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## Complex number
The The sum and product of two complex numbers are: - (
*a*+*ib*) + (*c*+*id*) = (*a*+*c*) +*i*(*b*+*d*) - (
*a*+*ib*) · (*c*+*id*) =*ac*-*bd*+*i*(*bc*+*ad*)
## HistoryThe earliest fleeting reference to square roots of negative numbers occurred in the work of the Greek mathematician and inventor Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid. They became more prominent when in the 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians (see Tartaglia, Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. This was doubly unsettling since not even negative numbers were considered to be on firm ground at the time. The term "imaginary" for these quantities was coined by René Descartes in the 17th century and was meant to be derogatory. The existence of complex numbers was not completely accepted until the geometrical interpretation (see below) had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss. The formally correct definition using pairs of real numbers was given in the 19th century. ## Definition
Formally we may define complex numbers as ordered pairs of real numbers ( - (
*a*,*b*) + (*c*,*d*) = (*a*+*c*,*b*+*d*) - (
*a*,*b*) · (*c*,*d*) = (*ac*-*bd*,*bc*+*ad*)
C (or in blackboard bold).
We identify the real number
## Geometry
A complex number can also be viewed as a point or a position vector on the two dimensional Cartesian coordinate system. This representation is sometimes called an -
*z*=*x*+*iy*=*r*(cos φ +*i*sin φ).
r cis φ, where r is called the absolute value of z and φ is called the complex argument of z.
However, Euler's formula states that e^{i φ} = cisφ. The exponential form gives us a better insight then the shorthand rcisφ, which is almost never used in serious mathematical articles.
By simple trigonometric identities,
we see that
Multiplication with ## Absolute value, conjugation and distance
Recall that the One can check readily that the absolute value has three important properties: - |
*z*+*w*| ≤ |*z*| + |*w*| - |
*z w*| = |*z*| |*w*| - |
*z / w*| = |*z*| / |*w*|
z and w. By defining the distance function d(z, w) = |z - w| we turn the complex numbers into a metric space and we can therefore talk about limits and continuity. The addition, subtraction, multiplication and division of complex numbers are then continuous operations. Unless anything else is said, this is always the metric being used on the complex numbers.
The complex conjugate of the complex number z about the real axis. The following can be checked:
- if and only if
*z*is real - if
*z*is non-zero
The ## Matrix representation of complex numbersWhile usually not useful, alternative representations of complex field can give some insight into their nature. One particularly elegant representation interprets every complex number as 2x2 matrix with real entries which stretches and rotates the points of the plane. Every such matrix has the form a and b. The sum and product of two such matrices is again of this form. Every non-zero such matrix is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a field. In fact, this is exactly the field of complex numbers. Every such matrix can be written as
i with
The absolute value of a complex number expressed as a matrix is equal to the square root of the determinant of that matrix. If the matrix is viewed as a transformation of a plane, then the transformation rotates points through an angle equal to the argument of the complex number and scales by a factor equal to the complex number's absolute value. The conjugate of the complex number ## Some properties## Real vector space
## Solutions of polynomial equations
A
Indeed, the complex number field is the algebraic closure of the real number field. It can be identified as the quotient ring of the polynomial ring **C**=**R**[*X*] / (*X*^{2}+ 1).
X^{2} + 1 is irreducible. The image of X in this quotient ring becomes the imaginary unit i.## Complex analysisThe study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example). Unlike real functions which are commonly represented as two dimensional graphs, complex functions have four dimensional graphs and may usefully be illustrated by color coding a three dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane. ## Applications
Complex numbers are used in signal analysis and other
fields
as a convenient description for periodically varying
signals. The absolute value | If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as the real part of complex valued functions of the form *f*(*t*) =*z**e*^{iωt}
z encodes the phase and amplitude as explained above.
In electrical engineering, this is done for varying voltages and currentss. The treatment of
resistors, capacitors and
inductors can then be unified by introducing imaginary
frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance.
(Electrical engineers and some physicists use the letter The residue theorem of complex analysis is often used in applied fields to compute certain improper integrals.
The complex number field is also of utmost importance in quantum mechanics
since the underlying theory is built on (infinite dimensional) Hilbert spaces over In Special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary.
In differential equations, it is common to
first find all complex roots In fluid dynamics, complex functions are used to describe potential flow in 2d. ## See alsoquaternions, complex geometry, local fields, phasors, Leonhard Euler, the most remarkable formula in the world, Hypercomplex number, De Moivre's formula, Complex numbers at Wikibooks ## Further Reading | |||||

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