It extends

complex number the complex numbers in such a manner the number range of the real numbers that also roots of negative numbers can be computed. This succeeds to i /math by introduction <of a new> number math \< mathrm> as solution of the equation <math> x^2 = -1< /math>. This number <math> \ mathrm i< /math>also imaginary unit one calls. The origin of the theory of the imaginary numbers, i.e. all numbers, whose square is a negative real number, goes on the Italian mathematician Raffaele Bombelli to in 16. Century back. The introduction thatimaginary unit <math> \ mathrm i< /math> as new number Leonhard Euler is attributed.

Complex numbers usually in the form <math> a+b \ cdot \ mathrm i< /math> represented, whereby <math> A< /math> and <math> b< /math> real numbers are and <math> \ mathrm i< /math> the imaginary unit is. On in such a way representedcomplex numbers can be used the usual arithmetic rules for real numbers, whereby <math> \ mathrm i^2< /math> always through <math> -1< /math> to be replaced can and in reverse.

In such a way designed number range of the complex numbers has a number of favourable characteristics, itself within many ranges thatNature and engineer sciences as extremely useful proved. One of the reasons for these positive characteristics is the algebraic compartmentation of the complex numbers. This means that each algebraic equation over the complex numbers possesses a solution. This characteristic is contentsthe fundamental principle of algebra. A further reason is a connection between trigonometric functions and the exponential function, which over the complex numbers can be manufactured.

For the quantity of the complex numbers the symbol becomes <math> \ mathbb C< /math> used.

Table of contents

definition

as complex numbers one designates the numbers of the form <math> a+b \ cdot \ mathrm i< /math> (and/or. in shortened notation <math> a+b \, \ mathrm i </math>), whereby for the addition

< math> (a+b \, \ mathrm i) + (c+d \, \ mathrm i) = (a+c) + (b+d) \, \ mathrm i< /math>

and the multiplication

< math> (a+b \, \ mathrm {i}) \ cdot (c+d \, \ mathrm {i}) = (AC-bd) + (ad+bc) \ cdot \ mathrm i< /math>

applies. The imaginaryUnit <math> \ mathrm i< /math> thereby a non-real number with the characteristic is <math> \ mathrm i^2=-1< /math>; above formula for the multiplication arises thereby as a result of simply Ausmultiplizieren and new grouping.

One calls <math> A< /math> the real part and <math> b< /math> the imaginary part of <math> A + b \, \ mathrm i< /math>.

A formal specifying would be for example the following: The complex numbers are a body< math> \ mathbb C< /math>, which contains the real numbers as subfield, as well as an element <math> \ mathrm i \ in \ mathbb C< /math>, that the equation <math> \ mathrm i^2=-1< /math> fulfilled, so that itself each elementby <math> \ mathbb C< /math> in clear way in the form <math> a+b \ mathrm i< /math> with <math> A, b \ in \ mathbb R< /math> to write leaves. Two pairs <math> (\ mathbb C, \ mathrm i)< /math> and <math> (\ tilde {\ mathbb C}, \ tilde {\ mathrm i})< /math> can be identified in clear way with one another.

to the notation

  • <math> a+b \, \ mathrm i< /math> - notationalso cartesian or algebraic form one calls. The designation cartesian explains itself from the representation in the complex and/or. Gauss' number level (S. further down).
  • In electro-technology the small i is already used for temporally variable rivers (see alternating current) can lead and to confounding with the imaginary unit i. Therefore within this range the letter j is used [z. B. Paperback of high-frequency engineering Bd.1..3; Meinke, basiclaugh, 1992].
  • In physics becomes between <math> i< /math> for alternating current and <math> \ mathrmi< /math> for the imaginary unit differentiated. This does not lead by the quite clear separation with the attentive reader to mistakes and in this form to a large extent both in the physical-experimental and in the physical-theoretical literature is used. See also:complex alternating current calculation
  • Complex numbers are frequently also underlined represented, in order to differentiate it from real numbers to.

