Complex number

A complex number is a mathematical object being used to define additional solutions in some equation S and with which the four usual operations extend from the real numbers. The complex numbers were introduced at the 16th century by the Italian mathematicians Jerome Cardan, Raphaël Bombelli and Nicolo Fontana, said Tartaglia, in order to express the solutions with the cubic equations in any general information by the Formules of Cardan joint, by using numbers of in particular square negative.

The whole of the sums and products of real numbers and the Imaginary number $\ scriptstyle {I}$ satisfies the properties of a structure of commutative body which contains the body of realities. It is called body of the complex numbers and notes $\ mathbb C$ . It is provided with the application module which generalizes the absolute Value of the real numbers but completely ordered in a compatible way with its structure of body cannot be .

It is only as from the 19th century that develops the geometrical aspect of the complex numbers, seen like elements or transformations of the plan, under the impulse of the Buée abbot and of Jean-Robert Argand (plane of Argand), then with work of Gauss and Cauchy.

In algebra, the Théorème of Alembert-Gauss identifies the degree of a complex Polynôme not no one with the number of its roots taken into account their order of multiplicity. The body of the complex numbers is thus algebraically closed.
In analysis, the exponential complexes makes it possible to simplify the study of the Fourier series then to define the Transformée of Fourier. The branch of the Analyze complexes relates to the study of the derivable functions with the complex direction, called holomorphic functions .
In physics, the complex numbers are used to describe the behavior of oscillating electric or the undulatory phenomena in electromagnetism.

Description

Notations of the complex numbers

The complex numbers, usually noted

z

, can thus be presented several manners:

Cartesian form,

algebraic: $Z = X + iy \,$
or vectorial: $Z = (X, there) \,$

form in polar coordinates:

geometrical $Z = \ rho.e^ {I \ theta} \,$
or vectorial: $Z = (\ rho, \ theta) = \ rho_ \ angle \ theta \,$
or trigonometrical: $Z = \ rho. (cos \ theta + I. sin \ theta) \,$

Cartesian form

A complex number

\ scriptstyle z

in general presents in Cartesian Coordonnées, like a sum

a + Bi \,

, where has and B is unspecified Real numbers and

\ scriptstyle {I}

(the imaginary Unité) is a particular number such as

\ scriptstyle {I} ^2 = -1

.
Reality has is called left real Z and notes

\ \ mathrm {Re} (Z)

\ Re (Z)

, reality B is its imaginary Partie and notes

\ \ mathrm {Im} (Z)

\ Im (Z)

.
Two complex numbers are equal if and only if they have the same real part and the same part imaginaire.

A complex number Z is known as pure Imaginaire or completely imaginary if its real part is null, in this case it is written in the form Z = Bi . A complex number of which the imaginary part is worth 0 is compared to a number réel.
The real number 0 is the only one which is at the same time real and imaginary pure, but the majority of the complex numbers are neither real nor imaginary pure.

The addition and the multiplication on the complex numbers have the same properties of Associativité, Commutativité and Distributivité that on the real numbers. The rules of calculation are thus written:

(+ Bi has) + (C + di) = (has + c) + (B + D) I \,

;

(+ Bi has) (C + di) = (ac - data base) + (AD + bc) I \,

In particular, this formula makes it possible to obtain the following equality:

(+ Bi has) ^2 = a^2 + 2 abi - b^2 \,

;

(has - Bi) ^2 = a^2 - 2 abi - b^2 \,

; -->

(+ Bi has) (- Bi has) = a^2 + b^2 \,

Since the sum a²+b² of two squares of real numbers is a strictly positive real number (except if has = B = 0 ), it exists a reverse with any complex number not no one with the equality:

\ frac {1} {a+bi} = \ frac {has - Bi} {a^2+b^2} \,

This fraction reveals two important expressions for the complex number $\ scriptstyle has + Bi \,$ :

* its Combined

\ overline {a+bi} =a-bi \,

is also a complex number;

* its module

\ left|a+bi \ right|= \ sqrt {a^2 + b^2}

is a positive real number.

