Complex number

The complex numbers enhance the speed range of the real numbers such that the equation is solvable.

This is achieved by introducing a new number with the property. This number is called the imaginary unit. In the art of letter is used to prevent confusion with a dependent ( denoted by ) of the current time instead.

The origin of the theory of imaginary numbers, that is, all numbers whose square is a negative real number, dates back to the Italian mathematician Gerolamo Cardano and Rafael Bombelli in the 16th century. The introduction of the imaginary unit as a new number is attributed to Leonhard Euler.

Complex numbers can be represented in the form where and are real numbers and imaginary unit. In the complex numbers as shown can be the usual rules for computing with real numbers apply, which can be replaced constantly and vice versa. For the set of complex numbers, the symbol ( Unicode U 2102: ℂ ) will be used.

The so- constructed number field of complex numbers forms an extension field of real numbers and has a number of advantageous properties, which have been found in many areas of natural and engineering sciences to be extremely useful. One of the reasons for these positive characteristics is the algebraic closure of the complex numbers. This means that every algebraic equation of positive degree over the complex numbers has a solution, which is not true for real numbers. This property is the content of the Fundamental Theorem of Algebra. Another is a relationship between the exponential and trigonometric functions, which can be prepared by the complex numbers. Furthermore, each on an open set once complex differentiable function there is also infinitely differentiable - unlike in the Analysis of real numbers. The properties of functions with complex arguments are the subject of the theory of functions, also known as complex analysis.

5.1 amount
5.2 metric

6.1 Polar form
6.2 Complex conjugation
6.3 Conversion Formulas 6.3.1 From the algebraic form in the polar form 6.3.1.1 Calculation of the angle in the interval [0, 2π )

6.4.1 Trigonometric form
6.4.2 exponential

7.1 magnitude 7.1.1 Natural exponents
7.1.2 Any complex exponents

9.1 pairs of real numbers 9.1.1 First properties
9.1.2 with respect to the representation in the form a bi

Definition

The complex numbers can be used as numerical range in terms of a set of numbers for which the basic arithmetic operations addition, multiplication, subtraction and division are explained, defined with the following properties:

The real numbers are included in the complex numbers. That is, each real number is a complex number.
The associative law and the commutative law apply for the addition and multiplication of complex numbers.
The distributive law.
For each complex number, there exists a complex number, so.
For each non-zero complex number, there is a complex number, such that.
There exists a complex number with the property.
In all speed ranges with the features mentioned above, the complex numbers are minimal.

The last requirement is equivalent to saying that any complex number can be in the form ( or in abbreviated notation as well ) with real numbers and represent. The imaginary unit is not a real number. The existence of such a numerical range will be demonstrated in the section on the construction of the complex numbers.

Using the concepts of body and isomorphism can formulate it this way: There is minimal body that contain the field of real numbers and an element with the property. In such a body, each one and only one presentation element as complex numbers with real The isomorphic to each such body.

The coefficients are called the real and imaginary parts of. For two notations have been established:

Notation

The notation in the form is also called (after René Descartes named ) Cartesian or algebraic form. The term Cartesian is explained by the representation in the complex number plane or Gaussian (see below). It is also the representation; General in the standard DIN 1302:1999 mathematical symbols and concepts she comes, however, not before.

In electrical engineering, the little i is already used for time-varying currents ( see alternating current) and can lead to confusion with the imaginary unit i. It can therefore be used in this range according to DIN 1302, the letter j.

In physics, a distinction is made between the current in alternating current and for the imaginary unit. This does not go through the very clear separation in the attentive reader to confusion and is widely used in this form in both the physical and experimental as well as in the physical- theoretical literature; handwritten this delicacy is not to be respected. See also: Complex AC circuit analysis

Complex numbers can be represented in accordance with DIN 1304-1 and DIN 5483-3 underlined to distinguish them from real numbers.

Computing in algebraic form

Addition

For the addition of two complex numbers and is

Subtraction

Analog for addition (see above ) also works subtraction

Multiplication

For multiplication applies accordingly

This formula arises with the definition by simply multiplying out and regroup.

