \begin \overline &= \overline \pm \overline, \\ \overline &= \overline \cdot \overline, \\ \overline &= \overline/\overline. \end 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 a =x+yi and b =u+vi are most easily added by separately adding their real and imaginary parts. That is to say: a + b =(x+yi) + (u+vi) = (x+u) + (y+v)i. Similarly, subtraction can be performed as a - b =(x+yi) - (u+vi) = (x-u) + (y-v)i. Multiplication of a complex number a =x+yi and a real number can be done similarly by multiplying separately and the real and imaginary parts of : ra=r(x+yi) = rx + ryi. In particular, subtraction can be done by negating the subtrahend (that is multiplying it with ) and adding the result to the minuend: a - b =a + (-1)\,b. Using the visualization of complex numbers in the complex plane, 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 In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) 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 eq ...
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
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
s and are congruent.


Multiplication and square

The rules of the distributive property, the commutative properties (of addition and multiplication), and the defining property apply to complex numbers. It follows that (x+yi)\, (u+vi)= (xu - yv) + (xv + yu)i. In particular, (x+yi)^2=x^2-y^2 + 2xyi.


Reciprocal and division

Using the conjugation, the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
of a nonzero complex number can always be broken down to \frac=\frac = \frac=\frac=\frac -\fraci, 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 \frac = w\cdot \frac = (u+vi)\cdot \left(\frac -\fraci\right)= \frac .


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 \begin \cos a \cos b & - \sin a \sin b & = & \cos(a + b) \\ \cos a \sin b & + \sin a \cos b & = & \sin(a + b) . \end we may derive z_1 z_2 = r_1 r_2 (\cos(\varphi_1 + \varphi_2) + i \sin(\varphi_1 + \varphi_2)). 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 (2+i)(3+i)=5+5i. Since the real and imaginary part of are equal, the argument of that number is 45 degrees, or (in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are
arctan In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spe ...
(1/3) and arctan(1/2), respectively. Thus, the formula \frac = \arctan\left(\frac\right) + \arctan\left(\frac\right) holds. As the
arctan In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spe ...
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 \frac = \frac \left(\cos(\varphi_1 - \varphi_2) + i \sin(\varphi_1 - \varphi_2)\right).


Square root

The square roots of (with ) are \pm (\gamma + \delta i), where \gamma = \sqrt and \delta = (\sgn b)\sqrt, where is the signum function. This can be seen by squaring \pm (\gamma + \delta i) to obtain . Here \sqrt 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 \sqrt= \sqrt, where .


Exponential function

The
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
\exp \colon \Complex \to \Complex ; z \mapsto \exp z can be defined for every complex number by the
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
\exp z= \sum_^\infty \frac , which has an infinite radius of convergence. The value at of the exponential function is Euler's number e = \exp 1 = \sum_^\infty \frac1\approx 2.71828. If is real, one has \exp z=e^z.
Analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ...
allows extending this equality for every complex value of , and thus to define the complex exponentiation with base as e^z=\exp z.


Functional equation

The exponential function satisfies the functional equation e^=e^ze^t. This can be proved either by comparing the power series expansion of both members or by applying
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ...
from the restriction of the equation to real arguments.


Euler's formula

Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
states that, for any real number , e^ = \cos y + i\sin y . The functional equation implies thus that, if and are real, one has e^ = e^x(\cos y + i\sin y) = e^x \cos y + i e^x \sin y , which is the decomposition of the exponential function into its real and imaginary parts.


Complex logarithm

In the real case, the natural logarithm can be defined as the
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...
\ln \colon \R^+ \to \R ; x \mapsto \ln x 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 z\in \Complex^\times is written in polar form z = r(\cos \varphi + i\sin \varphi ) with r, \varphi \in \R , then with \ln z = \ln r + i \varphi as complex logarithm one has a proper inverse: \exp \ln z = \exp(\ln r + i \varphi ) = r \exp i \varphi = r(\cos \varphi + i\sin \varphi ) = z . 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 \ln z = \left\, one has to use a
branch cut In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point ...
and to restrict the codomain, resulting in the bijective function \ln \colon \; \Complex^\times \; \to \; \; \; \R^+ + \; i \, \left(-\pi, \pi\right] . If z \in \Complex \setminus \left( -\R_ \right) 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 In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
outside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number z \in -\R^+ , where the principal value is .


