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In mathematics, a complex number is an element of a
number system A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
that extends the real numbers with a specific element denoted , called the imaginary unit 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. 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 The mathematical sciences are a group of areas of study that includes, in addition to mathematics, those academic disciplines that are primarily mathematical in nature but may not be universally considered subfields of mathematics proper. Statist ...
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 In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
s, even those that have no solutions in real numbers. More precisely, the
fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
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 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 ...
, and
distributive law In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic, ...
s. 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 also form a
real vector space Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
of dimension two, with as a standard basis. This standard basis makes the complex numbers a
Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, 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 In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D the ...
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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
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 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 In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
of the polynomial ring in the indeterminate , by the ideal generated by the polynomial (see 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 of all complex numbers is denoted by \Complex (
blackboard bold Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled. The symbols usually denote number sets. One way of pro ...
) 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 of ...
and electrical engineering, is used instead of as is frequently used to represent electric current. In these cases, complex numbers are written as , or .


Visualization

A complex number can thus be identified with an ordered pair (\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 Jean-Robert Argand (, , ; July 18, 1768 – August 13, 1822) was an amateur mathematician. In 1806, while managing a bookstore in Paris, he published the idea of geometrical interpretation of complex numbers known as the Argand diagram and is know ...
. 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 In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors a ...
, 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 ( 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 that uses the distance of the point from the origin (), and the angle subtended between the
positive real axis In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
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 In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
, 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 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 tha ...
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: \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 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 In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sine wave, sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and Phase (waves), initial phase (''θ'') are time-inva ...
, often used in electronics to represent a phasor with amplitude and phase , it is written as z = r \angle \varphi .


Complex graphs

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 In complex analysis, domain coloring or a color wheel graph is a technique for visualizing complex functions by assigning a color to each point of the complex plane. By assigning points on the complex plane to different colors and brightness, d ...
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 (without trigonometric functions) of a general
cubic equation In algebra, a cubic equation in one variable is an equation of the form :ax^3+bx^2+cx+d=0 in which is nonzero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
, when all three of its roots are real numbers, contains the square roots of
negative numbers In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed ma ...
, 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 In algebra, ''casus irreducibilis'' (Latin for "the irreducible case") is one of the cases that may arise in solving polynomials of degree 3 or higher with integer coefficients algebraically (as opposed to numerically), i.e., by obtaining roots th ...
'' ("irreducible case"). This conundrum led Italian mathematician Gerolamo Cardano 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 The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
, which shows that with complex numbers, a solution exists to every
polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
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 Rafael Bombelli (baptised Baptism (from grc-x-koine, βάπτισμα, váptisma) is a form of ritual purification—a characteristic of many religions throughout time and geography. In Christianity, it is a Christian sacrament of initia ...
. 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 roots of negative numbers 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 considered, apparently in error, the volume of an impossible
frustum In geometry, a (from the Latin for "morsel"; plural: ''frusta'' or ''frustums'') is the portion of a solid (normally a pyramid or a cone) that lies between two parallel planes cutting this solid. In the case of a pyramid, the base faces are ...
of a pyramid 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 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 solution A solution in radicals or algebraic solution is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of a polynomial equation, and relies only on addition, subtraction, multiplication, divisi ...
s for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (see
Niccolò Fontana Tartaglia Niccolò Fontana Tartaglia (; 1499/1500 – 13 December 1557) was an Italian mathematician, engineer (designing fortifications), a surveyor (of topography, seeking the best means of defense or offense) and a bookkeeper from the then Republi ...
, Gerolamo Cardano). 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 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. 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 ''Elements of Algebra'' is an elementary mathematics textbook written by mathematician Leonhard Euler around 1765 in German. It was first published in Russian as "''Universal Arithmetic''" (''Универсальная арифметика''), tw ...
'', 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 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 In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity) states that for any real number and integer it holds that :\big(\cos x + i \sin x\big)^n = \cos nx + i \sin nx, where is the imaginary unit (). ...
: (\cos \theta + i\sin \theta)^ = \cos n \theta + i\sin n \theta. In 1748, Euler went further and obtained Euler's formula of complex analysis: \cos \theta + i\sin \theta = e ^ 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 Danish
Norwegian Norwegian, Norwayan, or Norsk may refer to: *Something of, from, or related to Norway, a country in northwestern Europe * Norwegians, both a nation and an ethnic group native to Norway * Demographics of Norway *The Norwegian language, including ...
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 Caspar Wessel (8 June 1745, Vestby – 25 March 1818, Copenhagen) was a Danish– Norwegian mathematician and cartographer. In 1799, Wessel was the first person to describe the geometrical interpretation of complex numbers as points in the comp ...
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 Jean-Robert Argand (, , ; July 18, 1768 – August 13, 1822) was an amateur mathematician. In 1806, while managing a bookstore in Paris, he published the idea of geometrical interpretation of complex numbers known as the Argand diagram and is know ...
independently issued a pamphlet on complex numbers and provided a rigorous proof of the
fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
.
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
had earlier published an essentially
topological 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 ...
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 Norwegian, Norwayan, or Norsk may refer to: *Something of, from, or related to Norway, a country in northwestern Europe * Norwegians, both a nation and an ethnic group native to Norway * Demographics of Norway *The Norwegian language, including ...
Niels Henrik Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
and
Carl Gustav Jacob Jacobi Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasiona ...
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 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,
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
, Henri Poincaré,
Hermann Schwarz Karl Hermann Amandus Schwarz (; 25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis. Life Schwarz was born in Hermsdorf, Silesia (now Jerzmanowa, Poland). In 1868 he married Marie Kummer, ...
,
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
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 Wilhelm Wirtinger (19 July 1865 – 16 January 1945) was an Austrian mathematician, working in complex analysis, geometry, algebra, number theory, Lie groups and knot theory. Biography He was born at Ybbs on the Danube and studied at the Unive ...
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 both their real and imaginary parts are equal, that is, if and . Nonzero complex numbers written in
polar form In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
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 In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fiel ...
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 In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
'' 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 Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
. 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 In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
. 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 In electrical engineering, the maximum power transfer theorem states that, to obtain ''maximum'' external power from a power source with internal resistance, the resistance of the load must equal the resistance of the source as viewed from its ...
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 Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
(that is multiplying it with ) and adding the result to the
minuend Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
: 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 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 Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
.


