TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a complex number is an element of a
number system A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduc ...
that contains the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s and a specific element denoted , called the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad area ...
, and satisfying the
equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

. Moreover, every complex number can be expressed in the form , where and are real numbers. Because no real number satisfies the above equation, was called an
imaginary number An imaginary number is a complex number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...
by
René Descartes René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, Mathematics, mathematician, and scientist who invented analytic geometry, linking the previously sep ...

. For the complex number , 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. Statistic ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s, even those that have no solutions in real numbers. More precisely, the
fundamental theorem of algebra The fundamental theorem of algebra states that every non- constant single-variable polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, s ...
asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation $\left(x+1\right)^2 = -9$ has no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions and . Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule combined with the
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
,
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
and distributive laws. Every nonzero complex number has a
multiplicative inverse Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

. This makes the complex numbers a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
that has the real numbers as a subfield. The complex numbers form also a
real vector space Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ... of dimension two, with as a standard basis In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ... . This standard basis makes the complex numbers a Cartesian plane A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ... , called the complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ... . 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... which is identified to the horizontal axis of the complex plane. The complex numbers of absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ... one form the unit circle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ... . The addition of a complex number is a translation Translation is the communication of the meaning Meaning most commonly refers to: * Meaning (linguistics), meaning which is communicated through the use of language * Meaning (philosophy), definition, elements, and types of meaning discusse ... in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... is the reflection symmetry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ... with respect to the real axis. The complex absolute value is a Euclidean norm Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called pa ... . In summary, the complex numbers form a rich structure that is simultaneously an algebraically closed field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... , a commutative algebra Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ... over the reals, and a Euclidean vector space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ... of dimension two. Definition A complex number is a number of the form , where and are real numbers Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...

, and is an indeterminate satisfying . For example, is a complex number. This way, a complex number is defined as a
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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 In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studie ...
of the
polynomial ring In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
in the indeterminate , by the
ideal Ideal may refer to: Philosophy * Ideal (ethics) An ideal is a principle A principle is a proposition or value that is a guide for behavior or evaluation. In law Law is a system A system is a group of Interaction, interacting ...
generated by the polynomial (see
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
).

Notation

A real number can be regarded as a complex number , whose imaginary part is 0. A purely
imaginary number An imaginary number is a complex number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...
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 , $\mathcal\left(z\right)$, or $\mathfrak\left(z\right)$; the imaginary part of a complex number is denoted by , $\mathcal\left(z\right)$, or $\mathfrak\left(z\right).$ 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 Image:Blackboard bold.svg, 250px, An example of blackboard bold letters Blackboard bold is a typeface style that is often used for certain symbols in mathematics, mathematical texts, in which certain lines of the symbol (usually vertical or near-v ...

) or (upright bold). In some disciplines, particularly in
electromagnetism Electromagnetism is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in ...

and
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics The field of electronics is a branch of physics and electrical enginee ...

, is used instead of as is frequently used to represent
electric current An electric current is a stream of charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, ...
. In these cases, complex numbers are written as , or .

Visualization

A complex number can thus be identified with an
ordered pair In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

$\left(\Re \left(z\right),\Im \left(z\right)\right)$ 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 ArgandJean-Robert Argand (, , ; July 18, 1768 – August 13, 1822) was an amateur mathematician. In 1806, while managing a bookstore Image:Libraria Carturesti Carusel - Interior ziua.jpg, 250px, Cărturești Carusel, a bookshop in a historical building ...
. Another prominent space on which the coordinates may be projected is the two-dimensional surface of a sphere, which is then called
Riemann sphere In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

.

