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Complex Number System


This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the real line which is identified to the horizontal axis of the complex plane. The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with respect to the real axis. The complex absolute value is a Euclidean norm.




complex number system



The real number a is called the real part of the complex number a + bi; the real number b is called its imaginary part. To emphasize, the imaginary part does not include a factor i; that is, the imaginary part is b, not bi.[4][5]


In some disciplines, particularly in electromagnetism and electrical engineering, j is used instead of i as i is frequently used to represent electric current.[8] In these cases, complex numbers are written as a + bj, or a + jb.


The definition of the complex numbers involving two arbitrary real values immediately suggests the use of Cartesian coordinates in the complex plane. The horizontal (real) axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical (imaginary) axis, with increasing values upwards.


A charted number may be viewed either as the coordinatized point or as a position vector from the origin to this point. The coordinate values of a complex number z can hence be expressed in its Cartesian, rectangular, or algebraic form.


An alternative option for coordinates in the complex plane is the polar coordinate system that uses the distance of the point z from the origin (O), and the angle subtended between the positive real axis and the line segment Oz in a counterclockwise sense. This leads to the polar form


The absolute value (or modulus or magnitude) of a complex number z = x + yi is[11] r = z = x 2 + y 2 . =\sqrt x^2+y^2. If z is a real number (that is, if y = 0), then r = x. That is, the absolute value of a real number equals its absolute value as a complex number.


When visualizing complex functions, both a complex input and output are needed. Because each complex number is represented in two dimensions, visually graphing a complex function would require the perception of a four dimensional space, which is possible only in projections. Because of this, other ways of visualizing complex functions have been designed.


Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.


Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician Rafael Bombelli.[18] A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.[19]


The impetus to study complex numbers as a topic in itself first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolò Fontana Tartaglia, Gerolamo Cardano). It was soon realized (but proved much later)[21] that these formulas, even if one were interested only in real solutions, sometimes required the manipulation of square roots of negative numbers. As an example, Tartaglia's formula for a cubic equation of the form x3 = px + q[c] gives the solution to the equation x3 = x as


In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be re-expressed by the following de Moivre's formula:


In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée,[30][31] Mourey,[32] Warren,[33][34][35] Français and his brother, Bellavitis.[36][37]


The English mathematician G.H. Hardy remarked that Gauss was the first mathematician to use complex numbers in 'a really confident and scientific way' although mathematicians such as Norwegian Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.[38]


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.


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 is nonzero, and i2 + 12 = 0 is a non-trivial sum of squares. Thus, complex numbers are naturally thought of as existing on a two-dimensional plane.


This property can be used to convert a fraction with a complex denominator to an equivalent fraction with a real denominator by expanding both numerator and denominator of the fraction by the conjugate of the given denominator. This process is sometimes called "rationalization" of the denominator (although the denominator in the final expression might be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.


Conjugation is also employed in inversive geometry, a branch of geometry studying reflections more general than ones about a line. In the network analysis of electrical circuits, the complex conjugate is used in finding the equivalent impedance when the maximum power transfer theorem is looked for.


Using the visualization of complex numbers in the complex plane, addition has the following geometric interpretation: the sum of two complex numbers a and b, interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices O, and the points of the arrows labeled a and b (provided that they are not on a line). Equivalently, calling these points A, B, respectively and the fourth point of the parallelogram X the triangles OAB and XBA are congruent.


It seems natural to extend this formula to complex values of x, but there are some difficulties resulting from the fact that the complex logarithm is not really a function, but a multivalued function.


When the underlying field for a mathematical topic or construct is the field of complex numbers, the topic's name is usually modified to reflect that fact. For example: complex analysis, complex matrix, complex polynomial, and complex Lie algebra.


There are various proofs of this theorem, by either analytic methods such as Liouville's theorem, or 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 C . \displaystyle \mathbb C . For example, any non-empty complex square matrix has at least one (complex) eigenvalue.


It can be shown that any field having these properties is isomorphic (as a field) to C . \displaystyle \mathbb C . For example, the algebraic closure of the field Q p \displaystyle \mathbb Q _p of the p-adic number also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields).[51] Also, C \displaystyle \mathbb C 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 C \displaystyle \mathbb C contains many proper subfields that are isomorphic to C \displaystyle \mathbb C .


The preceding characterization of C \displaystyle \mathbb C describes only the algebraic aspects of C . \displaystyle \mathbb C . That is to say, the properties of nearness and continuity, which matter in areas such as analysis and topology, are not dealt with. The following description of C \displaystyle \mathbb C as a topological field (that is, a field that is equipped with a topology, which allows the notion of convergence) does take into account the topological properties. C \displaystyle \mathbb C contains a subset P (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:


The only connected locally compact topological fields are R \displaystyle \mathbb R and C . \displaystyle \mathbb C . This gives another characterization of C \displaystyle \mathbb C as a topological field, since C \displaystyle \mathbb C can be distinguished from R \displaystyle \mathbb R because the nonzero complex numbers are connected, while the nonzero real numbers are not.[52] 350c69d7ab


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