construction of the complex numbers

thereby the above axiomatic definition a sense has, must be proven that it at all a body <math> \ mathbb C< /math> withgives the necessary characteristics. This carries the following construction out.

the quantity of the pairs of real numbers

the construction takes first math \ mathrm i /math to no <purchase> to the imaginary< unit>: In the two-dimensional real vector space< math> \ R^2< /math> the arranged real Zahlenpaare <math> z= (A, b)< /math>becomes apart from the addition

<math> (A, b) + (C, D) = (A + C, b + D) \,< /math>

(that is the usual Vektoraddition) a multiplication through

< math> (A, b) \ cdot (C, D) = (A \ cdot C - b \ cdot D, \, A \ cdotD + b \ cdot C)< /math>

defined.

After this definition one writes <math> \ Bbb C= \ to R^2< /math>, and <math> (\ mathbb {C}, +, \ cdot)< /math> becomes a body, the body of the complex numbers.

first characteristics

  • the illustration <math> \ R \ tons \ Bbb C, \, A \ mapsto (A, 0)< /math> is oneBody imbedding of <math> \ R< /math> in <math> \ Bbb C< /math>, is able those we the real number <math> A< /math> with the complex number <math> (A, 0)< /math> identify.

Concerning the addition is:

  • the number <math> 0= (0,0)< /math> the Nullelement in <math> \ Bbb C< /math> and
  • the number <math> - z= (- A, - b)< /math> the inverse element in <math> \ BbbC< /math>.

Concerning the multiplication is:

  • the number <math> 1= (1,0)< /math> the neutral element (the one element) of <math> \ Bbb C< /math> and
  • the inverse (reciprocal one) too <math> z= (A, b) \ neq 0< /math> is <math> z^ {- 1} = \ left (\ frac {A} {a^2+b^2}, \, \ frac {- b} {a^2+b^2} \ right)< /math>.

purchase for representation in the form A + b i

through <math> \ mathrm i= (0,1)< /math> the imaginary unit i is specified; to these math <\> mathrm i^2=-1 /math< applies>.

Each complex number <math> z= (A, b) \ in \ Bbb C< /math> the clear representation of the form possesses

< math> z = (A, b) = (A, 0) + (b, 0) \ cdot (0,1) = A + b \, \ mathrm {i}< /math>

with <math> A, b \ in \ R< /math>; thisis the usual way of writing for the complex numbers.

arithmetic rules in the algebraic form

addition, subtraction

similar to the addition

< math> (A + b \, \ mathrm i) + (C + D \, \ mathrm i) = (A +c) + (b + D) \, \ mathrm i< /math>

functioned also subtraction

< math> (A + b \, \ mathrm i) - (C + D \, \ mathrm i) = (A C) + (b - D) \, \ mathrm i< /math>.

multiplication

That real part of the product consists the sum of the two mixed products of the product of the real parts minus the product of the imaginary parts, the imaginary part of the product is „real part times imaginary part “:

<math>
(a+b \ \ mathrm i cdot) \ cdot (c+d \ cdot \ mathrm i) = (A \ cdot C- b \ cdot D) + (A \ cdot d+b \ cdot C) \ cdot \ mathrm i

< /math>

division

the quotient of two complex numbers <math> a+b \, \ mathrm i< /math> and <math> c+d \, \ mathrm i< /math> with <math> c+d \, \ mathrm i \ neq 0< /math> can be computed, by one the break with the complex conjugated< math> CD\, \ mathrm i< /math> the denominator extends. The denominator becomes real thereby:

<math> \ frac {a+b \, \ mathrm i} {c+d \, \ mathrm i} = \ frac {(a+b \, \ mathrm i) (CD \, \ mathrm i)}{(c+d \, \ mathrm i) (CD \, \ mathrm i)} = \ frac {ac+bd} {c^2+d^2} + \ frac {UC-ad} {c^2+d^2} \ cdot \ mathrm i< /math>

arithmetical examples

addition:

<math> (3+2 \ mathrm i) + (5+5 \ mathrm i) = (3+5) + (2+5) \ mathrm i = 8+ 7 \ mathrm i \ </math>

Subtraction:

<math> (5+5 \ mathrm i) - (3+2 \ mathrm i) = (5-3) + (5-2) \ mathrm i = 2 + 3 \ mathrm i \ </math>

Multiplication:

<math> (2+5 \ mathrm i) \ cdot (3+7 \ mathrm i) = (2 \ cdot 3 - 5 \ cdot 7) + (2 \ cdot 7 + 5 \ cdot 3) \ mathrm i= -29 + 29 \ mathrm i< /math>

Division:

<math> {(2+5 \ mathrm {i}) \ more over (3+7 \ mathrm {i})} = {(2+5 \ mathrm {i}) \ more over (3+7 \ mathrm {i})} \ cdot {(3-7 \ mathrm {i}) \ more over (3-7 \ mathrm {i})} = {6-14 \ mathrm {i} +15 \ mathrm {i} - 35 \ mathrm {i} ^2 \ more over 9+21 \ mathrm {i} - 21 \ mathrm {i} - 49 \ mathrm {i} ^2} = {41+ \ mathrm {i} \ more over 9+49} = {41+ \ mathrm {i} \ more over 58}< /math>

further characteristics

  • of the bodies <math> \ Bbb C< /math> the complexNumbers is on the one hand a torso of <math> \ R< /math>, on the other hand a two-dimensional <math> \ R< /math> - vector space.
  • The body extension <math> \ Bbb C:\R< /math> =2 /math is <from> the degree math [\ Bbb C:\R<]>; math \ <Bbb> C /math is <more exact> isomorphic to the quotient body <math> \ R [X]/(X^2+1)< /math>, whereby <math> X^2+1< /math> the minimum polynomial of <math> \ mathrm {i}< /math> over <math> \ R< /math> is. Furthermore math <\> Bbb formsC< /math> already the algebraic conclusion of <math> \ R< /math>.
  • As <math> \ R< /math> - vector space possesses <math> \ Bbb C< /math> the basis <math> \ {1, \ mathrm {i} \}< /math>. Besides math <\> Bbb is C< /math> like each body also a vector space over itself, thus a linear <math> \ Bbb C< /math> - vector space with basis <math> \ {1 \}< /math>.
  • <math> \ mathrm {i}< /math> and <math> - \ mathrm {i}< /math>exactly the solutions of the quadratic equation are <math> x^2 + 1 = 0< /math>. In this sense math <\> mathrm {i knows}< /math> as “root from -1 " to be understood.
  • <math> \ Bbb C< /math> is contrary to <math> \ R< /math> no arranged body, D. h. there are nonewith the body structure compatible order relation “<“on <math> \ Bbb C< /math>. Of two different complex numbers one cannot say therefore, which of both the larger and/or. the smaller number is.

number complex number

Gauß'sche Ebene mit einer komplexen Zahl in kartesischen und in Polarkoordinaten
Gauss' level with a complex numberin cartesian and in polar coordinates

during itself the quantity <math> \ mathbb {R}< /math> the real numbers at number lines to illustrate, knows one leaves the quantity <math> \ mathbb {to C}< /math> the complex numbers as level (complex level, Gauss number level) illustrate. This corresponds “doubled nature” from <math> \ BbbC< /math> as two-dimensional real vector space. The subset of the real numbers forms therein the horizontal axle, the subset of the purely imaginary numbers (D. h. with real part) the senkrechte axle forms 0. A complex number <math> z = (A, b) = a+b \, \ mathrm {i}< /math> possesses thenthe horizontal coordinate <math> A< /math> and the vertical coordinate <math> b< /math>.