Division

The quotient of two complex numbers and can be calculated by expanding the break with the complex conjugate of the denominator. The denominator becomes real ( and is just the square of the modulus of ):

Worked examples

Addition:

Subtraction:

Multiplication:

Division:

Other properties

The body of the complex numbers is a part of an upper body, on the other hand, a two-dimensional vector space. Isomorphism is also referred to as a natural identification.
The field extension of degree; precisely is isomorphic to the factor ring, the minimal polynomial of over is. Furthermore, already forms the algebraic degree of.
As a vector space has the base. In addition, as every body is also a vector space over itself, ie a one-dimensional vector space with basis.
And are exactly the solutions of the quadratic equation. In this sense, can be thought of as " root " (but also ).
In contrast to no secondary body, ie, there are no acceptable to the body structure relative to linear order. Of two different complex numbers, one can not determine appropriate (based on the addition and multiplication ) so that of the two is greater or smaller number.

Magnitude and Metrics

Amount

The magnitude of a complex number is the length of its vector in the Gaussian plane and can be, for example,

Calculated from its real part and imaginary part. Than a length, the amount is real and non-negative.

Example:

Metrics

Induced by the distance function metric provides the complex vector space with its standard topology. It is consistent with the topology of the product as to the restriction of the standard metric matches.

Both rooms are fully as their under these metrics. Both rooms can be the topological concept of continuity extend to analytic terms such as differentiation and integration.

Complex number plane

While the set of real numbers can be illustrated by points on a number line, one can represent the set of complex numbers as points in a plane ( complex plane, gaussian number plane ). This corresponds to the " dual nature " of real is a two-dimensional vector space. The subset of the real numbers is formed in the horizontal axis, the portion of the purely imaginary numbers (i.e., the real part 0 ) is the vertical axis. A complex number then has the horizontal coordinate and the vertical coordinate.

As defined by the addition of complex numbers corresponding to the vector addition, where one identifies the points in the complex plane and their position vectors. Multiplication is the Gaussian plane, a rotation and expansion, which will become clearer below after the introduction of the polar form. Especially in physics geometrically intuitive level is often referred to as the complex plane considered and, where the notation of complex numbers in polar form in preference to the vector representation.

Polar form

If, instead of the Cartesian coordinates and polar coordinates, can be the complex number in the following, based on the Euler's relation called polar form

Represent the results from and. The presentation with the help of the complex exponential function is called while also exponential ( polar form), the representation by means of the expression trigonometric representation ( polar form). Because of Euler's relation both representations are equivalent and equally significant alternative spellings of the polar form. Furthermore, there is for them, especially in the practice, which further shortened spellings

Where is the sum and the mold is indicated by the operator as the angle Versordarstellung.

In the complex plane this corresponds to the Euclidean vector length (ie the distance from the origin 0) and the included angle with the real axis in the number. Usually, however, it is called here, the amount of (or its module ) (case ) and the angle of the argument (or the phase ) of ( writing ).

Because, while the same number can be assigned, the polar representation is not immediately clear. Therefore you usually restricted to the interval, ie, in order then to speak instead of the argument itself from its main value. The number, however, is any argument could assign, and for the purpose of a clear representation you can actually in this case it set to 0.

The argument is also the imaginary part of the complex natural logarithm

By choosing a region defined at all branch of the logarithm is therefore also an argument function determines (and vice versa).

All values form the unit circle of the complex numbers with the amount that these numbers are also called unimodular and form the circle group.

The fact that the multiplication of complex numbers (except zero) corresponds to rotation dilations can be mathematically expressed as follows: The multiplicative group of complex numbers without the zero can not equal as a direct product of the group of rotations, the circle group, and the dilations by a factor zero of the multiplicative group conceive. The former group can be parameterized by the argument, temptation just corresponds to the amounts.

Complex conjugation

If you turn the sign of the imaginary part of a complex number in order, we obtain the complex conjugate to (sometimes written ).

Conjugation is a ( involutory ) Körperautomorphismus as it is compatible with addition and multiplication, i.e. applies to all

In the polar representation of the complex conjugate has an unchanged amount just the negative angle of Man, the conjugation in the complex plane that is identified as the reflection in the real axis. In particular, the real numbers are exactly mapped to itself under conjugation.

The product of a complex number, and its complex conjugate gives the square of the sum:

The complex numbers thus form a trivial example of a C * - algebra.

The sum of a complex number and its complex conjugate yields the 2- times its real part:

The difference between a complex number and its complex conjugate yields the times its imaginary part:

Conversion formulas

From the algebraic form in the polar form

For in algebraic form

For the argument can be defined as 0, but remains mostly undefined. For the argument in the interval using the arc cosine or arc tangent of the can by

Be determined. Slightly more complicated ( as the case must be treated separately and the tangent of its range by running twice in the interval ) is the calculation variant

For everyone. Many programming languages and spreadsheets provide a variant of arctangent function is available ( often called atan2 called ) that gets passed two values and the result depending on the sign of and the appropriate quadrant maps.