The application of conjugation is a involutive Automorphisme : $\ overline {Z + z'} = \ bar {Z} + \ bar {z'}$ , $\ qquad \ overline {Z \ times z'} = \ bar {Z} \ times \ bar {z'}$ and $\ bar {\ bar {Z}} = z$ .
The module application is a absolute value because it is strictly positive apart from 0, under-additive $\ left (|Z + z'| \ it |Z|+|z'|\ right)$ and multiplicative $\ left (|Z z'| = |Z|\ times |z'|\ right)$ .

Realities are the only complex numbers which equal to their are combined. Positive realities are the only complexes equal to their module.
Number 0 is the only complex number whose module is worth 0.

Polar form

Complex plan

In a Plan

\ mathcal P

provided with a reference mark orthonormé

(O refines; \ vec {U}, \ vec {v})

, the image of a complex number

z = has + Bi \,

is the point M coordinates (has, b) , its vectorial image is the vector

\ overrightarrow {OM}

. The number Z is called affix point M or vector

\ overrightarrow {OM}

The module $|Z|\,$ is then the Length segment $\, \$ .
If Z is different from 0, its image is distinct from the origin O reference mark. One then calls argument of Z and one notes $\ mathrm {arg} (Z) \,$ any measurement $\ theta \,$ of the Angle $\ left (\ vec {U}, \ overrightarrow {OM} \ right)$ , well defined except for a multiple of $2 \ pi$ .

For example, strictly positive realities have a multiple argument of $2 \ pi$ , strictly negative realities have as an argument an odd multiple of $\ pi$ .
imaginary pure the nonnull ones have an adequate argument with $\ frac {\ pi} {2}$ or $- \ frac {\ pi} {2}$ modulo $2 \ pi$ , according to the sign of their imaginary part.

The plan $\ mathcal P$ , provided with its orthonormé reference mark and the actions of the complex numbers by addition and multiplication, is called plane complex . Since all the complex plans are Canonique lies isomorphous, one speaks about the plan complexes without specifying more.

Polar coordinates

The module and the argument of a complex number correspond to the polar Coordonnées

(R, \ theta)

of its image in the complex plan. By writing the Cartesian coordinates starting from the polar coordinates, any complex number not no one can thus be written under a trigonometrical form

z = R \ cos (\ theta) + I R \ sin (\ theta) \,

with

\, R > 0

The Formula of Euler $e^ {I \ theta} = \ cos (\ theta) + I \ sin (\ theta) \$ makes it possible to compact this writing under a exponential form $z = R e^ {I \ theta} \,$ .
Combined simply $\ bar {Z} = R e^ {is then written - I \ theta} = R \ cos (- \ theta) + I R \ sin (- \ theta) \,$ .

This writing moreover is adapted to the calculation of the product of two complex numbers because of the multiplicative properties of the function Exponentielle:

$\ left (R e^ {I \ theta} \ right) \ left (r' e^ {I \ theta'} \ right) = (R r') e^ {I (\ theta + \ theta')}\,$ ,
$\ left (R e^ {I \ theta} \ right) ^ {- 1} = \ frac {1} {R} e^ {- I \ theta} = \ frac {1} {R} \ cos (- \ theta) + I \ frac {1} {R} \ sin (- \ theta) \,$ .

Geometrical interpretation of the operations

That is to say Z and z' two complex number of respective images

M

and

M'

the image $M$ \, of the sum $z+z' \,$ is defined by the relation $\ overrightarrow {OM$ } = \ overrightarrow {OM} + \ overrightarrow {OM'} .
the action of a complex number per addition is interpreted geometrically like a translation according to the vector image.
Is $\ lambda$ a real number, the image $M_1$ of the product $\ lambda Z \,$ is defined by the relation $\ overrightarrow {OM_1} = \ lambda \ overrightarrow {OM}$ .
the action of the real number $\ lambda$ by scalar multiplication is interpreted geometrically like a Homothétie of center O and report/ratio $\ lambda$ on the complex level.
If Z is of module 1 and argument $\ theta$ , the image $M$ \, of the product $zz' \,$ is defined by the relations lengths $OM$ =OM' \, and of angles $\ left (\ overrightarrow {OM'}, \ overrightarrow {OM$ } \ right) = \ theta.
the action of a complex number of module 1 by multiplication is geometrically interpreted like a Rotation of center the origin and angle the argument.