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 In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, while multiplication (see
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
) corresponds to multiplying their magnitudes and adding the angles they make with the real axis. Viewed in this way, the multiplication of a complex number by corresponds to rotating the position vector
counterclockwise Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite sen ...
by a quarter
turn Turn may refer to: Arts and entertainment Dance and sports * Turn (dance and gymnastics), rotation of the body * Turn (swimming), reversing direction at the end of a pool * Turn (professional wrestling), a transition between face and heel * Turn, ...
() about the origin—a fact which can be expressed algebraically as follows: $(a + bi)\cdot i = ai + b(i)^2 = -b + ai .$

Polar complex plane

Modulus and argument

An alternative option for coordinates in the complex plane is the
polar coordinate system In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

that uses the distance of the point from the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
(), and the angle subtended between the
positive real axisIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
and the line segment in a counterclockwise sense. This leads to the polar form :$z=re^=r\left(\cos\varphi +i\sin\varphi\right)$ of a complex numbers, where is the
absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of , and $\varphi$is the
argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ...
. 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...
, the absolute value of a complex number is the distance to the origin of the point representing the complex number in the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
. The ''argument'' of (in many applications referred to as the "phase" ) is the angle of the
radius In classical geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative ...

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 mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

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 . 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 The function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automati ...

: $\varphi = \operatorname\left(\operatorname(z),\operatorname(z) \right).$ Together, and give another way of representing complex numbers, the ''polar form'', as the combination of modulus and argument fully specify the position of a point on the plane. Recovering the original rectangular co-ordinates from the polar form is done by the formula called ''trigonometric form'' $z = r(\cos \varphi + i\sin \varphi ).$ Using
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler's ...

this can be written as $z = r e^ \text z = r \exp i \varphi.$ Using the function, this is sometimes abbreviated to $z = r \operatorname\mathrm \varphi.$ In angle notation, often used in
electronics The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons The electron is a subatomic particle In physical sciences, subatomic particles are smaller than ...
to represent a
phasor In and , a phasor (a of phase vector), is a representing a whose (''A''), (''ω''), and (''θ'') are . It is related to a more general concept called ,Bracewell, Ron. ''The Fourier Transform and Its Applications''. McGraw-Hill, 1965. p2 ...
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 A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', t ...
, which is possible only in projections. Because of this, other ways of visualizing complex functions have been designed. In
domain coloring In complex analysis of the function . Hue represents the argument, brightness the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investiga ...

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 . This provides a simple way to visualize the functions without losing information. The picture shows zeros for and poles at
Riemann surface In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s are another way to visualize complex functions. Riemann surfaces can be thought of as deformations of the complex plane; while the horizontal axes represent the real and imaginary inputs, the single vertical axis only represents either the real or imaginary output. However, Riemann surfaces are built in such a way that rotating them 180 degrees shows the imaginary output, and vice versa. Unlike domain coloring, Riemann surfaces can represent
multivalued function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s like .

History

trigonometric functions In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

) of a general
cubic equation roots A root is the part of a plant that most often lies below the surface of the soil but can also be aerial or aerating, that is, growing up above the ground or especially above water. Root or roots may also refer to: Art, entertainment, a ...

, when all three of its roots are real numbers, contains the square roots of
negative numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, a situation that cannot be rectified by factoring aided by the
rational root test In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
, if the cubic is irreducible; this is the so-called ''
casus irreducibilis In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
'' ("irreducible case"). This conundrum led Italian mathematician
Gerolamo Cardano Gerolamo (also Girolamo or Geronimo) Cardano (; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501 (O. S.)– 21 September 1576 (O. S.)) was an Italian polymath A polymath ( el, πολυμαθής, ', "having learn ...

