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Topic: Hypercomplex numbers

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	FRACTINT hypercomplex type
		It is not possible to fully generalize the complex numbers to four dimensions without sacrificing some of the algebraic properties shared by real and complex numbers.
		Hypercomplex numbers fail the rule that says all non-zero elements have multiplicative inverses - that is, if z is not 0, there should be a number 1/z such that (1/z)*(z) = 1.
		Hypercomplex numbers were brought to our attention by Clyde Davenport, author of "A Hypercomplex Calculus with Applications to Relativity", ISBN 0-9623837-0-8.
	spanky.triumf.ca /www/fractint/hypercomplex_type.html (191 words)

	Hypercomplex number
		Hypercomplex numbers are extensions of the complex numbers, such as quaternions, octonions and sedenions.
		Whereas complex numbers can be viewed as points in a plane, hypercomplex numbers can be viewed as points in some higher-dimensional Euclidean space (4 dimensions for the quaternions, 8 for the octonions, 16 for the sedenions).
		The Clifford algebras are another family of hypercomplex numbers.
	www.ebroadcast.com.au /lookup/encyclopedia/hy/Hypercomplex_numbers.html (104 words)

	NationMaster - Encyclopedia: Hypercomplex number
		In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1.
		A bicomplex number is a number written in the form, a + bi1 + ci2 + dj, where i1, i2 and j are imaginary units.
		The arithmetical operations of numbers, such as addition, subtraction, multiplication and division, are generalized in the branch of mathematics called abstract algebra, the study of abstract number systems such as groups, rings and fields.
	www.nationmaster.com /encyclopedia/Hypercomplex-number (2489 words)

	tScholars.com \| Hypercomplex number (Site not responding. Last check: )
		The term hypercomplex number has been used in mathematics for the elements of algebras that extend or go beyond complex number arithmetic.
		Hypercomplex numbers have had a long lineage of devotees including Hermann Hankel, Georg Frobenius, Eduard Study, and Elie Cartan.
		A special case are the bicomplex numbers which are isomorphic to tessarines, conic quaternions (from MusÃ¨s' hypernumbers), and are also contained in the 'hypercomplex number' definition by Kantor and Solodovnikov.
	www.tscholars.com /encyclopedia/Hypercomplex_number (982 words)

	Hypercomplex.ru
		However, they pay attention to the fact that there are others generalizations of number, for example, hypercomplex numbers, waiting for interpretation of their link with physics and geometry.
		The problem of physical and geometrical interpretation of hypercomplex numbers is an actual task all the more because spaces associated with hypercomplex numbers belong to the class of Finsler spaces, the more general class than Riemannian manifolds.
		It is fundamentally important, that the points of Finsler spaces in some cases may be expressed in terms of hypercomplex numbers, algebras with special exclusive properties.
	hypercomplex.xpsweb.com /index.php?lang=en&ref=about (567 words)

	Hypercomplex Math in Fractal Programming (Part 1)
		Hypercomplex numbers are four-dimensional in the form of h(4) = hr+hi+hj+hk.
		The way hypercomplex math is done (using Fractal Creation's method), a sort of matrix math is used to combine the first two fields with the last two extension fields.
		A complex function is called twice with different combinations of the hypercomplex fields, and the separate complex return values are recombined into a single hypercomplex variable.
	www.mysticfractal.com /hypercomplex.html (458 words)

	Question Corner -- The Hypercomplex Numbers
		The hypercomplex numbers are a generalization of the complex numbers.
		Multiplication by complex numbers with a modulus ("length") of 1 corresponds to a rotation of the plane.
		The only dimensions in which there are hypercomplex numbers which allow for a notion of division are dimensions 4 and 8.
	www.math.toronto.edu /mathnet/questionCorner/hypercomplex.html (637 words)

	[No title]
		The hope is to inspire mathematical work on quadratics on hypercomplex number systems and related algebras, and practical work formulating system structure in terms of hypercomplex numbers and related algebras.
		This particular magician system, when used as the algebra G for an hypercomplex number space, yields the complex numbers; and the equation for an externally-driven magician system is then just the standard quadratic iteration in the complex plane.
		For instance, suppose one wants to look for system states in which all magicians keep their numbers within certain specified bounds (this occurs, for example, with biological systems, in which a "homeostatic" state is defined as one in which the levels of all important substances remain within their natural range).
	goertzel.org /books/complex/ch7.html (9513 words)

	page1
		For the case where T is a complex number (2D), this is the famous Mandelbrot set, shown to the right.
		This same image is the 1 vs. i or j axes in the hypercomplex tetrabrot, or 1 vs. i,j,or k in the quaternion tetrabrot.
		There are a few ways to extend complex numbers to higher dimensions, creating "hypercomplex" numbers.
	www.javaspider.com /jfract (151 words)

