complex number

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complex number:

see numbernumber,
entity describing the magnitude or position of a mathematical object or extensions of these concepts. The Natural Numbers

Cardinal numbers describe the size of a collection of objects; two such collections have the same (cardinal) number of objects if their
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complex number

[′käm‚pleks ′nəm·bər]
Any number of the form a + bi, where a and b are real numbers, and i 2= -1.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

complex number

any number of the form a + ib, where a and b are real numbers and i = &#221A--1
Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005

complex number

A number of the form x+iy where i is the square root of -1, and x and y are real numbers, known as the "real" and "imaginary" part. Complex numbers can be plotted as points on a two-dimensional plane, known as an Argand diagram, where x and y are the Cartesian coordinates.

An alternative, polar notation, expresses a complex number as (r e^it) where e is the base of natural logarithms, and r and t are real numbers, known as the magnitude and phase. The two forms are related:

r e^it = r cos(t) + i r sin(t) = x + i y where x = r cos(t) y = r sin(t)

All solutions of any polynomial equation can be expressed as complex numbers. This is the so-called Fundamental Theorem of Algebra, first proved by Cauchy.

Complex numbers are useful in many fields of physics, such as electromagnetism because they are a useful way of representing a magnitude and phase as a single quantity.
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References in periodicals archive ?
For the computation of the principal matrix pth root [T.sup.1/p] in step 1, we have modified the Schur-Newton Algorithm 3.3 in [8] in order to run it with complex arithmetic.
* generating s incomplete factorizations which have to be done in complex arithmetic even if A, the seed matrix, has real coefficients;
* performing matrix-vector multiplications (with [??] and [[??].sup.H]) or solving triangular systems (i.e., those with matrices [??] and [[??].sup.H]) in complex arithmetic.
Complex arithmetic is used just for solving diagonal linear systems and performing operations with vectors.
Moreover, order k > 1 approximations require the solution of a k-banded linear system in complex arithmetic per iteration, which can represent a not negligible computational cost if we have convergence in almost the same number of iterations of the underlying diagonal corrections.
Additional Key Words and Phrases: Accuracy, complex arithmetic, floating point, function evaluation, mathematical library, multiple precision, portable software
The ZM package is a collection of Fortran subroutines that performs floating-point multiple-precision evaluation of complex arithmetic and elementary functions.
Brent's MP package [Brent 1978] did not support complex arithmetic, and Bailey's more recent MP package [Bailey 1993; 1995] provides complex arithmetic and some complex elementary functions and contains a Fortran 90 module defining multiple-precision derived types.
Complex Arithmetic Timing (seconds per call on a 604 Macintosh) 50 S.D.
The complex arithmetic and function routines return results that are nearly always correctly rounded.

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