In accordance with definition the addition of complex numbers corresponds to the Vektoraddition. The multiplication is in the Gauss level a rotation straining, which after introduction of the polar representation further down will become clearer. Particularly in thatPhysics is understood frequent the geometrically descriptive level as the number complex number and given to the notation of the complex numbers the preference/advantage before the vector representation.

polar form and exponential form

each complex number <math> z=a+b \, \ mathrm {i}< /math> can z = in

<> the form mathr \ cdot e^ {\ mathrm {i} \ varphi} = r \ cdot (\ cos \ varphi + \ mathrm {i} \ cdot \ sin \ varphi)< /math>

, there math <A> = r are represented \ cdot \ cos \ varphi< /math> and <math> b = r \ cdot \ sin \ varphi< /math>.

  • The representation <math> r \ cdot (\ cos \ varphi + \ mathrm {i} \ cdot \ sin \ varphi)< /math>polar form or trigonometric form is called.
  • The representation <math> r \ cdot e^ {\ mathrm {i} \ varphi}< /math> with the help of the complex exponential function also exponential form is called.

The Euler identity are polar form and exponential form are able meaning. For the polar form there are also the alternative ways of writing

< math> r \ cdot \ operator name {cis} \, \ varphi= r \ \ operator name cdot {E} \, (\ varphi) =r \, (\ cos \ varphi + \ mathrm {i} \ cdot \ sin \ varphi)< /math>.

In the number complex number thereby math <r> /math< corresponds> the Euclidean vector length (D. h. the distance to the origin 0) and <math> \ varphi< /math> the angle of the number included with the real axle <math> z< /math>.

Usuallybecomes <math> r< /math> the amount or module of <math> z< /math> (Way of writing <math>|z|</math>) mentioned, <math> \ varphi< /math> becomes an argument (or also angle or phase) of <math> z< /math> called. There <math> \ varphi< /math> and <math> \ varphi+2 \ pi< /math> correspond to the same angle, is not not clear the polar representation first. Therefore one <math> \ varphi< /math>usually on the interval <math> [- \ pi; \ pi]< /math> and then 0 /math speak of <the argument> of math z \< neq>; the number <math> 0< /math> any argument could be assigned.

All values <math> e^ {\ mathrm {i} \ varphi}< /math> form the Einheitskreis of the complex numbers with the amount <math> 1< /math>.

complexes conjugation

Eine komplexe Zahl z = a+bi mit ihrer komplex konjugierten
a complex number of z = a+b i with their complex conjugated

turn one the sign of the imaginary part <math> b< /math> a complex number <math> z = a+b \, \ mathrm {i}< /math> over, one receives too <math> the z< /math> conjugates complex number <math> \ bar z=a-b \, \ mathrm {i}< /math> (sometimesalso <math> z^*< /math> written).

The conjugation <math> \ Bbb C \ tons \ Bbb C, \, z \ mapsto \ without z< /math> a body automorphism (involutorischer automorphism) is, since it with addition and multiplication is compatible, D. h. for all <math> y, z \ in \ Bbb C< /math> math

< \> overline applies {y+z} = \ bar y+ \ without z, \ quad \ overline {y \ cdot for z} = \ without y \ cdot \ without z< /math>.

In thatPolar representation has the complex conjugated number <math> \ without z< /math> with unchanged amount the straight negative angle of <math> z< /math>. One can identify the conjugation in the number complex number thus as the reflection at the real axle. In particular become exact under the conjugationthe real numbers itself shown.

The product of a complex number <math> z=a+b \ mathrm {i}< /math> with their complex conjugated one <math> \ without z< /math> the square of the amount results in:

<math> z \ cdot \ without z = (a+b \ mathrm {i}) (A-B \ mathrm {i}) = a^2 + b^2=|z|^2. </math>

conversion formulas

work on []

From the algebraic form to the polar form

for <math> z=a+b \ mathrm {i}< /math> in algebraic form is

<math> r = |z| = \ {a^2 sqrt + b^2}< /math> ;

for <math> z \ neq 0< /math> the argument is determined as follows:

<math> \ varphi = \ begin {cases} \ arctan \ frac {b} {A} & \ mathrm {f \ ddot ur} \ A> 0 \ \

