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# Topic: Complex number

###### In the News (Wed 17 Apr 19)

 Complex Numbers - MSN Encarta The complex numbers are the field C of numbers of the form x+iy, where x and y are real numbers and i is the imaginary unit equal to the square root of -1, sqrt(-1). Complex analysis, which combines complex numbers with ideas from calculus, has been widely applied to subjects as different as the theory of numbers and the design of airplane wings. The study of complex functions was continued by French mathematician Augustin Louis Cauchy, who in 1825 generalized the real definite integral of calculus to functions of a complex variable. encarta.msn.com /encyclopedia_761574697/Complex_Numbers.html   (834 words)

 Complex number Complex numbers are a field, and thus have addition, subtraction, multiplication, and division operations. The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be represented by the transpose of the matrix corresponding to z. The complex number field is relevant in the mathematical formulation of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. www.algebra.com /algebra/homework/complex/Complex_number.wikipedia   (3325 words)

 SparkNotes: Complex Numbers: Terms and Formulae Complex Conjugate - The complex conjugate of a given complex number a + bı is a - bı. Complex Plane - A plane with two perpendicular axes, the real axis and the imaginary axis, on which a complex number a + bı is plotted at the coordinate (a, b). In the complex plane, it is the distance between the plot of a complex number and the origin. www.sparknotes.com /math/precalc/complexnumbers/terms.html   (404 words)

 Complex number - Draft - Citizendium When complex numbers are represented as points in the plane, the resulting diagrams are known as Argand diagrams, after Robert Argand. The geometric representation of complex numbers turns out to be very useful, both as an aid to understanding the properties of complex numbers and as a tool in applying complex numbers to geometrical and physical problems. Complex numbers are frequently used in electrical engineering, but in that discipline it is usual to use j instead, reserving i for electrical current. en.citizendium.org /wiki/Complex_number/Draft   (2187 words)

 Complex numbers: the complex plane, addition and subtraction Real numbers are to be considered as special cases of complex numbers; they're just the numbers x + yi when y is 0, that is, they're the numbers on the real axis. The complex number z = 3 + i is located 3 units to the right of the imaginary axis and 1 unit above the real axis, while w = –1 + 2i is located 1 unit left and 2 units up. Note in the last example that the four complex numbers 0, z = 3 + i, w = –1 + 2i, and z + w = 2 + 3i are the corners of a parallelogram. www.clarku.edu /~djoyce/complex/plane.html   (772 words)

 Algebraic Structure of Complex Numbers from Interactive Mathematics Miscellany and Puzzles For, without (1) and (2), the theory of complex numbers would not deliver the closure to the branch of algebra that drove much of its development, viz., the search for the roots of polynomial equations. In the complex plane the axes also are referred to as real and imaginary, although both are real enough to the extent that the only way to distinguish between the two is by means of orientation: the rotation from the real to the imaginary axis proceeds counterclockwise. Complex numbers for which the real part is 0, i.e., the numbers in the form yi, for some real y, are said to be purely imaginary. www.cut-the-knot.org /arithmetic/algebra/ComplexNumbers.shtml   (1253 words)

 Split complex number The split-complex numbers do not form a normed algebra in the usual sense of the word since the "norm" is not positive-definite. In the twentieth-century the split-complex numbers became a common platform to describe the Lorentz boosts of special relativity, in a spacetime plane, because a velocity change between frames of reference is nicely expressed by a hyperbolic rotation. Split-complex numbers and their higher-dimensional relatives (coquaternions / split-quaternions and split-octonions) were at times referred to as "Musean numbers", since they are a subset of the hypernumber program developed by Charles MusÃ¨s. www.algebra.com /algebra/homework/complex/Split-complex_number.wikipedia   (1813 words)

 Maths - Complex Numbers - Martin Baker We can look at complex numbers as extensions or generalisations of other algebras (for example they extend real numbers and they are a subset of quaternions and clifford algebras). Complex numbers are two dimensional in that they contain two scalar values and they can represent points in 2D vector space. Is there any advantages in using complex numbers to represent the complete state of a solid object, in other words it has a state variable that includes both the position. www.euclideanspace.com /maths/algebra/realNormedAlgebra/complex/index.htm   (1819 words)

 Springer Online Reference Works Algebraically speaking, a complex number is an element of the (algebraic) extension However, multiplication and division of complex numbers, which must be performed according to (2) and (3), do not have immediate analogues in vector algebra (see [4], [5]). Leibniz said that  "complex numbers are a fine and wonderful refuge of the divine spirit, as if it were an amphibian of existence and non-existence" . eom.springer.de /c/c024140.htm   (1292 words)

 Complex number - Psychology Wiki In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for one of the square roots of negative one (−1). In mathematics, the adjective "complex" means that the field of complex numbers is the underlying number field considered, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be described by the transpose of the matrix corresponding to z. psychology.wikia.com /wiki/Complex_number   (3456 words)

 Complex and Imaginary Numbers Complex numbers act much like a bridge between two villages that are located on opposite sides of a river. If a complex number's real part a equals 0—so it has the form bi for some real b—we say the number is purely imaginary (or, more simly, imaginary). Also, the right side of each formula is always defined and corresponds to a complex number of the form [2]. www.riskglossary.com /articles/imaginary_numbers.htm   (661 words)

 complex number Complex numbers can be represented graphically on an Argand diagram, which uses rectangular Cartesian coordinates in which the x-axis represents the real part of the number and the y-axis the imaginary part. Thus the number z = a + ib is plotted as the point (a, b). Complex numbers have applications in various areas of science, such as the theory of alternating currents in electricity. www.tiscali.co.uk /reference/encyclopaedia/hutchinson/m0006657.html   (265 words)