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". Work on the problem of general polynomials ultimately led to the
fundamental theorem of algebra The fundamental theorem of algebra states that every non- constant single-variable polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, s ...
, which shows that with complex numbers, a solution exists to every
polynomial equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of degree one or higher. Complex numbers thus form an
algebraically closed field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, where any polynomial equation has a
root In vascular plant Vascular plants (from Latin ''vasculum'': duct), also known as Tracheophyta (the tracheophytes , from Greek τραχεῖα ἀρτηρία ''trācheia artēria'' 'windpipe' + φυτά ''phutá'' 'plants'), form a large grou ...
. Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician Rafael Bombelli. A more abstract formalism for the complex numbers was further developed by the Irish mathematician
William Rowan Hamilton Sir William Rowan Hamilton LL.D, DCL, MRIA (4 August 1805 – 2 September 1865) was an Irish mathematician, Andrews Professor of Astronomy at Trinity College Dublin, Trinity College Dublin, and Dunsink Observatory#Directors, Royal Astronomer ...
, who extended this abstraction to the theory of
quaternions In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion a ...
. The earliest fleeting reference to
square root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

s of
negative number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s can perhaps be said to occur in the work of the Greek mathematician
Hero of Alexandria Hero of Alexandria (; grc-gre, Ἥρων ὁ Ἀλεξανδρεύς, ''Heron ho Alexandreus'', also known as Heron of Alexandria ; c. 10 AD – c. 70 AD), was a Greek mathematician and engineer who was active in his native city of Alexandria, R ...

in the 1st century , where in his '' Stereometrica'' he considered, apparently in error, the volume of an impossible
frustum In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

of a
pyramid A pyramid (from el, πυραμίς ') is a structure A structure is an arrangement and organization of interrelated elements in a material object or system A system is a group of Interaction, interacting or interrelated elements that act ...

to arrive at the term $\sqrt = 3i\sqrt$ in his calculations, although negative quantities were not conceived of in
Hellenistic mathematics Greek mathematics refers to mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...
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 An algebraic solution or solution in radicals is a closed-form expression In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra ...
s for the roots of
cubic Cubic may refer to: Science and mathematics * Cube (algebra) In arithmetic and algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathema ...

and quartic
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s were discovered by Italian mathematicians (see
Niccolò Fontana Tartaglia Niccolò Fontana Tartaglia (; 1499/1500 – 13 December 1557) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
,
Gerolamo Cardano Gerolamo (also Girolamo or Geronimo) Cardano (; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501 (O. S.)– 21 September 1576 (O. S.)) was an Italian polymath A polymath ( el, πολυμαθής, ', "having learn ...

). It was soon realized (but proved much later) that these formulas, even if one was interested only in real solutions, sometimes required the manipulation of square roots of negative numbers. As an example, Tartaglia's formula for a cubic equation of the form gives the solution to the equation as $\tfrac\left(\left(\sqrt\right)^+\left(\sqrt\right)^\right).$ At first glance this looks like nonsense. However, formal calculations with complex numbers show that the equation has three solutions: $-i, \frac, \frac.$ Substituting these in turn for $\sqrt^$ in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of . Of course this particular equation can be solved at sight but it does illustrate that when general formulas are used to solve cubic equations with real roots then, as later mathematicians showed rigorously, the use of complex numbers is unavoidable. Rafael Bombelli was the first to address explicitly these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic trying to resolve these issues. The term "imaginary" for these quantities was coined by
René Descartes René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, Mathematics, mathematician, and scientist who invented analytic geometry, linking the previously sep ...

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 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 300px, Both groups are equal to 5. Apples are frequently used to explain arithmetic in textbooks for children. Elementary mathematics consists of mathematics Mathematics (from Ancient ...
, he introduces these numbers almost at once and then uses them in a natural way throughout. In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730
Abraham de Moivre Abraham de Moivre (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex number In mathematics, a complex number is a number that can be expressed in the form , where and are r ...

noted that the complicated identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be simply re-expressed by the following well-known formula which bears his name,
de Moivre's formula In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
: $(\cos \theta + i\sin \theta)^ = \cos n \theta + i\sin n \theta.$ In 1748
Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

went further and obtained
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler's ...