Calculation of the angle in the interval [0, 2π )

The calculation of the angle in the interval may be carried out in principle in such a way that the angle is calculated first as described above in the interval, and then to be increased if it is negative:

(see polar coordinates).

From the polar form in the algebraic form

As above, A is the real part and the imaginary part of those complex numbers b dar.

Multiplication, division and addition in the polar form

For multiplication in polar form, the amounts are multiplied and the phases added. The division of the sum of the dividend is divided by the sum of the divisor and the divisor is subtracted, the phase of the phase of the dividend. For the addition of a slightly more complicated formula exists:

Trigonometric form

Exponential

With and above.

Arithmetic operations 3rd stage

The arithmetic operations of the third stage include exponentiation, square root ( square root ) and logarithms.

Potencies

Natural exponent

For natural numbers, the -th power calculated in the polar form to

Or for the algebraic form by using the binomial theorem

Any complex exponents

The general definition of a power base with complex and complex exponent is

Where stands for the principal value of the complex logarithm (see below) so that the formula also provides a major value. In the case, however, agree or all possible results are consistent with this principal value and the function is unique.

Root

In computing with roots of the known calculation rules for non-negative real numbers do not apply. No matter which of the two possible values for one or determines is obtained, for example,

To calculate the - th roots of the complex number is the formula

Wherein the values of passes. So a number has complex -th roots. This is a root term in ambiguous.

Logarithms

The complex natural logarithm (other than the real) is not unique. A complex number w is called the logarithm of the complex number z, if

W, any number of any one of, for example, a logarithm is thus works with the main values , i.e., values of a particular strip of the complex plane.

The mean value of the natural logarithm of the complex number

With and

In other words: The mean value of the natural logarithm of the complex number z

Where the principal value of the argument of z.

Pragmatic calculation rules

The easiest way to perform calculations as follows:

Addition and subtraction of complex numbers are performed ( in algebraic form) componentwise.
The multiplication of complex numbers can be performed depending on the setting advantageous in algebraic form or in exponential form ( multiplying the respective amounts and adding the arguments ( angle) ).
In the division of complex numbers in exponential form their amounts are divided and their arguments (angle) is subtracted or added in algebraic form the quotient by the conjugate denominator.
When Raising a complex number with a real exponent its magnitude is potentiated and its argument (angle) multiplied by the exponent; the use of the algebraic form (with Newton's binomial theorem ) is in most cases complicated (particularly for high powers ).
In the square root ( square root ) of a complex number with a real exponent square root their amount and their argument (angle ) divided by the exponent. This gives rise to the first solution. In a - th root yield solutions which are distributed in the angle of the origin of the Gaussian plane. See Root ( mathematics). A square root can also be quite easily calculated in Cartesian form.

Construction of the complex numbers

In this section it is shown that in fact a field of complex numbers exists that satisfies the required properties in the above definition. Various designs are possible, but lead to the same body, up to isomorphism.

Pairs of real numbers

The first construction assumes no respect to the imaginary unit: In 2- dimensional real vector space of ordered pairs of real numbers is next to the addition

(which is the ordinary vector addition ) a multiplication by

Defined.

According to this definition, to write, and to a body, the body of the complex numbers.

First properties

The figure is a body of embedding in the basis of which we identify the real number of the complex number.

The summation is:

The number of the zero element in and
The number in the inverse element.

With respect to the multiplication:

The number of the neutral element ( the identity element ) and of
The inverse ( reciprocal ) is too.

Relation to representation in the form a bi

Through the imaginary unit is defined; applies to this.

Each complex number has the unique representation of the form

With; This is the usual notation for the complex numbers.

Polynomials: adjunction

Another construction of the complex numbers is the factor ring

The polynomial ring in one variable over the reals. The number i corresponds to the image of the indeterminate, the real numbers can be identified with the constant polynomials.

This design principle is also applicable in other contexts, this is called adjunction.

Matrices

The set of matrices of the form

Also forms a model of the complex numbers. The real or imaginary component unit represented by the unit matrix or the matrix. Therefore:

This quantity is a subspace of the vector space of real matrices.

Real numbers correspond to diagonal matrices

Belonging to the matrices of linear maps are provided and are not both zero rotation dilations in the room. It is exactly the same rotation dilations as in the interpretation of multiplication by a complex number in the Gaussian number plane.