\ arctan \ frac ba+ \ pi& \ mathrm {f \ ddot ur} \ A< 0, b \ geq0 \ \ \ arctan \ fracba \ pi& \ mathrm {f \ ddot ur} \ A< 0, b< 0 \ \ \ pi/2& \ mathrm {f \ ddot ur} \ a=0, b> 0 \ \ - \ to pi/2& \ mathrm {f \ ddot ur} \ a=0, b< 0 \ end {cases}< /math>

<math> {} = \ begin {cases} \ arccos \ frac ar& \ mathrm {f \ ddot ur} \ b \ geq0 \ \

\ arccos \ left (- \ frac acre \ right) - \ pi& \ mathrm {f \ ddot ur} \ b< 0 \ end {cases} </math>

The computation variant over the arc tangents needs its definitions by cases, there the special case <math> a=0< /math> specially to be treated and there the tangent must the same value twice in the interval <math> [0, 2 \ pi]< /math> assumes. The use of the arccos version gets along with fewer definitions by cases, since only the problem of the double angles is to be treated. The newer programming languages make however usually a ArcTan function available, those the valuedepending upon signs of A and b assigns to the suitable quadrant (frequently with name atan2).

from the polar form to the algebraic form

<math> A = r \ cdot \ cos \ varphi< /math>
<math> b = r \ cdot \ sin \ varphi< /math>

A places like further abovethe real part and b the imaginary part of that complex number.

multiplication and division in the polar form

with the multiplication are multiplied the amounts and the phases are added:

<math>
r \ cdot (\ cos \ varphi + \ mathrm {i} \ cdot \ sin \ varphi) \; \ cdot \; s\ cdot (\ cos \ psi + \ mathrm {i} \ cdot \ sin \ psi) = r \ cdot s \ cdot \ left [\ cos (\ varphi+ \ psi) + \ mathrm {i} \ cdot \ sin (\ varphi+ \ psi) \ right]

< /math> During the division the amount of the divisor is divided by the amount dividends, and the phasethe divisor from the phase dividends subtracts:

<math>
 \ frac {p \ cdot (\ cos \ phi + \ mathrm {i} \ cdot \ sin \ phi)}{r \ cdot (\ cos \ psi + \ mathrm {i} \ cdot \ sin \ psi)}= \ frac {p} {r} \ cdot \ left [\ cos (\ phi \ psi) + \ mathrm {i} \ cdot \ sin (\ phi \ psi) \ right]

< /math>

Multiplication in the exponential form

the amounts are multiplied here and the phases are added:

<math> (r \ cdot e^ {\ mathrm {i} \ varphi}) \ cdot (s \ cdot e^ {\ mathrm {i} \ psi}) = (r \ cdot s) \ e^ {\ mathrm {i cdot} (\ varphi+ \ psi)}</math>

roots

when counting on roots largest caution is appropriate, since the well-knownArithmetic rules for real numbers do not apply here. No matter, which the two possible values <math> \ mathrm i< /math> or <math> - \ mathrm i< /math> one for <math> \ sqrt {- 1}< /math> , receives one specifies z. B.

<math> 1 = \ sqrt {1} = \ sqrt {(- 1) \ cdot (- 1)} \ neq \ {- 1} \ cdot \ sqrt {- 1 sqrt} = -1.</math>

pragmatic arithmetic rules

can simplest be accomplished the computations as follows:

  • Addition and subtraction of complex numbers are accomplished (in the algebraic form) komponentenweise.
  • With the multiplication of complex numbers their amounts are multiplied and their arguments (angles) are added.
  • During the division of complex numbers their amounts are divided and their arguments (angles) are subtracted.
  • When strengthening complex numbers their amounts are strengthened and their arguments (angles) by the exponent are multiplied.
  • With extracting the root (draw root) complex numbers their amounts are extracted the rootand their arguments (angles) by the exponent divide. Thereby the first solution develops. With one <math> n< /math> - root develops math <n> /math< ten> Solutions, those in the angle of <math> 2 \ pi/n< /math> around the origin of the Gauss' level are distributed.See root (mathematics)

historical

The impossibility of the solution indicated above was already very early noticed and emphasized with the treatment of the quadratic equation, z. B. already in around 820 the n. Chr. wrote algebra of the Muhammed ibn Mûsâ Alchwârizmî. But with thatnearest and indisputable conclusion that this kind of equation is not solvable, one did not stop.