of
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
: $\cos \theta + i\sin \theta = e ^$ by formally manipulating complex
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \cdots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
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 Denmark, Danish–Norway, Norwegian mathematician Caspar Wessel in 1799, although it had been anticipated as early as 1685 in John Wallis, 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 ArgandJean-Robert Argand (, , ; July 18, 1768 – August 13, 1822) was an amateur mathematician. In 1806, while managing a bookstore Image:Libraria Carturesti Carusel - Interior ziua.jpg, 250px, Cărturești Carusel, a bookshop in a historical building ...
independently issued a pamphlet on complex numbers and provided a rigorous proof of the Fundamental theorem of algebra#History, fundamental theorem of algebra. Carl Friedrich Gauss had earlier published an essentially topology, topological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1". It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology.
If one formerly contemplated this subject from a false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, -1, $\sqrt$ positive, negative, or imaginary (or even impossible) units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness. — Gauss (1831)
In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée, C. V. Mourey, Mourey, John Warren (mathematician), Warren, Jacques Frédéric Français, Français and his brother, Giusto Bellavitis, 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 Norway, Norwegian Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise. Augustin Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of #Complex analysis, 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, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others. Important work (including a systematization) in complex multivariate calculus has been started at beginning of the 20th century. Important results have been achieved by Wilhelm Wirtinger in 1927.

Relations and operations

Equality

Complex numbers have a similar definition of equality to real numbers; two complex numbers and are equal if and only if both their real and imaginary parts are equal, that is, if and . Nonzero complex numbers written in polar form are equal if and only if they have the same magnitude and their arguments differ by an integer multiple of .

Ordering

Unlike the real numbers, there is no natural ordering of the complex numbers. In particular, there is no linear ordering on the complex numbers that is compatible with addition and multiplication. Hence, the complex numbers do not have the structure of an ordered field. One explanation for this is that every non-trivial sum of squares in an ordered field#nontrivialSquareSum, ordered field is nonzero, and is a non-trivial sum of squares. Thus, complex numbers are naturally thought of as existing on a two-dimensional plane.

Conjugate

The ''complex conjugate'' of the complex number is given by . It is denoted by either or . This unary operation on complex numbers cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division. Geometrically, is the reflection symmetry, "reflection" of about the real axis. Conjugating twice gives the original complex number $\overline=z,$ which makes this operation an involution (mathematics), involution. The reflection leaves both the real part and the magnitude of unchanged, that is $\operatorname(\overline) = \operatorname(z)\quad$ and $\quad , \overline, = , z, .$ The imaginary part and the argument of a complex number change their sign under conjugation $\operatorname(\overline) = -\operatorname(z)\quad \text \quad \operatorname \overline \equiv -\operatorname z \pmod .$ For details on argument and magnitude, see the section on #Polar form, Polar form. The product of a complex number and its conjugate is known as the ''absolute square''. It is always a non-negative real number and equals the square of the magnitude of each: $z\cdot \overline = x^2 + y^2 = , z, ^2 = , \overline, ^2.$ This property can be used to convert a fraction with a complex denominator to an equivalent fraction with a real denominator by expanding both numerator and denominator of the fraction by the conjugate of the given denominator. This process is sometimes called "rationalisation (mathematics), 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: $\overline = \overline \pm \overline,$ $\overline = \overline \cdot\overline,\quad \overline = \overline/\overline.$ Conjugation is also employed in inversive geometry, a branch of geometry studying reflections more general than ones about a line. In the Network analysis (electrical circuits), network analysis of electrical circuits, the complex conjugate is used in finding the equivalent impedance when the maximum power transfer theorem is looked for.

Two complex numbers and are most easily addition, added by separately adding their real and imaginary parts of the summands. That is to say: $a + b =(x+yi) + (u+vi) = (x+u) + (y+v)i.$ Similarly, subtraction can be performed as $a - b =(x+yi) - (u+vi) = (x-u) + (y-v)i.$ Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers and , interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices , and the points of the arrows labeled and (provided that they are not on a line). Equivalently, calling these points , , respectively and the fourth point of the parallelogram the triangles and are Congruence (geometry), congruent. A visualization of the subtraction can be achieved by considering addition of the negative subtrahend.