By composition of a homothety and a rotation, the action of a complex number Z not no one by multiplication is geometrically interpreted like a direct similarity of center the origin, of $report/ratio|Z|\,$ and of angle $\ mathrm {arg} (Z) \,$ .

the image of combined $\ bar {Z}$ of $z \,$ is the symmetrical of $M$ compared to the x-axis.

Construction

See also: Construction of the complex numbers

There exist several current manners to build the body of the complex numbers starting from the unit of the real numbers and its elementary arithmetic operations. In addition to the objects all thus defined are isomorphous, constructions presented hereafter clarify three important characteristics:

the body of realities is clearly identified like a subset of the body of the complexes and the operations of addition and multiplication are preserved in the new structure. Real number 1 remains neutral for the multiplication.
There canonically exists a complex number $\ selected scriptstyle i$ whose square is worth $-1$ , although its opposite checks also this property.
Two real parameters are necessary and sufficient to describe all the complex numbers, which underlines the structure of real vector space of dimension 2 with a bases canonical.

Vector of the Euclidean plan

One can define a complex number as a vector of the plan

\ R^2

provided with his canonical base.

Each complex number is thus represented by a couple $(has, b)$ number réels.
The addition corresponds to that of the vectors, i.e. the addition of the coordinates term in the long term:

(has, b) + (C, d) = \ left (has + C, B + D \ right) \,

The multiplication is defined “arbitrarily” by:

(has, b) \ times (C, d) = \ left (ac - data base, AD + bc \ right) \,

The whole of realities is identified with the line

\ R \ times \ {0 \}

and the element

\ scriptstyle i

is the second basic vector

(0, 1)

. The module of a complex number corresponds finally to the euclidian norm of the associated vector and the argument is a measurement of the angle formed by the vector associated with the first basic vector.

This definition has the advantage of simplicity, since it requires few prérequis mathematical. It moreover is adapted to the geometrical representation of the complex numbers. On the other hand, the Associativeness of the multiplication and the existence of a Inverse are tiresome to show.

Stamp similarity

It is interesting to define a complex number as a matrix of direct similarity

\ left (
  \begin{matrix}
    has & - B \ \
    B & has \ \
   \end{matrix}
\ right)

with real coefficients, because the matric operations precisely induce the desired algebraic structure. Moreover, the module and the argument become respectively the report/ratio and a measurement of the angle of the similitude.
It should however be checked that the whole of these matrices is Stable by product:

\ begin {pmatrix} a&-b \ \ b&a \ end {pmatrix} \ begin {pmatrix} c&-d \ \ d&c \ end {pmatrix} = \ begin {pmatrix} ac-bd&-AD-bc \ \ ad+bc&ac-data base \ end {pmatrix},

what justifies in the passing the Commutativité product and ensures isomorphism between this structure and that previously definite.
The whole of realities is identified then with the whole of the diagonal matrices of the form

\ left (
  \begin{matrix}
    has & 0 \ \
    0 & have \ \
   \end{matrix}
\ right)

, the unit being represented by the matrix identity. The element

\ scriptstyle i

indicates classically the matrix

\ left (\ begin {matrix}
    0 & -1 \ \
    1 & 0 \ \
   \ end {matrix} \ right)

.
The determinant corresponds to the square of the module, which involves that all the nonnull elements are invertible and the method of the cofacteurs shows stability by reverse.

This point of view provides a natural construction which can be adapted to obtain the real algebra of the Quaternion S. It gives moreover a geometrical interpretation of the multiplication of the complex numbers like composition of similarities of the plan. The conjugation is finally represented by the transposition of matrices.