Multiplication and square

The rules of the
distributive property In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmet ...
, 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 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 In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
\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 tha ...
). 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 \frac = \arctan\left(\frac\right) + \arctan\left(\frac\right) 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 \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 \exp \colon \Complex \to \Complex ; z \mapsto \exp z can be defined for every complex number by the power series \exp z= \sum_^\infty \frac , which has an infinite
radius of convergence In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
. The value at of the exponential function is
Euler's number The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of as approaches infinity, an expressi ...
e = \exp 1 = \sum_^\infty \frac1\approx 2.71828. If is real, one has \exp z=e^z. Analytic continuation 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 In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
e^=e^ze^t. 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

Euler's formula 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 \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 In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
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 In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...
\ln z = \left\, one has to use a branch cut and to restrict the
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
, resulting in the
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
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 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 In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...
. 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 In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity) states that for any real number and integer it holds that :\big(\cos x + i \sin x\big)^n = \cos nx + i \sin nx, where is the imaginary unit (). ...
: 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 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 : \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 In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fiel ...
, 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 \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, complex polynomial, 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 The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
, of
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
and
Jean le Rond d'Alembert Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the '' Encyclopéd ...
. Because of this fact, \Complex is called an algebraically closed field. 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 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 ...
ones such as the
winding number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of t ...
, or a proof combining
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
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.


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 In abstract algebra, the transcendence degree of a field extension ''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of ...
over \Q, the
prime field In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive ide ...
of \Complex, is the cardinality of the continuum. * Third, it is
algebraically closed 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 ...
(see above). It can be shown that any field having these properties is isomorphic (as a field) to \Complex. For example, the
algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...
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 In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series : \begin x^ &+ 2x^ + x^ + 2x^ + x^ + x^5 + \cdots\\ &=x^+ 2x^ + x^ + 2x^ + x^ + ...
. However, specifying an isomorphism requires the axiom of choice. 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 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 In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is w ...
(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 Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
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 Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
, while the nonzero real numbers are not.


Formal construction


Construction as ordered pairs

William Rowan Hamilton 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 In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic, ...
(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 over the real numbers. The set of complex numbers is defined as the
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
\R (X^2+1). This extension field contains two square roots of , namely (the
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s of) and , respectively. (The cosets of) and form a basis of \mathbb (X^2 + 1) as a real vector space, 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 (as fields). Accepting that \Complex is algebraically closed, since it is an
algebraic extension In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ext ...
of \mathbb in this approach, \Complex is therefore the
algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...
of \R.