Multiplication and square

The rules of the distributive property, the commutative property, 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 Multiplicative inverse, 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 $\begin \cos a \cos b & - \sin a \sin b \,& = \,& \cos(a + b) \\ \cos a \sin b & + \sin a \cos b \,& = \,& \sin(a + b) . \end$ 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 Turn may refer to: Arts and entertainment Dance and sports * Turn (dance and gymnastics), rotation of the body * Turn (swimming), reversing direction at the end of a pool * Turn (professional wrestling), a transition between face and heel * 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 \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

). 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 pi, . 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 $\pm \left(\gamma + \delta i\right)$, where $\gamma = \sqrt$ and $\delta = (\sgn b)\sqrt,$ where is the sign function, signum function. This can be seen by squaring $\pm \left(\gamma + \delta i\right)$ to obtain . Here $\sqrt$ is called the absolute value, 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 In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \cdots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
$\exp z= \sum_^\infty \frac ,$ which has an infinite radius of convergence. The value at of the exponential function is Euler's number $e = \exp 1 = \sum_^\infty \frac1\approx 2.71828.$ If is real, one has $\exp z=e^z.$ Analytic continuation allows extending this equality for every complex value of , and thus to define the complex exponentiation with base as $e^z=\exp z.$

Functional equation

The exponential function satisfies the functional equation $e^=e^ze^t.$ This can be proved either by comparing the power series expansion of both members or by applying analytic continuation from the restriction of the equation to real arguments.

Euler's formula

Euler's formula Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler's ...

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 function, 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 $z = r(\cos \varphi + i\sin \varphi )$ 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

$\ln z = \left\,$ one has to use a branch cut and to restrict the codomain, resulting in the bijective function $\ln \colon \; \Complex^\times \; \to \; \; \; \R^+ + \; i \, \left(-\pi, \pi\right] .$ If $z \in \Complex \setminus \left\left( -\R_ \right\right)$ is not a non-positive real number (a positive or a non-real number), the resulting
principal value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

. 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
: $z^=(r(\cos \varphi + i\sin \varphi ))^n = r^n \, (\cos n\varphi + i \sin n \varphi).$ The nth root, th roots of a complex number are given by $z^ = \sqrt[n]r \left( \cos \left(\frac\right) + i \sin \left(\frac\right)\right)$ for . (Here $\sqrt\left[n\right]r$ 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 multivalued function, -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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
.See , pages 15–16. Briefly, this means that the following facts hold: first, any two complex numbers can be added and multiplied to yield another complex number. Second, for any complex number , its additive inverse is also a complex number; and third, every nonzero complex number has a Multiplicative inverse, 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, 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 total order, 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 analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
, complex matrix (mathematics), matrix, complex
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, 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 states that every non- constant single-variable polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, s ...
'', of Carl Friedrich Gauss and Jean le Rond d'Alembert. Because of this fact, $\Complex$ is called an
algebraically closed field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. This property does not hold for the rational number, field of rational numbers $\Q$ (the polynomial does not have a rational root, since square root of 2, √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 (complex analysis), Liouville's theorem, or topology, topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of ''odd'' degree has at least one real root. Because of this fact, theorems that hold ''for any algebraically closed field'' apply to $\Complex.$ For example, any non-empty complex square matrix has at least one (complex) eigenvalue.

