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 numbe=
rs and their operations, and conversely expressing in terms of complex numb=
ers some geometric properties and constructions. For example, the real numb=
ers form the real line which is identified to the horizontal axis of the co=
mplex 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, a=
nd the multiplication by a complex number is a similarity centered at the o=
rigin. The complex conjugation is the reflection symmetry with respect to t=
he real axis. The complex absolute value is a Euclidean norm.

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 pa=
rt does not include a factor i; that is, the imaginary part is b, not bi.[4=
][5]

complex number system
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In some disciplines, particularly in electromagnetism and electrical engine=
ering, 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 i=
mmediately 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 po=
sition vector from the origin to this point. The coordinate values of a com=
plex number z can hence be expressed in its Cartesian, rectangular, or alge=
braic form.

An alternative option for coordinates in the complex plane is the polar coo=
rdinate system that uses the distance of the point z from the origin (O), a=
nd 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 =3D x + =
yi is[11] r =3D | z | =3D x 2 + y 2 . z If z is a real number (that is, if =
y =3D 0), then r =3D |x|. That is, the absolute value of a real number equa=
ls its absolute value as a complex number.

When visualizing complex functions, both a complex input and output are nee=
ded. Because each complex number is represented in two dimensions, visually=
 graphing a complex function would require the perception of a four dimensi=
onal space, which is possible only in projections. Because of this, other w=
ays of visualizing complex functions have been designed.



Work on the problem of general polynomials ultimately led to the fundamenta=
l theorem of algebra, which shows that with complex numbers, a solution exi=
sts to every polynomial equation of degree one or higher. Complex numbers t=
hus 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 com=
plex numbers were developed by the Italian mathematician Rafael Bombelli.[1=
8] A more abstract formalism for the complex numbers was further developed =
by the Irish mathematician William Rowan Hamilton, who extended this abstra=
ction to the theory of quaternions.[19]

The impetus to study complex numbers as a topic in itself first arose in th=
e 16th century when algebraic solutions for the roots of cubic and quartic =
polynomials were discovered by Italian mathematicians (see Niccol=C3=B2 Fon=
tana Tartaglia, Gerolamo Cardano). It was soon realized (but proved much la=
ter)[21] that these formulas, even if one were interested only in real solu=
tions, sometimes required the manipulation of square roots of negative numb=
ers. As an example, Tartaglia's formula for a cubic equation of the form x3=
 =3D px + q[c] gives the solution to the equation x3 =3D x as

In the 18th century complex numbers gained wider use, as it was noticed tha=
t formal manipulation of complex expressions could be used to simplify calc=
ulations involving trigonometric functions. For instance, in 1730 Abraham d=
e Moivre noted that the identities relating trigonometric functions of an i=
nteger multiple of an angle to powers of trigonometric functions of that an=
gle could be re-expressed by the following de Moivre's formula:

In the beginning of the 19th century, other mathematicians discovered indep=
endently the geometrical representation of the complex numbers: Bu=C3=A9e,[=
30][31] Mourey,[32] Warren,[33][34][35] Fran=C3=A7ais and his brother, Bell=
avitis.[36][37]

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

Later classical writers on the general theory include Richard Dedekind, Ott=
o H=C3=B6lder, Felix Klein, Henri Poincar=C3=A9, Hermann Schwarz, Karl Weie=
rstrass and many others. Important work (including a systematization) in co=
mplex multivariate calculus has been started at beginning of the 20th centu=
ry. Important results have been achieved by Wilhelm Wirtinger in 1927.

Unlike the real numbers, there is no natural ordering of the complex number=
s. In particular, there is no linear ordering on the complex numbers that i=
s compatible with addition and multiplication. Hence, the complex numbers d=
o not have the structure of an ordered field. One explanation for this is t=
hat every non-trivial sum of squares in an ordered field is nonzero, and i2=
 + 12 =3D 0 is a non-trivial sum of squares. Thus, complex numbers are natu=
rally 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 numerat=
or and denominator of the fraction by the conjugate of the given denominato=
r. This process is sometimes called "rationalization" of the denominator (a=
lthough the denominator in the final expression might be an irrational real=
 number), because it resembles the method to remove roots from simple expre=
ssions in a denominator.

Conjugation is also employed in inversive geometry, a branch of geometry st=
udying reflections more general than ones about a line. In the network anal=
ysis of electrical circuits, the complex conjugate is used in finding the e=
quivalent impedance when the maximum power transfer theorem is looked for.

Using the visualization of complex numbers in the complex plane, addition h=
as the following geometric interpretation: the sum of two complex numbers a=
 and b, interpreted as points in the complex plane, is the point obtained b=
y 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). Equivalentl=
y, calling these points A, B, respectively and the fourth point of the para=
llelogram X the triangles OAB and XBA are congruent.

It seems natural to extend this formula to complex values of x, but there a=
re 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 fiel=
d of complex numbers, the topic's name is usually modified to reflect that =
fact. For example: complex analysis, complex matrix, complex polynomial, an=
d complex Lie algebra.

There are various proofs of this theorem, by either analytic methods such a=
s 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 satis=
fies these three properties, so these two fields are isomorphic (as fields,=
 but not as topological fields).[51] Also, C \displaystyle \mathbb C  is is=
omorphic to the field of complex Puiseux series. However, specifying an iso=
morphism requires the axiom of choice. Another consequence of this algebrai=
c 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 analy=
sis and topology, are not dealt with. The following description of C \displ=
aystyle \mathbb C  as a topological field (that is, a field that is equippe=
d with a topology, which allows the notion of convergence) does take into a=
ccount the topological properties. C \displaystyle \mathbb C  contains a su=
bset P (namely the set of positive real numbers) of nonzero elements satisf=
ying the following three conditions:

The only connected locally compact topological fields are R \displaystyle \=
mathbb R  and C . \displaystyle \mathbb C . This gives another characteriza=
tion of C \displaystyle \mathbb C  as a topological field, since C \display=
style \mathbb C  can be distinguished from R \displaystyle \mathbb R  becau=
se the nonzero complex numbers are connected, while the nonzero real number=
s are not.[52]
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