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Topic: Complex plane

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Complex number

Complex conjugate

Imaginary unit

Field (mathematics)

Real number

Absolute value

Riemann sphere

Polynomial

Riemann surface

Mathematics

Norm (mathematics)

Nyquist plot

Sphere

Bernhard Bolzano

OpenType

	WWW interactive mathematics server (Site not responding. Last check: 2007-11-07)
		Graphic complex inequalities, recognize a region of the complex plane described by inequalities.
		Tir complexe, repérer un nombre complexe en cliquant sur le plan complexe.
		Inégalités complexes graphiques, reconnaître une région du plan complexe décrite par des inégalités.
	webwork.math.ohio-state.edu /wims/wims.cgi (4774 words)

	Creatures
		Like many other fractal artists who carve the complex plane, I am interested in the symbolic implications of this new world of visuals, a world which was not possible seven years ago.
		In short, each C in a given window of the plane is used in the function, which is iterated a given number of times.
		One of the first discoveries made: it is possible to generate shapes in which the fractal parts (the complex, fuzzy areas) are shifted to occupy distinct localized areas or configurations.
	www.ventrella.com /Alife/Creatures/creatures.html (1599 words)

	Complex numbers: the complex plane, addition and subtraction
		The standard symbol for the set of all complex numbers is C, and we'll also refer to the complex plane as C.
		For instance, the sum of 5 + 3i and 4 + 2i is 9 + 5i.
		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)

	PlanetMath: complex (Site not responding. Last check: 2007-11-07)
		is algebraically closed: every polynomial with complex coefficients, and especially every polynomial with real coefficients, (and with positive degree) has at least one complex zero (which might be real as well).
		The ordinate (vertical) axis is known as the imaginary axis, since it consists of all complex numbers with real part equal to zero.
		This is version 34 of complex, born on 2001-11-08, modified 2007-01-30.
	planetmath.org /encyclopedia/Complex.html (444 words)

	PlanetMath: topology of the complex plane (Site not responding. Last check: 2007-11-07)
		"topology of the complex plane" is owned by matte.
		Cross-references: Euclidean metric, clear, metric, induced, complex plane, topology
		This is version 1 of topology of the complex plane, born on 2003-05-25.
	planetmath.org /encyclopedia/TopologyOfTheComplexPlane.html (84 words)

	Complex number - Psychology Wiki (Site not responding. Last check: 2007-11-07)
		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.
	psychology.wikia.com /wiki/Complex_number (3490 words)

	The Geometer's Sketchpad® - Visualizing Complex Functions
		A great leap forward in the history and development of the complex numbers comes some 250 years after Cardano first proposes them, in the near-simultaneous realizations of Argand, Gauss, and Wessel that a complex number can be interpreted as, and represented by, a geometric point in two dimensions.
		This simple correspondence introduces not only the ability to visualize the complex numbers---or, now, the complex plane---in a simple and straightward sense, but to bring geometric techniques to the interpretation and analysis of the mathematics of that complex plane.
		This paper's purpose is to explore the potential for using these software tools to investigate and model complex numbers and their operations, and to visualize functions defined on the complex plane.
	www.dynamicgeometry.com /general_resources/recent_talks/complex_ictmt6 (492 words)

	Algebraic Structure of Complex Numbers
		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 (1204 words)

	Geometry of Complex Numbers
		Complex numbers are ordered pairs of real numbers, so they can be represented by points in the plane.
		In this section we show the effect that algebraic operations on complex numbers have on their geometric representations.
		Addition of complex numbers is analogous to addition of vectors in the plane.
	math.fullerton.edu /mathews/c2003/ComplexGeometryMod.html (475 words)

	SparkNotes: Complex Numbers: Introduction to Complex 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.
		These two axes, the real and imaginary axes, form the complex plane, in which complex numbers in rectangular form (a, b) are plotted the same way as points are plotted using rectangular coordinates.
	www.sparknotes.com /math/precalc/complexnumbers/section1.html (329 words)

	The Complex Plane
		Complex numbers are defined as having two parts: a real part and an imaginary part, in the form a+bi.
		In the complex plane, the same region is defined by Re Likewise, the upper half-plane is denoted as Im We can also define a vertical strip by, for example, 0 < Re This would be in the right half-plane between 0 and 2 and extending infinitely up and down.
		Now that we've defined what we mean by the complex plane, there are a few basic ideas that you must know to help you in your study of the Riemann Hypothesis.
	www.math.unl.edu /~jcliber/sectionIV.html (884 words)