Classify equivalence of polynomials

A complex number can finally be seen like a Polynôme real of unspecified

\ scriptstyle i

, where the square

\ scriptstyle i^2

is identified with the constant polynomial of value -1, therefore with the identifications

\ scriptstyle i^3 = - i

\ scriptstyle i^4 = 1

…
Formally, that amounts comparing the whole of the complex numbers to the Espace quotient

\ R (X^2 + 1)

, in which two polynomials belong to same the Classe of equivalence if and only if they have the same remainder of Euclidean Division by X ² + 1.
The irreducible character of the polynomial X ² + 1 ensures the structure of body directly. Realities are represented by the constant polynomials and degree 2 of the dividing polynomial is the dimension of the whole like real vector space.

This very seemingly sophisticated design is perhaps that which describes best the invention of the complex numbers, far from the geometry, starting from one only algebraic generator and of only one relation. The formalism (more recent) of the quotient of a Euclidean ring (here the ring of the real polynomials to unspecified) by one of its ideal irreducible is at the base of the construction of the algebraic extensions of body.

Structure of the body of the complexes

The square roots of a complex number are written easily when this one is in trigonometrical form: those of

z = R e^ {I \ theta}

are

z_1 = \ sqrt {R} e^ {I \ frac {\ theta} {2}}

and

z_2 = \ sqrt {R} e^ {I \ left (\ frac {\ theta} {2} + \ pi \ right)}

and is opposite one of the autre.
The existence of two square roots, in the body of the complex numbers, for any complex number not no one (including for any strictly negative reality) is a property which is not checked by restriction on the body of realities, since no strictly negative reality can be obtained like the square of a real number .

Developments in mathematics

Complex analysis

Article principal : Analyze complexes.

The complex numbers were initially conceived to answer an algebraic problem. However, to extend the definitions of the analysis to the field of the complex numbers proves quite as fertile. For example the usual definition of the dérivée :

\ lim_ {H \ rightarrow 0} \ frac {F (z+h) - F (Z)}{H}

(with use of the multiplication and the subtraction complexes) makes it possible to obtain a new concept of derivable function, of complex variable with complex values called holomorphic Fonction . This concept proves more restrictive than its during reality, in particular, any holomorphic function sees its derivative being holomorphic, and even, any holomorphic function is analytical, i.e. admits a development in whole series of each point of its field of holomorphy.

In theory of integration, by using the concept of integral along a way, one obtains the integral Théorème of Cauchy, which ensures that the integral of a holomorphic function, on a field checking certain topological properties, along a closed loop, is null. This crucial property makes it possible to obtain the concept of primitive of a holomorphic function, always on an adapted field. Some of these topological conditions can be abandoned, thanks to the concept of Point singular, leading to the Remainder theorem.

Holomorphic dynamics

Differential equations in the complex field

The study of the holomorphic differential equations has the same basic results as that of the equations on functions of real variable, and in particular the Théorème of Cauchy-Lipschitz, which gives the existence and the unicity of a solution to a problem of Cauchy; or results of linear algebra on spaces of solutions of the linear differential equations.

However, the study of the equations at the singular points is definitely more fertile than the simple studies of connection of the real case: the topology of the plan complexes in the vicinity of a singular point makes that there is an infinity in manner of approaching it, and the study of the connections of the solutions obtained with all the methods of pleasing approach to the concept of Monodromie. This concept is then used within a more general framework: the differential Theory of Welshman.

Analyzes of Fourier

August 1st

Hypercomplexes numbers

In topology

By identifying the vector space $\ R^ {2n}$ with the vector space $\ mathbb {C} ^n$ , the multiplication by $\ scriptstyle {I}$ defines an application without Point fixes on the Sphère S of odd size.
the addition of a point “ad infinitum” to the complex plan defines the homeomorphic Sphère of Riemann in the usual sphere S², which can be seen like the first projective Espace complexe.
projection of the S³ sphere, seen like sphere unit of space $\ mathbb {C} ^2$ , on the sphere of Riemann by quotient of the action of the Cercle unit S¹ then constitutes the Fibration of Hopf.
complex projective spaces of even size generate rationally the ring of Cobordisme directed.