History

The impossibility of the above solution has been noted and highlighted very early in the treatment of the quadratic equation, for example, already in the written order of the algebra 820 AD Muhammad ibn Musa Alchwârizmî. But at the nearest and indisputable conclusion that this type of equation is not solvable, it did not stop.

In a sense, already Gerolamo Cardano of Italians (1501-1576) went out in his 1545 book published Artis magnae sive de regulis algebraicis liber unus about it. He treated there, the task is to find two numbers whose product is 40 and whose sum is 10. He emphasizes that the applicable stage of this equation:

Has no solution, but adds some remarks added by the general solution of the quadratic equation

For and the values (-10 ) and 40 starts. Thus, if it were possible, the resulting expression

To make sense, and in a way that one might expect this character to the same rules as with a real number, so the terms would

Be a solution indeed.

For the square root of negative numbers and more generally for all of an arbitrary real number and a positive real number composite numbers

Has come to be the term imaginary number since the mid-17th century.

In contrast, known as the ordinary number of real numbers. Such a juxtaposition of the two terms can be found in 1637 published La Géométrie of Descartes and dive there probably for the first time.

Today, people call only the expression that is formed by the root of a negative number, as imaginary number and the amount of numbers formed by two kinds of numbers as complex numbers. One can therefore say that Cardano was expecting for the first time in the modern sense with complex numbers and thus has established a number of considerations.

Since the computation with these as " pointless " respected figures first appeared as a mere game, one was more surprised that this " game" very often yielded valuable results already known results or to give a more satisfactory form allowed. So Leonhard Euler in his Introductio came, for example, in analysin infinitorum to some remarkable equations containing only real numbers and, without exception, proved to be correct, but in other ways could not be so easily won.

So it happened that you did not reject these figures as absurd but dealt more and more with them. Nevertheless, this area of mathematics surrounded still something mysterious, puzzling and unsatisfactory. Only through the treatise Essai sur la représentation de la Analytique direction from 1797 the Norwegian- Danish land surveyor Caspar Wessel ( 1745-1818 ), the Enlightenment was initiated on these numbers. This work, which he submitted to the Danish Academy, initially took no heed. A similar fate befell the works of other mathematicians, so that these considerations had to be made several times yet.

As first defined Augustin- Louis Cauchy 1821 in his textbook Cours d'analyze a function of a complex variable in the complex plane and proved many basic sets of function theory.

General attention they found only, and Carl Friedrich Gauss in an article in the goddess 's scholarly ads the same ideas developed in 1831, apparently without the knowledge of any predecessors.

Today, these things make no conceptual or practical difficulties. Due to the simplicity of the definition, the previously discussed importance and applications in many fields of science, the complex numbers are according to the real numbers in nothing. The concept of " imaginary " numbers, in the sense of imaginary or unreal numbers, thus has proven over the centuries as an oblique view.

Importance

Complex numbers in physics

Complex numbers play a central role in fundamental physics. In quantum mechanics, the state of a physical system is considered as an element of a ( projective ) Hilbert space over the complex numbers. Complex numbers are used in the definition of differential operators in the Schrödinger equation and the Klein-Gordon equation. For the Dirac equation requires a number range extension of the complex numbers, the quaternions. Alternatively, a formulation with Pauli matrices is possible, but have the same structure as the algebraic quaternions.

Complex numbers have an important role as a calculation tool in physics and engineering. This allows in particular to simplify vibration operations because thus be replaced by products of exponentials the complicated relationships in the context of products of sine and cosine functions, only the exponent must be added the treatment of differential equations. To add not more than, for example, in the complex AC circuit analysis suitable imaginary in the real output equations, which are then ignored in the analysis of the calculation results again. This harmonic oscillations ( real) are complemented to circular motion in the complex plane in the intermediate calculation that have more symmetry and are therefore easier to treat.

In optics, refractive and absorptive effects of a substance in a complex, wavelength dependent permittivity ( dielectric constant ) or the complex index of refraction are combined, which in turn is fed back to the electric susceptibility.

In fluid dynamics, complex numbers are used to explain and understand planar potential flows. Any complex function of a complex argument always represents a plane potential flow - the locus corresponds to the complex argument in the Gaussian number plane, the flow potential of the real part of the function, and the streamlines the contours of the imaginary part of the function with the opposite sign. The vector field of the flow velocity corresponds to the complex conjugate of the first derivative of the function. By experimenting with various overlays of parallel flow, sources, sinks, dipoles and vortices can represent the flow around different contours. Distort let this flow images by conformal mapping - the complex argument is replaced by a function of the complex argument. For example, can the flow around a circular cylinder ( parallel flow dipole eddy ) in the flow around an airfoil -like profile ( Joukowski profile ) distort and study the role of the supporting vertebra of an aircraft wing. How useful is this method of learning and understanding, for accurate calculation it is not sufficient in general.