 Help for Understanding and Using Complex Numbers The two webpages Complex Numbers and Trig for Today's Students and Distributive Law for Complex Numbers, one or both should be read first, and followed by the easy consequences B2 to B10. The idea of introducing complex numbers geometrically stems from a 1976 lectures of the late Richard Feynman, one of three public lectures given in fall 1976. Here is the complex number based proof, another consequence of assuming the distributive law and obtaining two different ways to compute the product of a complex number with its complex conjugate. whyslopes.com /etc/ComplexNumbers   (3087 words)

 PlanetMath: complex number The complex numbers form an algebraically closed field. There is a standard metric on the complex numbers, defined by This is version 4 of complex number, born on 2001-10-23, modified 2002-08-26. planetmath.org /encyclopedia/ComplexNumber.html   (107 words)

 7.4.4 Complex Number Objects Python's complex number objects are implemented as two distinct types when viewed from the C API: one is the Python object exposed to Python programs, and the other is a C structure which represents the actual complex number value. The C structure which corresponds to the value portion of a Python complex number object. Most of the functions for dealing with complex number objects use structures of this type as input or output values, as appropriate. www.python.org /doc/1.5.2p2/api/complexObjects.html   (168 words)

 Complex number - Wikipedia, the free encyclopedia In mathematics, the adjective "complex" means that the field of complex numbers is the underlying number field considered, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be described by the transpose of the matrix corresponding to z. Complex numbers are used in signal analysis and other fields as a convenient description for periodically varying signals. en.wikipedia.org /wiki/Complex_number   (3460 words)

 Imaginary number - Wikipedia, the free encyclopedia The number a is the real part of the complex number, and b is the imaginary part. Although Descartes originally used the term "imaginary number" to mean what is currently meant by the term "complex number", the term "imaginary number" today usually means a complex number with a real part equal to 0, that is, a number of the form bi. Geometrically, imaginary numbers are found on the vertical axis of the complex number plane, allowing them to be presented orthogonal to the real axis. en.wikipedia.org /wiki/Imaginary_number   (678 words)

 HSE Complex number - Wikibooks, collection of open-content textbooks The complex congugate of a + ib is a - ib. The conjugate of 3 - 9i is 3 + 9i. The conjugate of 9i - 20 is -20 - 9i. en.wikibooks.org /wiki/HSE_Complex_number   (1930 words)

 Complex number Info - Encyclopedia WikiWhat.com   (Site not responding. Last check: ) The complex numbers are an extension of the real numbers, in which all polynomials have roots. Every complex number can be represented in the form x+iy, where x and y are real numbers called the real part and the imaginary part of the complex number respectively. 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. www.wikiwhat.com /encyclopedia/c/co/complex_number.html   (1594 words)

 Complex Numbers When a number system is extended the arithmetic operations must be defined for the new numbers, and the important properties of the operations should still hold. Definition: The conjugate (or complex conjugate) of the complex number a + bi is a - bi. So, a number divided by itself will be 1, where 1 is the multiplicative identity; i.e., 1 times any number is that number. www.uncwil.edu /courses/mat111hb/Izs/complex/complex.html   (1390 words)

 Complex number Summary Complex numbers are generally expressed in the form a + bi, where a represents any real number (rational or irrational) and b represents the re... The set of complex numbers includes all the numbers we commonly work with in school mathematics (whole numbers, fractions, decimals, square roots, etc.), plus many more numbers that are generally not encountered until the study of higher mathematics. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is an imaginary number, called the imaginary unit, with the property i 2= −1. www.bookrags.com /Complex_number   (198 words)

 SparkNotes: Complex Numbers: Introduction to Complex Numbers A complex number is a number of the form a + bı, where a and b are real numbers. Complex numbers can be plotted on the complex plane. The rectangular form of the complex number z is the ordered pair (a, b), such that the first coordinate is the real part, and the second coordinate is the coefficient of the imaginary unit of the imaginary part. www.sparknotes.com /math/precalc/complexnumbers/section1.html   (340 words)

 SETS AND GROUPS LECTURE Two Complex numbers gradually established themselves as a valuable extension to the real number system and although doubts about their existence slowly disappeared, the legacy of the old terminology 'real' and 'imginary' still survives. By the middle of the last century the theory of complex variables was a thriving and central branch of mathematics. We mentioned earlier that, just as the real numbers can be used to represent points on a line, the complex numbers are a useful model for studying points in the plane. www.warwick.ac.uk /ETS/projects/maths/complex.htm   (2007 words)

 What is complex number? - a definition from WhatIs.com A complex number is a quantity of the form v + iw, where  v and w are real numbers, and i represents the unit imaginary numbers equal to the positive square root of -1. The set C of all complex numbers corresponds one-to-one with the set R R of all ordered pairs of real numbers.The set C  also corresponds one-to-one with the points on a geometric plane. Real numbers are used to denote electrical resistance, imaginary numbers are used to denote reactance, and complex numbers are used to represent impedance. whatis.techtarget.com /definition/0,,sid9_gci283967,00.html   (174 words)

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