Exponentiation

If is real and complex, the exponentiation is defined as x^z=e^, 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 z^t=\left\\mid k\in \mathbb Z\right\}


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: z^=(r(\cos \varphi + i\sin \varphi ))^n = r^n \, (\cos n\varphi + i \sin n \varphi). The th roots of a complex number are given by z^ = \sqrt \left( \cos \left(\frac\right) + i \sin \left(\frac\right)\right) for . (Here \sqrt 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 (z^n)^ \ne z, since the left-hand side consists of values, and the right-hand side is a single value.


Properties


Field structure

The set \Complex of complex numbers is a field. 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 Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
complex number. Moreover, these operations satisfy a number of laws, for example the law of
commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
of addition and multiplication for any two complex numbers and : \begin z_1 + z_2 & = z_2 + z_1 ,\\ z_1 z_2 & = z_2 z_1 . \end 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, \Complex 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 Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
on \Complex. 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 Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
, complex
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
, and complex Lie algebra.


Solutions of polynomial equations

Given any complex numbers (called coefficients) , the equation a_n z^n + \dotsb + a_1 z + a_0 = 0 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, \Complex is called an
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
. This property does not hold for the field of rational numbers \Q (the polynomial does not have a rational root, since √2 is not a rational number) nor the real numbers \R (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 \Complex. For example, any non-empty complex square matrix has at least one (complex)
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
.


Algebraic characterization

The field \Complex 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 \Q, the prime field of \Complex, is the
cardinality of the continuum In 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 \mathfrak c (lowercase fraktur "c") or , \math ...
. * Third, it is algebraically closed (see above). It can be shown that any field having these properties is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
(as a field) to \Complex. For example, the algebraic closure of the field \Q_p of the -adic number also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields). Also, \Complex is isomorphic to the field of complex Puiseux series. However, specifying an isomorphism requires the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
. Another consequence of this algebraic characterization is that \Complex contains many proper subfields that are isomorphic to \Complex.


Characterization as a topological field

The preceding characterization of \Complex describes only the algebraic aspects of \Complex. That is to say, the properties of nearness and continuity, which matter in areas such as
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
and
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, are not dealt with. The following description of \Complex as a topological field (that is, a field that is equipped with a
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, which allows the notion of convergence) does take into account the topological properties. \Complex 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 \Complex. Moreover, \Complex has a nontrivial involutive automorphism (namely the complex conjugation), such that is in for any nonzero in \Complex. 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 \Complex. The only connected locally compact topological fields are \R and \Complex. This gives another characterization of \Complex as a topological field, since \Complex can be distinguished from \R because the nonzero complex numbers are connected, while the nonzero real numbers are not.


Formal construction


Construction as ordered pairs

William Rowan Hamilton Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
introduced the approach to define the set \Complex of complex numbers as the set \mathbb^2 of of real numbers, in which the following rules for addition and multiplication are imposed: \begin (a, b) + (c, d) &= (a + c, b + d)\\ (a, b) \cdot (c, d) &= (ac - bd, bc + ad). \end 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 \Complex 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 (x+y) z = xz + yz must hold for any three elements , and of a field. The set \R of real numbers does form a field. A polynomial with real coefficients is an expression of the form a_nX^n+\dotsb+a_1X+a_0, where the are real numbers. The usual addition and multiplication of polynomials endows the set \R /math> of all such polynomials with a ring structure. This ring is called the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
over the real numbers. The set of complex numbers is defined as the quotient ring \R (X^2+1). This extension field contains two square roots of , namely (the cosets of) and , respectively. (The cosets of) and form a basis of \mathbb (X^2 + 1) as a real
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, which means that each element of the extension field can be uniquely written as a linear combination in these two elements. Equivalently, elements of the extension field can be written as ordered pairs of real numbers. The quotient ring is a field, because is irreducible over \R, so the ideal it generates is maximal. The formulas for addition and multiplication in the ring \R modulo the relation , correspond to the formulas for addition and multiplication of complex numbers defined as ordered pairs. So the two definitions of the field \Complex are
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
(as fields). Accepting that \Complex is algebraically closed, since it is an algebraic extension of \mathbb in this approach, \Complex is therefore the algebraic closure of \R.