Matrix representation of complex numbers

Complex numbers can also be represented by
matrices 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'' (franchis ...
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 In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preservi ...
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 of the corresponding matrix, and the conjugate of a complex number with the
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
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 In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
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, "Mat ...
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 A three-dimensional graph may refer to * A graph (discrete mathematics) In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense ...
to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.


Complex exponential and related functions

The notions of convergent series and continuous functions in (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said to
converge Converge may refer to: * Converge (band), American hardcore punk band * Converge (Baptist denomination), American national evangelical Baptist body * Limit (mathematics) * Converge ICT, internet service provider in the Philippines *CONVERGE CFD s ...
if and only if its real and imaginary parts do. This is equivalent to the
(ε, δ)-definition of limit Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches z ...
s, 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 \operatorname(z_1, z_2) = , z_1 - z_2, is a complete metric space, which notably includes the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
, 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 function In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and ...
s: the '' exponential function'' , also written , is defined as the
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
\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 In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the un ...
sinh and cosh, also carry over to complex arguments without change. For the other trigonometric and hyperbolic functions, such as tangent, 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. '' Euler's formula'' states: \exp(i\varphi) = \cos \varphi + i\sin \varphi for any real number , in particular \exp(i \pi) = -1 , which is Euler's identity. Unlike in the situation of real numbers, there is an infinitude 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 In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...
, 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 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 In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
if it satisfies the
Cauchy–Riemann equations In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differenti ...
. 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 . The second summand b \overline z is real-differentiable, but does not satisfy the
Cauchy–Riemann equations In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differenti ...
. 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 function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
s, 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 In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior. The category ''essential singularity'' is a "left-over" or default group of isolated singularities that a ...
, such as at .


Applications

Complex numbers have applications in many scientific areas, including signal processing, control theory,
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 of ...
, fluid dynamics,
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 chemistr ...
, cartography, and
vibration analysis Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic, such as the motion of a pendulum—or random, such ...
. Some of these applications are described below.


Geometry


Shapes

Three non-collinear points u, v, w in the plane determine the shape 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 The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. This ...
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 set In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values wi ...
s have the same rules, except where c remains constant.


Triangles

Every triangle has a unique
Steiner inellipse In geometry, the Steiner inellipse,Weisstein, E. "Steiner Inellipse" — From MathWorld, A Wolfram Web Resource, http://mathworld.wolfram.com/SteinerInellipse.html. midpoint inellipse, or midpoint ellipse of a triangle is the unique ellipse i ...
– 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 In mathematics, Marden's theorem, named after Morris Marden but proved about 100 years earlier by Jörg Siebeck, gives a geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its deriva ...
: Denote the triangle's vertices in the complex plane as , , and . Write the
cubic equation In algebra, a cubic equation in one variable is an equation of the form :ax^3+bx^2+cx+d=0 in which is nonzero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
(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 In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
containing
roots of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
, it can be shown that it is not possible to construct a regular
nonagon In geometry, a nonagon () or enneagon () is a nine-sided polygon or 9-gon. The name ''nonagon'' is a prefix hybrid formation, from Latin (''nonus'', "ninth" + ''gonon''), used equivalently, attested already in the 16th century in French ''nonogo ...
using only compass and straightedge – a purely geometric problem. Another example is the
Gaussian integer In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
s; 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 numbers.


Improper integrals

In applied fields, complex numbers are often used to compute certain real-valued
improper integral In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative infinity; or in some instances as both endpoin ...
s, by means of complex-valued functions. Several methods exist to do this; see
methods of contour integration In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. ...
.


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, an ...
s, it is common to first find all complex roots of the characteristic equation of a
linear differential equation In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b ...
or equation system and then attempt to solve the system in terms of base functions of the form . Likewise, in
difference equations In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
, 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 In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matr ...
is a useful tool for computing matrix powers and
matrix exponential In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential give ...
s. However, it often requires the use of complex numbers, even if the matrix is real (for example, a rotation matrix). Complex numbers often generalize concepts originally conceived in the real numbers. For example, the
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
generalizes the
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
,
hermitian matrices In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...
generalize
symmetric matrices In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with re ...
, and unitary matrices generalize orthogonal matrices.