Algebraic characterization

The field $\Complex$ has the following three properties: * First, it has characteristic (algebra), characteristic 0. This means that for any number of summands (all of which equal one). * Second, its transcendence degree over $\Q$, the prime field of $\Complex,$ is the cardinality of the continuum. * Third, it is algebraically closed (see above). It can be shown that any field having these properties is isomorphic (as a field) to $\Complex.$ For example, the algebraic closure of the field $\Q_p$ of the p-adic number, -adic number also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields). Also, $\Complex$ is isomorphic to the field of complex Puiseux series. However, specifying an isomorphism requires the axiom of choice. 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 neighborhood (topology), nearness and continuity (topology), continuity, which matter in areas such as Mathematical analysis, analysis and topology, are not dealt with. The following description of $\Complex$ as a topological ring, topological field (that is, a field that is equipped with a topological space, topology, 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 involution (mathematics), 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 (topology), base, where ranges over the field and ranges over . With this topology is isomorphic as a ''topological'' field to $\Complex.$ The only connected space, connected locally compact topological ring, 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 space, connected, while the nonzero real numbers are not.

Formal construction

Construction as ordered pairs

William Rowan Hamilton Sir William Rowan Hamilton LL.D, DCL, MRIA (4 August 1805 – 2 September 1865) was an Irish mathematician, Andrews Professor of Astronomy at Trinity College Dublin, Trinity College Dublin, and Dunsink Observatory#Directors, Royal Astronomer ...
introduced the approach to define the set $\Complex$ of complex numbers as the set $\mathbb^2$ of of real numbers, in which the following rules for addition and multiplication are imposed: $\begin (a, b) + (c, d) &= (a + c, b + d)\\ (a, b) \cdot (c, d) &= (ac - bd, bc + ad). \end$ It is then just a matter of notation to express as .

Construction as a quotient field

Though this low-level construction does accurately describe the structure of the complex numbers, the following equivalent definition reveals the algebraic nature of $\Complex$ more immediately. This characterization relies on the notion of fields and polynomials. A field is a set endowed with addition, subtraction, multiplication and division operations that behave as is familiar from, say, rational numbers. For example, the distributive law $(x+y) z = xz + yz$ must hold for any three elements , and of a field. The set $\R$ of real numbers does form a field. A polynomial with real coefficients is an expression of the form $a_nX^n+\dotsb+a_1X+a_0,$ where the are real numbers. The usual addition and multiplication of polynomials endows the set $\R\left[X\right]$ of all such polynomials with a ring (mathematics), ring structure. This ring is called the
polynomial ring In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
over the real numbers. The set of complex numbers is defined as the
quotient ring In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studie ...
$\R\left[X\right]/\left(X^2+1\right).$ This extension field contains two square roots of , namely (the cosets of) and , respectively. (The cosets of) and form a basis of 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 polynomial, irreducible over $\R,$ so the ideal it generates is Maximal ideal, maximal. The formulas for addition and multiplication in the ring $\R\left[X\right],$ 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 isomorphism, isomorphic (as fields). Accepting that $\Complex$ is algebraically closed, since it is an algebraic extension of $\mathbb$ in this approach, $\Complex$ is therefore the algebraic closure of $\R.$

Matrix representation of complex numbers

Complex numbers can also be represented by matrix (mathematics), matrices that have the form: $\begin a & -b \\ b & \;\; a \end$ Here the entries and are real numbers. As the sum and product of two such matrices is again of this form, these matrices form a subring of the ring matrices. A simple computation shows that the map: $a+ib\mapsto \begin a & -b \\ b & \;\; a \end$ is a ring isomorphism from the field of complex numbers to the ring of these matrices. This isomorphism associates the square of the absolute value of a complex number with the determinant of the corresponding matrix, and the conjugate of a complex number with the transpose of the matrix. The geometric description of the multiplication of complex numbers can also be expressed in terms of rotation matrix, 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: $\begin r\cos t & - r\sin t \\ r\sin t & \;\; r\cos t \end$ 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 (mathematics), rotation of the angle .

Complex analysis

The study of functions of a complex variable is known as
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (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 graph of a function of two variables, three-dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.