Uses in physics and engineering

Representation of the periodic phenomena and analyzes of Fourier

Then, the trigonometrical form made it possible to simplify the modeling and the writing of many phenomena, for example the undulatory phenomena (in particular in connection with the electromagnetic waves, or in electronic and more precisely in the field of the electronic analysis of the circuits containing of self-inductances (coils or reels) noted L, of the noted capacities C and noted resistances R (examples, R+jLw or R-j/Cw and w=2πF, here one has (R, L respectively) and (R, C) in series and with the diagram in series: R*R + (jLw - j/Cw) * (jLw-j/Cw) = Z*Z from where Z = SQR (R*R + (jLw-j/Cw) * (jLw-j/Cw)) and with tangent of the angle = (Lw-1/Cw) /R).

In the field of electronics, I representing the imaginary one in mathematics, becomes J for the physicists and one can plot the diagram of Fresnel then and, this, some is the expression.

Indeed, let us take an unspecified parameter, has ( T ), which depends on time in a sinusoidal way. That means that the value of has varies between has and - has , with always the same period, say, ω₁, and that one can write has = has .cos (ω₁ T ). If one multiplies the value of has by the value of B ( T ), a parameter of the same form, but of different period ω₂, we obtain:

C=a \ cos (\ omega_1 T). B \ cos (\ omega_2 T) \,

what is very pretty, but not easy to handle… but in exponential writing, we obtain:

C' = (ab) e^ {I (\ omega_1+ \ omega_2) T} = ce^ {I \ omega_3t} \,

what is much simpler to handle… But C ′, is not the product of has by B ! It is the real part of C ′! Implicitly, we transformed has and B in complexes, and handled them (, multiplied here). While taking the real part of C ′, we return in the body of realities. The complexes then have no physical reality.

In fact, one is useful oneself owing to the fact that $\ mathbb {C}$ contains $\ mathbb {R}$ to simplify the writings. Indeed, if one must write that a parameter is worth R cos (θ), one needs two realities, R and θ. But with complexes, it is enough to a number, which is much simpler.

In electromagnetism always, but in a different context, one can write the electromagnetic field like a complex combination of the electric field and magnetic field. Pure artifice of calculation, one can associate one or the other of these fields with the “imaginary” part of the complex field obtained: that simplifies the operations largely.

One also uses the complexes for the analyzes of Fourier, very much used in many fields, like the treatment of the signal.

Mechanics of the fluids in the plan

In Mechanical of the fluids (hydro/aerodynamic), one reveals complex potentials and speeds. Indeed, for a flow with two dimensions, one can break up the speed of the fluid into v_x and v_y . However, it is shown that:

V_x = \ frac {\ partial \ phi} {\ partial X} = \ frac {\ partial \ psi} {\ partial there}

V_y = \ frac {\ partial \ partial phi} {\ there} = \ frac {\ partial \ psi} {\ partial X}

To satisfy these conditions (conditions of Cauchy - Riemann) is equivalent saying that there exists an analytical function such as

f (Z) = \ phi + I \ psi \,

where

z=x+iy \,

This still makes it possible to write:

\ frac {df} {dz} = \ frac {\ partial \ phi} {\ partial X} + I \ frac {\ partial \ psi} {\ partial X} = V_x - I V_y \,

One calls F ( Z ) the potential complex , and its derivative compared to Z , the speed complexes . Thanks to this function, one directly obtains the module of speed, and his direction (by taking the trigonometrical form). Especially, one can simply model a flow around an obstacle, in a simple and compact way. The function ψ must be constant along the profile of this obstacle, which allows a simple resolution of F , thanks to simple results of complex analysis.

Quantum mechanics

Another simplification for physicists: the quantum Mécanique requires the complex numbers. The quantum functions of waves are thus all complex (see Postulats of quantum mechanics). In this case, however, it is possible (according to nonquantum theories) that corresponds to the real structure of the universe: either with 4 dimensions (space time), but of 5 and more - in certain theories up to 11 - on quantum scales (small). In spite of our perception (adapted on larger scales), imaginary dimension could thus extremely well also correspond to a “reality physique ” and not to represent only one convenience of writing.

If as well is besides as one takes place to establish a difference, because it is noticed that the effective notations to generate objects are it as much to describe them with precision then (see Fractale, Complexité of Kolmogorov, Compression, Entropie of Shannon and even neumatic Notation in music.