In applied mathematics


Control theory

In control theory, systems are often transformed from the
time domain Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the c ...
to the complex
frequency domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
using the Laplace transform. The system's
zeros and poles In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. In some sense, it is the simplest type of singularity. Technically, a point is a pole of a function if ...
are then analyzed in the ''complex plane''. The
root locus In control theory and stability theory, root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly a gain within a feedback system. This is a technique used as a ...
, 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 In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stable are unstable; systems can also be mar ...
, * all in the left half plane, it will be stable, * on the imaginary axis, it will have
marginal stability In the theory of dynamical systems and control theory, a linear time-invariant system is marginally stable if it is neither asymptotically stable nor unstable. Roughly speaking, a system is stable if it always returns to and stays near a particul ...
. 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 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, di ...
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 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 and the complex number ''A'' encodes the phase and amplitude as explained above. This use is also extended into digital signal processing and
digital image processing Digital image processing is the use of a digital computer to process digital images through an algorithm. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing. It allo ...
, which use digital versions of Fourier analysis (and
wavelet A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
analysis) to transmit,
compress compress is a Unix shell compression program based on the LZW compression algorithm. Compared to more modern compression utilities such as gzip and bzip2, compress performs faster and with less memory usage, at the cost of a significantly lo ...
, restore, and otherwise process digital audio signals, still images, and video 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, the Fourier transform is used to analyze varying voltages and
currents Currents, Current or The Current may refer to: Science and technology * Current (fluid), the flow of a liquid or a gas ** Air current, a flow of air ** Ocean current, a current in the ocean *** Rip current, a kind of water current ** Current (stre ...
. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. This 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, or, more particularly, , which is generally in use to denote instantaneous electric current. Since the voltage 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 representation of the real-valued, measurable signal .


Fluid dynamics

In fluid dynamics, 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 spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg's
matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum j ...
– make use of complex numbers.


Relativity

In
special Special or specials may refer to: Policing * Specials, Ulster Special Constabulary, the Northern Ireland police force * Specials, Special Constable, an auxiliary, volunteer, or temporary; police worker or police officer Literature * ''Specia ...
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 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
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
s, which are a generalization of the tensors 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 In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced b ...
. It can be carried further to higher dimensions, yielding the quaternions \mathbb H and octonions \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 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 octonions, additionally to not being commutative, fails to be associative: for some octonions . Reals, complex numbers, quaternions and octonions are all
normed division algebra In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic f ...
s over \mathbb R. By Hurwitz's theorem they are the only ones; the
sedenion In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers; they are obtained by applying the Cayley–Dickson construction to the octonions, and as such the octonions are isomorphic to ...
s, the next step in the Cayley–Dickson construction, fail to have this structure. The Cayley–Dickson construction is closely related to the
regular representation In mathematics, and in particular the theory of group representations, the regular representation of a group ''G'' is the linear representation afforded by the group action of ''G'' on itself by translation. One distinguishes the left regular rep ...
of \mathbb C, thought of as an \mathbb R- algebra (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 Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...
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 In mathematics, a complex structure on a real vector space ''V'' is an automorphism of ''V'' that squares to the minus identity, −''I''. Such a structure on ''V'' allows one to define multiplication by complex scalars in a canonical fashion so ...
.
Hypercomplex number In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group represen ...
s 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 numbers, with respect to the usual absolute value metric. Other choices of
metrics 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 mathema ...
on \mathbb Q lead to the fields \mathbb Q_p of -adic numbers (for any prime number ), 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 In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers \Q is equivalent to either the usual real absolute value or a -adic absolute value. Definitions Raisi ...
. 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 field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...
s.


See also

* Algebraic surface * Circular motion using complex numbers *
Complex-base system In arithmetic, a complex-base system is a positional numeral system whose radix is an imaginary number, imaginary (proposed by Donald Knuth in 1955) or complex number (proposed by S. Khmelnik in 1964 and Walter F. Penney in 1965W. Penney, A "binar ...
*
Complex geometry In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
*
Dual-complex number In this article, we discuss certain applications of the dual quaternion algebra to 2D geometry. At this present time, the article is focused on a 4-dimensional subalgebra of the dual quaternions which we will call the ''planar quaternions''. Th ...
* Eisenstein integer * Euler's identity * Geometric algebra (which includes the complex plane as the 2-dimensional
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
subspace \mathcal_2^+) *
Unit complex number In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. ...


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