	The Complex Plane
		Complex numbers may be written as a + b times i, where a and b are real numbers and i represents the square root of minus 1.
		Multiplication of complex numbers is more complicated and does not have such a simple geometric representation in the complex plane.
		In the complex plane there are all sorts of paths by which a function might approach a limit.
	www.mcasco.com /compln.html (1838 words)

	Conformal Mapping
		Complex variables are combinations of real and imaginary numbers, which is taught in secondary schools.
		The use of complex variables to perform a conformal mapping is taught in college.
		Similarly, in the airfoil plane, the horizontal coordinate is B and the vertical coordinate is C, and every point A is specified by:
	www.grc.nasa.gov /WWW/K-12/airplane/map.html (699 words)

	Math::Complex - perldoc.perl.org
		Complex numbers are numbers that have both a real part and an imaginary part, and are usually noted:
		A graphical representation of complex numbers is possible in a plane (also called the complex plane, but it's really a 2D plane).
		Since there is a bijection between a point in the 2D plane and a complex number (i.e.
	perldoc.perl.org /Math/Complex.html (2066 words)

	Complex Numbers and Geometry (Site not responding. Last check: 2007-11-07)
		For any complex number, c = x + y i, where x and y are real numbers, we refer to x as the "real part of c" and y as the "complex part of c".
		The geometric rule for addition is implicit in the very way we plot complex numbers, since a general complex number is naturally written as a sum.
		The domain of both functions is equal to the complex plane, while their range is the real number line.
	campus.northpark.edu /math/PreCalculus/Transcendental/Trigonometric/Complex/index.html (2082 words)

	ITERATION INTO THE COMPLEX PLANE...
		The first one is based on the development of new fractal formulae, the second is based on mappings of classic formulae according to the 17 crystallographic groups and the third one is the use of kaleidoscopes and tessellations.
		The structures obtained iterating the complex variable values into a dynamical system are mapped all along the plane creating an amazing family of graphics with several applications to Art and Design.
		After a formula is introduced into a dynamical system, iterated enough times and mapped along the complex plane, the resulting values can be modified by external transformations, not related with the formulae.
	members.tripod.com /vismath7/proceedings/barrallo.htm (489 words)

	Making Order Out of Chaos: The Eulicidian and Complex Plane (Site not responding. Last check: 2007-11-07)
		The complex plane is used to graph fractal images.
		In order to understand this plane, you need to understand complex numbers (view the last section if you need to review).
		The complex plane is unique, because unlike the Eulicidian Plane, only one number is necessary to graph a point.
	library.thinkquest.org /12170/theory/plane.html (250 words)

	All Elementary Mathematics - Study Guide - Algebra - Complex numbers...
		Addition. A sum of complex numbers a+ bi and c+ di is called a complex number (a+ c) + (b+ d) i. So, at addition of complex numbers their abscissas and ordinates are added separately.
		Then, a complex number a+ bi will be represented by point P with abscissa a and ordinate b (see figure).
		Modulus of a complex number is a length of vector OP, representing this complex number in a coordinate (complex) plane.
	www.bymath.com /studyguide/alg/sec/alg26.html (658 words)

	Recent Papers of Robert L. Devaney
		In the dynamical plane, the dynamics of the map on each of these sets is always the same: on the Cantor set portion of the web, the map is conjugate to the shift map, while points in the open disks eventually escape to infinity.
		We describe the hyperbolic components of the parameter plane for the complex exponential family using two tools: a parameter plane kneading sequence and a dynamical plane kneading sequence.
		We prove that a saddle-node (parabolic) periodic point in a complex dynamical system often admits homoclinic points, and in the case that these homoclinic points are nondegenerate, this is accompanied by the existence of infinitely many baby Mandelbrot sets converging to the saddle-node parameter value in the corresponding parameter plane.
	math.bu.edu /people/bob/papers.html (2981 words)