Complex exponential and related functions

The notions of convergent series and continuous functions in (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said to convergent sequence, converge if and only if its real and imaginary parts do. This is equivalent to the (ε, δ)-definition of limits, where the absolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view, $\mathbb$, endowed with the metric (mathematics), metric $\operatorname(z_1, z_2) = , z_1 - z_2,$ is a complete metric space, which notably includes the triangle inequality $, z_1 + z_2, \le , z_1, + , z_2,$ for any two complex numbers and . Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the ''exponential function'' , also written , is defined as the infinite series $\exp z:= 1+z+\frac+\frac+\cdots = \sum_^ \frac.$ The series defining the real trigonometric functions sine and cosine, as well as the hyperbolic functions sinh and cosh, also carry over to complex arguments without change. For the other trigonometric and hyperbolic functions, such as tangent (function), 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 Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler's ...

'' 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 infinite set, 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 arg (mathematics), argument defined #Polar form, above, and ln the (real) natural logarithm. As arg is a
multivalued function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

, unique only up to a multiple of , log is also multivalued. The
principal value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of log is often taken by restricting the imaginary part to the interval (mathematics), interval . Complex exponentiation is defined as $z^\omega = \exp(\omega \log z),$ 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 Exponentiation#Failure of power and logarithm identities, 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$$\mathbb$ is called Holomorphic function, holomorphic if it satisfies the Cauchy–Riemann equations. For example, any Linear transformation#Definition and first consequences, $\mathbb$-linear map $\mathbb$$\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. Complex analysis shows some features not apparent in real analysis. For example, any two holomorphic functions and that agree on an arbitrarily small open subset of $\mathbb$ necessarily agree everywhere. Meromorphic functions, functions that can locally be written as with a holomorphic function , still share some of the features of holomorphic functions. Other functions have essential singularity, essential singularities, such as at .

Applications

Complex numbers have applications in many scientific areas, including signal processing, control theory,
electromagnetism Electromagnetism is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in ...

, fluid dynamics, quantum mechanics, cartography, and Vibration#Vibration analysis, vibration analysis. Some of these applications are described below.

Geometry

Shapes

Three collinearity, non-collinear points $u, v, w$ in the plane determine the Shape#Similarity classes, 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 shape#Similarity classes, similarity class of triangles with the same shape.

Fractal geometry

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

Triangles

Every triangle has a unique Steiner inellipse – an ellipse inside the triangle and tangent to the midpoints of the three sides of the triangle. The Focus (geometry), foci of a triangle's Steiner inellipse can be found as follows, according to Marden's theorem: Denote the triangle's vertices in the complex plane as , , and . Write the
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$\left(x-a\right)\left(x-b\right)\left(x-c\right)=0$, 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 $\mathbb$. ''Argumentum a fortiori, 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 (mathematics), field theory to the number field containing root of unity, roots of unity, it can be shown that it is not possible to construct a regular nonagon compass and straightedge constructions, using only compass and straightedge – a purely geometric problem. Another example is the Gaussian integers; that is, numbers of the form , where and are integers, which can be used to classify Fermat's theorem on sums of two squares, 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 integrals, by means of complex-valued functions. Several methods exist to do this; see methods of contour integration.

Dynamic equations

In differential equations, it is common to first find all complex roots of the Linear differential equation#Homogeneous equations with constant coefficients, characteristic equation of a linear differential equation or equation system and then attempt to solve the system in terms of base functions of the form . Likewise, in difference equations, the complex roots of the characteristic equation of the difference equation system are used, to attempt to solve the system in terms of base functions of the form .

Linear Algebra

Eigendecomposition of a matrix, Eigendecomposition is a useful tool for computing matrix powers and Matrix exponential, matrix exponentials. 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 generalizes the transpose, Hermitian matrix, hermitian matrices generalize Symmetric matrix, symmetric matrices, and Unitary matrix, unitary matrices generalize Orthogonal matrix, orthogonal matrices.