History

The first allusion to the roots of the negative numbers appears with work of the Greek and inventive Mathématicien Héron of Alexandria at first century BC, when it is interested in volume of the impossible trunk of a Pyramide, and this in spite of the fact that with the hellenistic time the negative numbers are inconceivable.

The complex numbers appear more clearly at the 16th century, when is established a formula to calculate for the polynomial roots of the cubic equations and polynomial quartics by the Italian mathematicians Niccolo Fontana Tartaglia and Gerolamo Cardano. It is carried out very early sometimes that these formulas, even if one is not interested whereas in the real solutions, require to handle the square root of negative numbers. For example, the cubic formula of Tartaglia gives the following solution to the equation X ³ − X = 0:

\ frac {1} {\ sqrt {3}} \ left (\ sqrt {- 1} ^ {1/3} + \ frac {1} {\ sqrt {- 1} ^ {1/3}} \ right).

At first sight that seems a nonsense. Although formal calculation with the complex numbers shows that the equation Z ³ = I has as a solution − I , ${\ scriptstyle \ frac {\ sqrt {3}} {2}} + {\ scriptstyle \ frac {1} {2}} i$ and ${\ scriptstyle \ frac {- \ sqrt {3}} {2}} + {\ scriptstyle \ frac {1} {2}} i$ . In substituent these results in ${\ scriptstyle \ sqrt {- 1} ^ {1/3}}$ the cubic formula of Tartaglia and while simplifying, one obtains 0,1 and − 1 as solutions of X ³ − X = 0.

This result is doubly astonishing because the concept even of negative numbers is not validated yet at the time. Name Imaginary number for these quantities is introduced by Rene Descartes in 1637, term qualified of “pejorative” so much their reality is contestable. An additional source of confusion lies in the fact that the equation $\ sqrt {- 1} ^2= \ sqrt {- 1} \ sqrt {- 1} =-1$ gives at the same time an algebraic identity $\ sqrt {has} \ sqrt {B} = \ sqrt {ab}$ , valid with positive realities has and B , but also with complex numbers of which one of the members (' or B has) is positive and the other negative one. The incorrect use of this identity (and the dependant identity $\ frac {1} {\ sqrt {has}} = \ sqrt {\ frac {1} {has}}$ ) if at the same time has and B is negative holds in particular Euler in failure. It is this difficulty which carries out the mathematicians of the time to agree to use the special symbol I in the place of $\ sqrt {- 1}$ to preserve EC error.

At the 18th century, in 1730, Abraham de Moivre states the well-known formula which door are name (Formule of Moivre):

$(\ cos \ theta + I \ sin \ theta) ^ {N} = \ cos N \ theta + I \ sin N \ theta \,$

Little time after, in 1748, Leonhard Euler gives, as for him, the following formula (Formule of Euler) used in Analyze complexes:

$\ cos \ theta + I \ sin \ theta = E ^ {I \ theta}. \,$

It is only in Hankel]] (1867) calls $\ cos \ phi + I
\ sin \ phi$ '' directional coefficient ''; * [[Karl Weierstrass|Weierstrass]], as for him, employs '' absolute value '' for '' module ''. After Cauchy and Gauss will follow many contributors of high ranking. Among those: * [[Ernst Kummer|Kummer]] (1844); * [[Leopold Kronecker|Kronecker]] (1845), * [[Scheffler]] (1845, 1851,1880); * [[Giusto Bellavitis|Bellavitis]] (1835, 1852); * [[George Peacock|Peacock]] (1845); * [[Augustus De Morgan|De Morgan]] (1849); * [[August Ferdinand Möbius|Möbius]] with which one must of many publications on the geometrical applications of the complex numbers; * [[Johann Peter Gustav Lejeune Dirichlet|Dirichlet]] to have extended the theory of the complex numbers and including to them [[prime number there|prime numbers], concept of [[congruence]], of [[Reciprocity (mathematics)|reciprocity]], etc, like in the case of them [[real number|real numbers]]. One [[Ring (mathematics)|ring]] or one [[Field (mathematics)|field]] complex is a whole of complex numbers stable by [[addition]], [[subtraction]] or [[Produced (mathematical)|multiplication]]. [[Carl Friedrich Gauss|Gauss]] studies the complex numbers of the form $a + bi$ , where '' has '' and '' B '' is just, or rational (where '' I '' is one of the two roots of $x^2 + 1 = 0$ ). Its pupil, [[Ferdinand Eisenstein]], studies the $a type + B \ omega$ , where $\ omega$ is a complex root of $x^3 - 1 = 0$ . Other classes (called [[cyclotomic Field|cyclotomic fields]]) of complex numbers are derived from [[the Root of the unit|roots of the unit]] $x^k - 1 = 0$ for the highest values of $k$ . This generalization is largely due to [[Ernst Kummer|Kummer]], which invents to also them [[ideal number|ideal numbers]], which were expressed like geometrical entities by [[Felix Klein]] in [[1893]]. The general theory of the fields is created by [[Welsh Évariste]], which studies the fields generated by the roots of any polynomial equation: : $\ F (X) = 0.$ Lastly, among the last contributors (after [[1884]]) of the general theory: * [[Karl Weierstrass|Weierstrass]]; * [[Hermann Schwarz|Schwarz]]; * [[Richard Dedekind|Dedekind]]; * [[Otto Hölder|Hölder]]; *l' abbot [[Berloty]]; * [[Henri Poincaré|Poincaré]]; * [[Eduard Study]]; * [[Alexander MacFarlane|MacFarlane]]. The correct formal definition, using pairs of real numbers was given to the {{XIXe century}}. to ==Voir aussi== {{Wikibooks|Al Complex numbers}} ===Articles connexes=== * [[Construction of the complex numbers]] * [[holomorphic Function]] * [[Root of complex number]] * [[Trigonometry complexes]] * [[split Complex number|Split complex numbers]] * [[Sphere of Riemann]] === external Bonds === * [http://villemin.gerard.free.fr/Wwwgvmm/Type/ImagComp.htm Complex numbers] * [http://share.devstude.net/index.php/Les_Nombres_Complexes ShareAn Shares easy knowledge, cards and easy ways on the complex numbers] {{Mathematical Units}} {{gate mathematics}} [[category: Nature of a number|Complex]] [[Category: Analysis complexes]] [[rear: عددعقدي]] [[bg: Комплексночисло]] [[bn: জটিলসংখ্যা]] [[Ca: Complex number]] [[Cs: Komplexní číslo]] [[da: Komplekse tal]] [[of: Komplexe Zahl]] [[el: Μιγαδικός αριθμός]] [[in: Complex number]] [[eo: Kompleksa nombro]] [[be: Número complejo]] [[and: Kompleksarv]] [[have: Zenbaki konplexu]] [[F: عددمختلط]] [[fi: Kompleksiluku]] [[fy: Kompleks getal]] [[He: מספרמרוכב]] [[hr: Kompleksni broj]] [[hu: Komplex számok]] [[is: Tvinntölur]] [[it: Number complesso]] [[ja: 複素数]] [[KB: 복소수]] [[lmo: Nümar cumpless]] [[lt: Kompleksiniai skaičiai]] [[ms: Nombor khayalan]] [[nl: Complex getal]] [[N: Komplekse tal]] [[No: Komplekst tall]] [[pl: Liczby zespolone]] [[Pt: Número complexo]] [[ro: Număr complex]] [[Ru: Комплексноечисло]] [[SNA: Nùmmuru cumplessu]] [[simple: Complex number]] [[sk: Komplexné číslo]] [[SSL: Kompleksno število]] [[Sr: Комплексанброј]] [[sv: Komplexa tal]] [[your: சிக்கலெண்]] [[HT: จำนวนเชิงซ้อน]] [[tr: Karmaşık sayı]] [[VI: Trường số phức]] [[yo: Nọ́mbà tósòro]] [[zh: 複數 (數學)]] [[zh-min-nan: Ho̍k-cha̍p-sò͘]] [[zh-yue: 複數
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