	Hypercomplex number: Definition and Links by Encyclopedian.com
		...Hypercomplex number Hypercomplex number Hypercomplex numbers are extensions of...points in a plane, hypercomplex numbers can be viewed as points in some higher-dimensional...16 for the sedenions).More precisely, they form finite- dimensional algebras over the real...
		...extend the complex numbers, they are hypercomplex numbers.
		These algebras all have a notion of norm and conjugate,...general idea being that the product of an element and its conjugate should equal the square of...
	www.encyclopedian.com /hy/Hypercomplex-number.html (296 words)

	Cayley-Dickson construction
		The algebras produced by this process are known as Cayley-Dickson algebras; since they extend the complex numbers, they are hypercomplex numbers.
		A complex number whose second component is zero is associated with a real number: the complex number $(a, 0)$ is the real number $a$ .
		These operators are direct extensions of their complex analogs: if $a$ and $b$ are taken from the real subset of complex numbers, the appearance of the conjugate in the formulas has no effect, so the operators are the same as those for the complex numbers.
	www.ebroadcast.com.au /lookup/encyclopedia/ca/Cayley-Dickson_algebra.html (795 words)

	NSDL Publication Data Processing Center - Full Resource Record Display
		The first type of hypercomplex numbers, called polar hypercomplex numbers, is characterized by the presence in an even number of dimensions greater or equal to 4 of two polar axes, and by the presence in an odd number of dimensions of one polar axis.
		The other type of hypercomplex numbers exists as a distinct entity only when the number of dimensions n of the space is even, and since the position of a point is specified with the aid of n/2-1 planar angles, these numbers have been called planar hypercomplex numbers.
		The book presents a detailed analysis of the hypercomplex numbers in 2, 3 and 4 dimensions, then presents the properties of hypercomplex numbers in 5 and 6 dimensions, and it continues with a detailed analysis of polar and planar hypercomplex numbers in n dimensions.
	grackle.cc.columbia.edu /cwis/SPT--FullRecord.php?ResourceId=4620 (416 words)

	Hypercomplex number Information
		In mathematics, hypercomplex numbers are extensions of the complex numbers constructed by means of abstract algebra, such as quaternions, split-quaternions, tessarines, coquaternions, octonions, split-octonions, biquaternions and sedenions.
		Whereas complex numbers can be viewed as points in a plane, hypercomplex numbers can be viewed as points in some higher-dimensional Euclidean space (4 dimensions for the quaternions, tessarines and coquaternions; 8 for the octonions and biquaternions; and 16 for the sedenions).
		But none of these extensions forms a field, essentially because the field of complex numbers is algebraically closed — see fundamental theorem of algebra.
	www.bookrags.com /wiki/Hypercomplex_number (206 words)

	Hyperdimensional Quadractic Julia Set Fractals
		This definition encompasses a huge number of functions, and analysis is generally restricted to a subset of complex functions.
		Hypercomplex numbers are a generalization of normal complex numbers, whereas quaternions are complex numbers designed for representing 3D rotations.
		Neither choice is perfect, as multiplication is non-commutative in quaternion algebra, whereas hypercomplex numbers may not have a multiplicative inverse.
	graphics.ucsd.edu /courses/rendering/2005/jkelly/index.html (743 words)

	:: Mathematics :: (Site not responding. Last check: )
		Numbers take on many forms in maths, these start with the natural numbers, often referred to as the counting numbers, as they can easily be counted and have a clear and defined starting point, that is, they start from 1 and continue to infinity.
		Numbers then extend to the rationals and reals, and finally on to complex numbers.
		While the real numbers can be illustrated with a line, where a number is a point on this line, the complex numbers have to be illustrated on a 2d plane, the Argand diagram.
	people.bath.ac.uk /ma3cspb/complex.htm (388 words)

	Springer Online Reference Works
		Historically, hypercomplex numbers arose as a generalization of complex numbers (cf.
		In trying to construct numbers whose role with respect to three-dimensional space corresponds to the role played by complex numbers with respect to the plane, it became clear that a full analogy is not possible; this gave rise to the development of the theory of systems of hypercomplex numbers.
		The definition of a system of hypercomplex numbers may include the requirement of associativeness of multiplication; one also identifies the concepts of an algebra and a hypercomplex system.
	eom.springer.de /h/h048390.htm (291 words)

	[No title] (Site not responding. Last check: )
		The hope is to inspire mathematical work on quadratics on hypercomplex number systems and related algebras, and practical work formulating system structure in terms of hypercomplex numbers and related algebras.
		This particular magician system, when used as the algebra G for an hypercomplex number space, yields the complex numbers; and the equation for an externally-driven magician system is then just the standard quadratic iteration in the complex plane.
		For instance, suppose one wants to look for system states in which all magicians keep their numbers within certain specified bounds (this occurs, for example, with biological systems, in which a "homeostatic" state is defined as one in which the levels of all important substances remain within their natural range).
	www.goertzel.org /books/complex/ch7.html (9513 words)