	Math Tutorial -- Complex Waves
		It is not immediately obvious that a complex exponential function provides the oscillatory behavior needed to represent a plane wave.
		The ``waviness'' in a complex exponential plane wave resides in the phase rather than in the magnitude of the wave function.
		We aren't used to having complex numbers show up in physical theories and it is hard to imagine how we would measure such a number.
	www.physics.nmt.edu /~raymond/classes/ph13xbook/node93.html (426 words)

	stereographic projection
		Stereographic projection is often used to reduce the size of the complex plane while preserving some of the characteristics of the plane, such as the angles between intersecting curves.
		In practice various lines are drawn from a point in such a way that they intersect both a sphere and a plane, although it is easy to imagine two spheres or two planes.
		In this representation, the unit circle corresponds to the equator of the sphere, whose axis lies perpendicular to the plane.
	delta.cs.cinvestav.mx /~mcintosh/comun/complex/node6.html (375 words)

	The educational encyclopedia, mathematics: complex numbers
		Complex numbers a complex number is a number with a real and an imaginary part, usually expressed in cartesian form
		Complex numbers arithmetic of complex numbers, identifying points in the complex plane, real and imaginary points, adding complex numbers, subtracting complex numbers, multiplication of complex numbers, finding the square roots of a complex number, finding conjugates of complex numbers, modulus, argument and polar form of a complex number, De Moivre's theorem
		Complex variables basic definitions and arithmetic, the complex plane, the polar form of a complex number, Euler's formula
	www.educypedia.be /education/mathcomplex.htm (346 words)

	Complex Numbers
		Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.
		Definition: The conjugate (or complex conjugate) of the complex number a + bi is a - bi.
		Conjugates are important because of the fact that a complex number times its conjugate is real; i.e., its imaginary part is zero.
	www.uncwil.edu /courses/mat111hb/Izs/complex/complex.html (1390 words)

	Complex numbers : Complex numbers and the complex plane
		Note that it is customary to denote a complex number by the letter z.
		If the imaginary part of a complex number is zero, the number is just a real number.
		So complex numbers can be plotted in a plane by using the x-axis for the real part and the y-axis for the imaginary part.
	scholar.hw.ac.uk /site/maths/topic5.asp?outline= (455 words)

	Complex No Intro - Vector and Complex Numbers:
		The addition of points in the plane is given by means of their rectangular coordinates while multiplication is given in terms of polar coordinates.
		The angle of a purely imaginary complex number z = a+ib = 0+ib = (0,b) is 90 degrees or 270 degrees (modulo 360 degrees), depending on the sign of the imaginary part b.
		Since the angle 180 degrees is associated with -1, and the angles 0 and 360 degrees are both associated with the number +1, the polar coordinate definition of multiplication of points in the plane agrees with (or yields) the law of signs for the multiplication of positive and negative numbers.
	whyslopes.com /etc/ComplexNumbers/complex.html (3347 words)

	The Basic Scoop on Complex Numbers
		Complex numbers show up in a variety of interesting, real-world applications, such as the analysis of electronic circuits, high-speed flow over a jet plane, and so on.
		Graphing complex numbers is done in a manner very similar to plotting a point (x, y) in rectangular coordinates; the main difference is that the Y-axis now represents IMAGINARY numbers, rather than the REALS, as when plotting points in rectangular coordinates.
		The amplitude is the length of the "arrow" or line which connects the complex nummber to the origin.
	www.geom.uiuc.edu /~laurence/welcome.html (1228 words)

	Conformal Mapping Properties (Site not responding. Last check: 2007-11-07)
		The symmetric point of the complex number z with respect to a line is the complex point that is the reflection of
		The symmetric point of the complex number z with respect to a circle is the complex point along the same ray emanating from the center of the circle, and such that the geometric mean of the distances from the center is the radius of the circle.
		It is a one-to-one mapping of the closed complex plane onto itself (the closed complex plane includes the point at infinity); thus, for each z,
	www.ee.cooper.edu /courses/course_pages/97_Fall/EE114/complex/node6.html (607 words)

	The Complex Plane
		Figure 2.2: Plotting a complex number as a point in the complex plane.
		A complex plane (or Argand diagram) is any 2D graph in which the horizontal axis is the real part and the vertical axis is the imaginary part of a complex number or function.
		in the complex plane can be viewed as a plot in Cartesian or rectilinear coordinates.
	ccrma.stanford.edu /~jos/mdft/Complex_Plane.html (241 words)

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