Matrix representation of complex numbers

Complex numbers can also be represented by matrices that have the form: \begin a & -b \\ b & \;\; a \end Here the entries and are real numbers. As the sum and product of two such matrices is again of this form, these matrices form a subring of the ring matrices. A simple computation shows that the map: a+ib\mapsto \begin a & -b \\ b & \;\; a \end is a ring isomorphism from the field of complex numbers to the ring of these matrices. This isomorphism associates the square of the absolute value of a complex number with the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the corresponding matrix, and the conjugate of a complex number with the transpose of the matrix. The geometric description of the multiplication of complex numbers can also be expressed in terms of rotation matrices by using this correspondence between complex numbers and such matrices. The action of the matrix on a vector corresponds to the multiplication of by . In particular, if the determinant is , there is a real number such that the matrix has the form: \begin \cos t & - \sin t \\ \sin t & \;\; \cos t \end In this case, the action of the matrix on vectors and the multiplication by the complex number \cos t+i\sin t are both the rotation of the angle .


Complex analysis

The study of functions of a complex variable is known as complex analysis and has enormous practical use in
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
employ techniques from complex analysis (see prime number theorem 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 In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a_0, a_1, a_2, \ldots) defines a series that is denoted :S=a_0 +a_1+ a_2 + \cdots=\sum_^\infty a_k. The th partial ...
and
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
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, \mathbb, endowed with the
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
\operatorname(z_1, z_2) = , z_1 - z_2, is a complete
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
, which notably includes the triangle inequality , z_1 + z_2, \le , z_1, + , z_2, for any two complex numbers and . Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the ''
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
'' , also written , is defined as the infinite series \exp z:= 1+z+\frac+\frac+\cdots = \sum_^ \frac. The series defining the real trigonometric functions sine and cosine, as well as the hyperbolic functions sinh and cosh, also carry over to complex arguments without change. For the other trigonometric and hyperbolic functions, such as
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
, things are slightly more complicated, as the defining series do not converge for all complex values. Therefore, one must define them either in terms of sine, cosine and exponential, or, equivalently, by using the method of
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ...
. ''
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
'' states: \exp(i\varphi) = \cos \varphi + i\sin \varphi for any real number , in particular \exp(i \pi) = -1 , which is
Euler's identity In mathematics, Euler's identity (also known as Euler's equation) is the equality e^ + 1 = 0 where : is Euler's number, the base of natural logarithms, : is the imaginary unit, which by definition satisfies , and : is pi, the ratio of the circ ...
. Unlike in the situation of real numbers, there is an
infinitude Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
of complex solutions of the equation \exp z = w for any complex number . It can be shown that any such solution – called complex logarithm of – satisfies \log w = \ln, w, + i\arg w, where arg is the argument defined above, and ln the (real) natural logarithm. As arg is a multivalued function, unique only up to a multiple of , log is also multivalued. The principal value of log is often taken by restricting the imaginary part to the interval . Complex
exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...
is defined as z^\omega = \exp(\omega \ln z), and is multi-valued, except when is an integer. For , for some natural number , this recovers the non-uniqueness of 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 a^ = \left(a^b\right)^c. 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'': \mathbb\mathbb is called holomorphic if it satisfies the Cauchy–Riemann equations. For example, any \mathbb-linear map \mathbb\mathbb can be written in the form f(z)=az+b\overline with complex coefficients and . This map is holomorphic
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
. The second summand b \overline z is real-differentiable, but does not satisfy the Cauchy–Riemann equations. Complex analysis shows some features not apparent in real analysis. For example, any two holomorphic functions and that agree on an arbitrarily small open subset of \mathbb necessarily agree everywhere. Meromorphic functions, functions that can locally be written as with a holomorphic function , still share some of the features of holomorphic functions. Other functions have essential singularities, such as at .


Applications

Complex numbers have applications in many scientific areas, including
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
,
control theory Control theory is a field of mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive ...
,
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
,
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
,
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
,
cartography Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an i ...
, and vibration analysis. Some of these applications are described below.


Geometry


Shapes

Three non-collinear points u, v, w in the plane determine the
shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie ...
of the triangle \. Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic as S(u, v, w) = \frac . The shape S of a triangle will remain the same, when the complex plane is transformed by translation or dilation (by an affine transformation), corresponding to the intuitive notion of shape, and describing similarity. Thus each triangle \ is in a similarity class of triangles with the same shape.