In applied mathematics

Control theory

In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The system's zeros and poles are then analyzed in the ''complex plane''. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane. In the root locus method, it is important whether zeros and poles are in the left or right half planes, that is, have real part greater than or less than zero. If a linear, time-invariant (LTI) system has poles that are * in the right half plane, it will be unstable, * all in the left half plane, it will be BIBO stability, stable, * on the imaginary axis, it will have marginal stability. If a system has zeros in the right half plane, it is a nonminimum phase system.

Signal analysis

Complex numbers are used in signal analysis and other fields 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, the absolute value of the corresponding is the amplitude and the Argument (complex analysis), argument is the phase (waves), 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, which utilize digital versions of Fourier analysis (and wavelet analysis) to transmit, Data compression, compress, restore, and otherwise process Digital data, digital Sound, 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
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, the Fourier transform is used to analyze varying voltages and Electric current, currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the Electrical impedance, 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
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, or, more particularly, , which is generally in use to denote instantaneous electric current. Since the voltage in an AC electric circuit, 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 signal, 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 – make use of complex numbers.

Relativity

In special relativity, special and general relativity, 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 Wick rotation, used in an essential way in quantum field theory.) Complex numbers are essential to spinors, 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. 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 ordered field, 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 algebras over $\mathbb R$. By Hurwitz's theorem (normed division algebras), Hurwitz's theorem they are the only ones; the sedenions, the next step in the Cayley–Dickson construction, fail to have this structure. The Cayley–Dickson construction is closely related to the regular representation of $\mathbb C,$ thought of as an $\mathbb R$-Algebra (ring theory), algebra (an -vector space with a multiplication), with respect to the basis . This means the following: the $\mathbb R$-linear map $\begin \mathbb &\rightarrow \mathbb \\ z &\mapsto wz \end$ for some fixed complex number can be represented by a matrix (once a basis has been chosen). With respect to the basis , this matrix is $\begin \operatorname(w) & -\operatorname(w) \\ \operatorname(w) & \operatorname(w) \end,$ that is, the one mentioned in the section on matrix representation of complex numbers above. While this is a linear representation of $\mathbb C$ in the 2 × 2 real matrices, it is not the only one. Any matrix $J = \beginp & q \\ r & -p \end, \quad p^2 + qr + 1 = 0$ has the property that its square is the negative of the identity matrix: . Then $\$ is also isomorphic to the field $\mathbb C,$ and gives an alternative complex structure on $\mathbb R^2.$ This is generalized by the notion of a linear complex structure. Hypercomplex numbers also generalize $\mathbb R,$ $\mathbb C,$ $\mathbb H,$ and $\mathbb.$ For example, this notion contains the split-complex numbers, which are elements of the ring $\mathbb R\left[x\right]/\left(x^2-1\right)$ (as opposed to $\mathbb R\left[x\right]/\left(x^2+1\right)$ 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
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metric (mathematics), metric. Other choices of metric (mathematics), metrics on $\mathbb Q$ lead to the fields $\mathbb Q_p$ of p-adic number, -adic numbers (for any prime number ), which are thereby analogous to . There are no other nontrivial ways of completing $\mathbb Q$ than $\mathbb R$ and $\mathbb Q_p,$ by Ostrowski's theorem. The algebraic closures $\overline$ of $\mathbb Q_p$ still carry a norm, but (unlike $\mathbb C$) are not complete with respect to it. The completion $\mathbb_p$ of $\overline$ turns out to be algebraically closed. By analogy, the field is called -adic complex numbers. The fields $\mathbb R,$ $\mathbb Q_p,$ and their finite field extensions, including $\mathbb C,$ are called local fields.

* Algebraic surface * Circular motion#Using complex numbers, Circular motion using complex numbers * Complex-base system * Complex geometry * Dual-complex number * Eisenstein integer * Euler's identity * Geometric algebra#Unit pseudoscalars, Geometric algebra (which includes the complex plane as the 2-dimensional Spinor#Two dimensions, spinor subspace $\mathcal_2^+$) * Unit complex number

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