In certain sense the Italian Gerolamo Cardano (1501 - 1576) in its 1545 book Artis magnae sive de published is already regulisalgebraicis rather unus gone beyond it. There it treats the task to find two numbers whose product 40 and their sum 10 are. He emphasizes that the equation which can be set for it:

<math> x (10-x) =40< /math> or <math> x^2-10x+40=0< /math>

no solution has, adds however some remarksin addition, by it into the general solution of the quadratic equation

<math> x_ {1.2} = - \ frac {p} {2} \ pm \ sqrt {\ frac {p^2} {4} - q} </math>

for <math> p< /math> and <math> q< /math> the values (−10) and 40 uses. If it thus possible would sqrt be the arising expression

< math> \ {25-40}< /math> or<math> \ sqrt {- 15}< /math>

to give a sense, so that one might count on this indication after same rules, as with a real number, then the expressions

<math> 5 + \ sqrt {- 15}< /math> or <math> 5 - sqrt \ {- 15}< /math>

indeed a solutionrepresent.

For the square root from negative numbers and more generally for all from any real number <math> \ alpha< /math> and any real number <math> \ beta< /math> compound number

<math> \ alpha + \ sqrt {- \ beta}< /math> or <math> \ alpha - sqrt \ {- \ beta}< /math>

has itself since the center 17. Century the designation imaginary number in-patriated.

In contrast to it usual number the real numbers were called. Such a confrontation of the two terms is in the 1637 appeared Geómetrie of Descartes and emerges there probably for the first time.

Todaycalls one only the expression, which is formed by the root from a negative number, imaginary number and the quantity of numbers, formed of both kinds of numbers, as complex numbers. One can therefore say that Cardanoto first time in the today's sense on complex numbers and on it a set of views counted employed.

Since counting on these appeared senseless as „“outstanding numbers first as bare play, one was the more surprised that this „play “or permitted already well-known results a more satisfying form supplied very frequently valuable results to give. Thus Leonhard Euler came for example into its Introductio into analysin infinitorum to some remarkable equations, only the real numbers contained and without exception asproved correctly, which could not be so simply won however on other way.

Thus it came the fact that one these numbers not when paradoxically rejected, but ever more with them was occupied. Nevertheless this area of mathematics surrounded still somewhatMysterious, puzzling and unsatisfactory. Only by the paper Essai sur la répresentation analytique de la direction from the year 1797 of the Norwegian-Danish land measurer Caspar Wessels (1785-1818) was initiated the clearing-up over these numbers. This work, which it with the DanishAcademy submitted, found at the beginning of no attention. Similarly it was issued work of other mathematicians, so that these views had to be employed still several times.

As the first defined Augustin Louis Cauchy 1821 in its text book Cours a d'analyse function of complex variables into the number complex numberand proved many fundamental sets of the function theory.

General attentions found it only then, and Carl Friedrich Gauss in the year 1831 in an article in the Götting taught announcements the same views developed, obviously without knowledge of any predecessors.

Todaydo not make these things any conceptual or actual difficulties. By the simplicity the complex numbers are inferior to the definition, the meaning and applications in many Wissenschaftsgebieten, already described, to the real numbers in nothing. The term of the “imaginary” numbers, in the sense ofconceited and/or. unreal numbers, thus in the course of the centuries as inclined view proved.

application

the complex numbers in the physics

complex numbers play in basic physics a central role. Thus fits in particularthe mathematical structure of the quantum theory accurately for the structure of the number complex number, which is not to be excluded there. There she finds use with the definition of differential operators in the Schroedinger equation and the Klein-Gordon-Equation. For the Dirac-Gleichung one needs a number range extension thatcomplex numbers, the quaternions. Alternatively a formulation with Pauli stencils is possible, which exhibit however the same algebraic structure as the quaternions.