Fractal geometry

The Mandelbrot set is a popular example of a fractal formed on the complex plane. It is defined by plotting every location c where iterating the sequence f_c(z)=z^2+c does not diverge when iterated infinitely. Similarly, Julia sets have the same rules, except where c remains constant.


Triangles

Every triangle has a unique Steiner inellipse – an ellipse inside the triangle and tangent to the midpoints of the three sides of the triangle. The foci of a triangle's Steiner inellipse can be found as follows, according to Marden's theorem: Denote the triangle's vertices in the complex plane as , , and . Write the cubic equation (x-a)(x-b)(x-c)=0, take its derivative, and equate the (quadratic) derivative to zero. Marden's theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse.


Algebraic number theory

As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in \mathbb. '' 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 theory. Compared to \overline, the algebraic closure of \mathbb, which also contains all algebraic numbers, \mathbb 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 theory to the number field containing roots of unity, 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 is the Gaussian integers; that is, numbers of the form , where and are integers, which can be used to classify sums of squares.


Analytic number theory

Analytic number theory 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 is related to the distribution of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s.


Improper integrals

In applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this; see methods of contour integration.


Dynamic equations

In
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s, it is common to first find all complex roots of the characteristic equation of a linear differential equation or equation system and then attempt to solve the system in terms of base functions of the form . Likewise, in difference equations, the complex roots 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 .


Linear algebra

Eigendecomposition is a useful tool for computing matrix powers and matrix exponentials. However, it often requires the use of complex numbers, even if the matrix is real (for example, a
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \ ...
). Complex numbers often generalize concepts originally conceived in the real numbers. For example, the conjugate transpose generalizes the transpose, hermitian matrices generalize symmetric matrices, and unitary matrices generalize orthogonal matrices.


In applied mathematics


Control theory

In
control theory Control theory is a field of mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive ...
, systems are often transformed from the time domain to the complex frequency domain using the
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
. The system's zeros and poles are then analyzed in the ''complex plane''. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane. In the root locus method, it is important whether zeros and poles are in the left or right half planes, that is, have real part greater than or less than zero. If a linear, time-invariant (LTI) system has poles that are * in the right half plane, it will be unstable, * all in the left half plane, it will be
stable A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...
, * on the imaginary axis, it will have marginal stability. If a system has zeros in the right half plane, it is a nonminimum phase system.


Signal analysis

Complex numbers are used in signal analysis and other fields 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 wave of a given
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, the absolute value of the corresponding is the
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
and the argument is the phase. 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 x(t) = \operatorname \ and X( t ) = A e^ = a e^ e^ = a e^ where ω represents the
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
and the complex number ''A'' encodes the phase and amplitude as explained above. This use is also extended into
digital signal processing Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are ...
and digital image processing, which use digital versions of Fourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital
audio Audio most commonly refers to sound, as it is transmitted in signal form. It may also refer to: Sound *Audio signal, an electrical representation of sound *Audio frequency, a frequency in the audio spectrum * Digital audio, representation of sou ...
signals, still images, and
video Video is an electronic medium for the recording, copying, playback, broadcasting, and display of moving visual media. Video was first developed for mechanical television systems, which were quickly replaced by cathode-ray tube (CRT) sy ...
signals. Another example, relevant to the two side bands of amplitude modulation of AM radio, is: \begin \cos((\omega + \alpha)t) + \cos\left((\omega - \alpha)t\right) & = \operatorname\left(e^ + e^\right) \\ & = \operatorname\left(\left(e^ + e^\right) \cdot e^\right) \\ & = \operatorname\left(2\cos(\alpha t) \cdot e^\right) \\ & = 2 \cos(\alpha t) \cdot \operatorname\left(e^\right) \\ & = 2 \cos(\alpha t) \cdot \cos\left(\omega t\right). \end


In physics


Electromagnetism and electrical engineering

In
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
is used to analyze varying
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
s and currents. The treatment of resistors,
capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of ...
s, and
inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
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 impedance. This approach is called phasor calculus. In electrical engineering, the imaginary unit is denoted by , to avoid confusion with , which is generally in use to denote
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The movi ...
, or, more particularly, , which is generally in use to denote instantaneous electric current. Since the
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
in an AC circuit is oscillating, it can be represented as V(t) = V_0 e^ = V_0 \left (\cos\omega t + j \sin\omega t \right ), To obtain the measurable quantity, the real part is taken: v(t) = \operatorname(V) = \operatorname\left V_0 e^ \right = V_0 \cos \omega t. The complex-valued signal is called the
analytic Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic or analytical can also have the following meanings: Chemistry * ...
representation of the real-valued, measurable signal .