Complex numbers in electrical engineering

In electrical engineering, the presentation of electrical quantities with the help of complex numbers has wide circulation. It is used in the calculation of time- sinusoidal varying sizes such as electric and magnetic fields. In the representation of a sinusoidal alternating voltage as a complex quantity and corresponding representations of resistors, capacitors and inductors, the calculation of the electric current, the active and reactive power in a circuit easier. The given by differential quotients or integrals coupling goes into a coupling by trigonometric functions; the calculation of the correlations can be so much easier. Also the interaction of several different sinusoidal voltages and currents that can at different times have their zero crossings can easily be represented in a complex bill. More details on this topic is available in the article about the complex AC circuit analysis.

In recent years, the digital signal processing has gained greatly in importance, whose foundation is the complex number calculation.

Body theory and algebraic geometry

The field of complex numbers is the algebraic closure of the field of real numbers.

The two algebraically closed fields with the same characteristics and the same transcendence degree over its prime field (which is determined by the characteristic) are ( ring theory) isomorphic. When a body of characteristic 0 with überabzählbarem transcendence degree of this is equal to the cardinality of the body. Body theory, form the complex numbers so the only algebraically closed field with characteristic 0 and the cardinality of the continuum. A construction of the body of the complex numbers is using this statement purely algebraically as an extension of the algebraic completion of the rational numbers to continuum many transcendental elements. Another construction provides an ultra Product: For this we form for each finite field its algebraic degree and form of them the Ultra product in terms of any free ultrafilter. It follows from the set of Łoś that this Ultra product an algebraically closed field with characteristic 0 is the cardinality of the continuum follows from set-theoretic considerations.

Under the slogan Lefschetz principle different sets are combined, which allow results of algebraic geometry that are proven over the complex numbers, to other algebraically closed field with characteristic 0 to transfer (which largely on the completeness of the theory of algebraically closed fields builds with characteristic 0 ). If the observation of the complex has the advantage that there topological and analytical methods can be used to obtain algebraic results. Above Ultra product design allows the transmission of results in the case of characteristic not equal to 0 on the complex numbers.

Spectral Theory and Functional Analysis

Many results of the spectral theory applies to complex vector spaces to a greater extent than for real. So take, for example, complex numbers as eigenvalues of real matrices ( then in each case together with the complex conjugate eigenvalue ). This can be explained by the fact that the characteristic polynomial of the matrix due to the algebraic closedness of over the complex numbers always splits into linear factors. On the other hand, there are real matrices with no real eigenvalues , while the spectrum of any bounded operator on a complex ( at least one-dimensional ) Banach space is never empty. In the spectral theory of Hilbert spaces can be rates that apply in the real case only for self-adjoint operators, often transferred to normal operators in the complex case.

Play in other parts of functional analysis the complex numbers a special role. For example, the theory of C *-algebras is usually operated in the complex domain, the harmonic analysis is concerned with representations of groups on complex Hilbert spaces.

Function theory and complex geometry

The study of differentiable functions on subsets of the complex numbers is the subject of the theory of functions. It is more rigid than the real analysis in many ways and leaves less pathologies. Examples include the statement that every differentiable in an area function is already infinitely differentiable, or the identity theorem for holomorphic functions.

The function theory often allows conclusions to be purely real statements, for example, can be some integrals using the residue theorem to calculate. An important application of these methods is the analytic number theory, which returns integers statements about statements about complex functions frequently. , In the form of Dirichlet series A prominent example is the connection between prime number theorem and Riemannian ζ function. In this context, the Riemann conjecture plays a central role.

The above-mentioned rigidity of holomorphic functions appear even more on global issues in appearance, that is, the study of complex manifolds. So there on a compact complex manifold no nonconstant global holomorphic functions; Statements such as the embedding theorem of Whitney are in the complex so wrong. This so-called " analytic geometry " (not to be confused with the classical analytic geometry of René Descartes! ) Is also closely related to algebraic geometry, many results can be transferred. The complex numbers are sufficiently large to capture the complexity of the algebraic varieties over arbitrary characteristic 0 ( Lefschetz principle) in a suitable way.