	Springer Online Reference Works
		Complex numbers, double numbers and dual numbers are also called complex numbers of hyperbolic, elliptic and parabolic types, respectively.
		These numbers are sometimes used to represent motions in the three-dimensional spaces of Lobachevskii, Riemann and Euclid (see, for instance, Helical calculus).
		This property of double and dual numbers is utilized for the description of the tangent space to an arbitrary functor in the category of schemes [1], [3].
	eom.springer.de /d/d033860.htm (399 words)

	How to Form Multidimensional Number Systems Analogous to Complex Numbers
		The number systems generated by my process, described below, are apparently a special subclass of the set of all the various hypercomplex number systems that may be devised.
		Because of the manner of their creation, and the reasoning behind that process (which I am presently writing about), I believe that these RADN (“rotating any-dimensional number”) systems are the true counterparts in higher dimensions of the complex numbers, in contrast to the many other possible systems built upon different multiplication tables.
		The fact that all the RADN systems are based on the same reasoning about the nature of numbers (which, as mentioned, I am now writing about), and that all exhibit the orderly rotation of vectors characteristic of the complex system, makes it possible to consider them in some ways as a unified group.
	www3.sympatico.ca /rr.rawlings/n-dimensional.number.systems.htm (667 words)

	Color Image Processing
		his research is using quaternion (hypercomplex) algebra to study the definitions and properties of linear vector methods for processing color images.
		Complex numbers, being two dimensional in nature, cannot be used in processing signals of more dimensions without either iterating over a set of complex variables, or folding one dimension into another.
		In particular, quaternions are a four dimensional extension to the standard complex number.
	home.att.net /~t.a.ell/Imaging.htm (416 words)

	Amazon.com: Hypercomplex Numbers: An Elementary Introduction to Algebras: Books: I. L. Kantor,A. S. Solodovnikov,A. ... (Site not responding. Last check: )
		After complex numbers appeared as an extension of the real number system, the question arose as to whether further extensions might be made and what would they look like.
		For n=1, we have the complex numbers, n = 2 are the quaternions and n = 3 are the octonions or Cayley numbers.
		The author also discusses hypercomplex systems in general and the "doubling" process which produces the complex numbers from the reals, the quaternions from the complex numbers and the octonions from the quaternions.
	www.amazon.com /Hypercomplex-Numbers-Elementary-Introduction-Algebras/dp/0387969802 (1199 words)

	Abstract algebra Hypercomplex numbers Reference & Education Book (Site not responding. Last check: )
		Hypercomplex numbers are numbers that use the square root of -1 to create more than 1 extra dimension.
		The most basic Hypercomplex number is the one used most often in vector mathematics, the Quaternion, which consists of 4 dimensions.
		The quaternion is a 4 dimensional number, but it can be used to diagram three dimensional vectors and can be used to turn them without the use of calculus.
	www.panload.com /page.php?id=1053 (491 words)

	MaxEnt 94 - Probability Theory As Logic Applied To Hypercomplex Two Dimensional Nuclear Magnetic Resonance Data (Site not responding. Last check: )
		In two dimensional hypercomplex nuclear magnetic resonance data there are two time domains or precession periods in which the spins evolve.
		Thus a single free induction decay may be thought of as a hypercomplex set of data, each element in the hypercomplex free induction decay consists of four numbers: the real-real, the real-imaginary, the imaginary-real, and the imaginary-imaginary.
		Consequently, in two dimensional hypercomplex data the sufficient statistic is no longer just a power spectrum; rather there are two sufficient statistics --- a power spectrum, and a term that is best described as a cross correlation term.
	omega.albany.edu:8008 /MaxEnt94/abstracts/bretthorst_abs.html (440 words)

	Clyde Davenport's Commutative Hypercomplex Math Page
		Fortunately, a wide audience should be able to follow the discussion, because the commutative hypercomplex math is derived directly from well-known, fundamental concepts, such as groups, rings, calculus, complex variables, matrices, complex function theory, and vector analysis.
		Accordingly, I do not claim original discovery of the commutative hypercomplex algebra [Davenport(1), 1991], but do claim origination of certain of its representations, interpretations, and the formulation of the function theory and analysis which is constructed upon it.
		The 4-D Cauchy-Riemann conditions have a number of interesting consequences that are extensions of those for the complex variable case [Davenport(8), 1991].
	home.usit.net /~cmdaven/hyprcplx.htm (4469 words)

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