Fluid dynamics

In
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
, complex functions are used to describe potential flow in two dimensions.


Quantum mechanics

The complex number field is intrinsic to the mathematical formulations of quantum mechanics, where complex
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
and Heisenberg's matrix mechanics – make use of complex numbers.


Relativity

In special and
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, some formulas for the metric on
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
become simpler if one takes the time component of the spacetime continuum to be imaginary. (This approach is no longer standard in classical relativity, but is used in an essential way in quantum field theory.) Complex numbers are essential to spinors, which are a generalization of the
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
s used in relativity.


Generalizations and related notions

The process of extending the field \mathbb R of reals to \mathbb C is known as the Cayley–Dickson construction. It can be carried further to higher dimensions, yielding the quaternions \mathbb H and
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s \mathbb which (as a real vector space) are of dimension 4 and 8, respectively. In this context the complex numbers have been called the binarions. Just as by applying the construction to reals the property of
ordering Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
is lost, properties familiar from real and complex numbers vanish with each extension. The quaternions lose commutativity, that is, for some quaternions , and the multiplication of
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s, additionally to not being commutative, fails to be associative: for some octonions . Reals, complex numbers, quaternions and octonions are all normed division algebras over \mathbb R. By Hurwitz's theorem they are the only ones; the sedenions, the next step in the Cayley–Dickson construction, fail to have this structure. The Cayley–Dickson construction is closely related to the regular representation of \mathbb C, thought of as an \mathbb R-
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
(an \mathbb-vector space with a multiplication), with respect to the basis . This means the following: the \mathbb R-linear map \begin \mathbb &\rightarrow \mathbb \\ z &\mapsto wz \end for some fixed complex number can be represented by a matrix (once a basis has been chosen). With respect to the basis , this matrix is \begin \operatorname(w) & -\operatorname(w) \\ \operatorname(w) & \operatorname(w) \end, that is, the one mentioned in the section on matrix representation of complex numbers above. While this is a linear representation of \mathbb C in the 2 × 2 real matrices, it is not the only one. Any matrix J = \beginp & q \\ r & -p \end, \quad p^2 + qr + 1 = 0 has the property that its square is the negative of the identity matrix: . Then \ is also isomorphic to the field \mathbb C, and gives an alternative complex structure on \mathbb R^2. This is generalized by the notion of a linear complex structure. Hypercomplex numbers also generalize \mathbb R, \mathbb C, \mathbb H, and \mathbb. For example, this notion contains the split-complex numbers, which are elements of the ring \mathbb R (x^2-1) (as opposed to \mathbb R (x^2+1) for complex numbers). In this ring, the equation has four solutions. The field \mathbb R is the completion of \mathbb Q, the field of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s, with respect to the usual absolute value
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
. Other choices of metrics on \mathbb Q lead to the fields \mathbb Q_p of -adic numbers (for any
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
), which are thereby analogous to \mathbb. There are no other nontrivial ways of completing \mathbb Q than \mathbb R and \mathbb Q_p, by Ostrowski's theorem. The algebraic closures \overline of \mathbb Q_p still carry a norm, but (unlike \mathbb C) are not complete with respect to it. The completion \mathbb_p of \overline turns out to be algebraically closed. By analogy, the field is called -adic complex numbers. The fields \mathbb R, \mathbb Q_p, and their finite field extensions, including \mathbb C, are called local fields.


See also

*
Algebraic surface In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
* Circular motion using complex numbers * Complex-base system * Complex geometry * Dual-complex number * Eisenstein integer *
Euler's identity In mathematics, Euler's identity (also known as Euler's equation) is the equality e^ + 1 = 0 where : is Euler's number, the base of natural logarithms, : is the imaginary unit, which by definition satisfies , and : is pi, the ratio of the circ ...
* Geometric algebra (which includes the complex plane as the 2-dimensional spinor subspace \mathcal_2^+) * Unit complex number


Notes


References


Works cited

* * * * *


Further reading

* * *


Mathematical

* * * * * *


Historical

* * * * — 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. {{DEFAULTSORT:Complex Number Composition algebras