In relativity theory space and time are linked to a four-dimensional space-time. Substitutes one in addition the time t by means of x 4 = i ct by 4. Space coordinate x 4, then results a form of the laws of nature, in which these four coordinates arise structurally completely equally. Thus one receives M of this space-time in particular for the Metrik

M = x 1 2 + x 2 2 + x 3 2 + x 4 2,

those the same fundamental role for space-time plays as the spatial distance for the usual area. This substitution becomes of someAuthors in text books uses, which treat special relativity theory or contain sections concerning this. The expression x = (x, y, z, i ct) is called also Vierervektor. In the practice however a formulation became generally accepted with real Vierervektoren, embedded in oneFormalism, which leads directly to general relativity theory. Depending upon definition of the underlying Metrik different forms are used, z. B. x = (, x, y, z ct) or y = (x, y, z, ct). One assumes however the existence of additional hidden dimensions of space-time, over of themStill one speculates to number and structure, so that the value of the substitution is still open x 4 = i ct in the long run.

Beyond that mathematics of the complex numbers of those of the real numbers is clearly superior regarding elegance and compartmentation. Sois, in order to call only one example, each complex differentiable function automatically infinitely often differentiably, differently than in mathematics of the real numbers.

It was shown that complex numbers more deeply in nature and also in mathematicsare embodied, when one could suspect at present their discovery. The fundamental question seems to be nearly less, why the quantum theory fits so well the complex numbers, but why we during the physical description of our everyday life world actually so wellget along with the real numbers.

complex numbers in applied mathematics

complex numbers have an important role in physics and technology as computing aid. Thus in particular the treatment from differential equations can be simplified to Schwingungsvorgängen, there itselfthus the complicated relations in connection with products of sine and/or. Cosine functions by products of exponential functions to replace leave, whereby the exponents must be only added. Thus one adds arbitrary however suitable imaginary parts in addition for example in the complex alternating current calculation into thosereal initial equations, which one ignores during the evaluation of the results of computation then again. It concerns thereby only a computing trick without philosophical background.

In the fluid dynamics complex numbers are used, in order to explain and understand even streamline motions. Everyonearbitrary complex function of a complex argument always represents an even Potenzialströmung - the geometrical place corresponds to the complex argument in the Gauss number level, the Strömungspotenzial the real part of the function, and the streamlines the ISO lines of the imaginary part of the function alsoreverse sign. The vector field of the flow rate corresponds conjugates complex first derivative to the function. By experimenting with different overlays of parallel flow, sources, lowering, dipoles and eddies one can represent the flow past different outlines. To distort these flow patterns let themselves throughmapping - which complex argument replaced by a function of the complex argument. For example the flow past a circular cylinder (parallel flow + dipole + eddies) can be distorted into the flow past a wing-similar profile (Jukowski profile) and the role of thebasic eddy at an airplane bearing area study. This method for learning and understanding is so useful, for exact computation is not sufficient it generally not.

Important also the application of complex numbers is with the computation of improper real integrals in the framework of the Residue theorem of the function theory.

complex numbers in the abstract mathematics

an important area of application in the abstract mathematics is the analytic number theory. One uses that the whole and the rational numbers, the one of the main study objects thatNumber theory are, into which complex numbers lie. In such a way won liberty permits the application of analytic methods, those if necessary. Conclusions on the whole and rational numbers permit.

Furthermore the complex numbers supply the starting point for the so-called. complex geometry, D. h.the study of complex variousnesses. This area is already important for itself taken very. In addition statements of complex geometry often supply referring to connections in algebraic geometry, which studies very similar things.

used topics

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