
	Where results make sense

Topic: Cayley numbers

Related Topics

Arthur Cayley

Sir George Cayley

Cayley Hamilton theorem

George Cayley

Cayley Dickson construction

Cayley Galt Tariff

Cayley Newbirth operation matrix

Cayley numbers

Cayley Omega process

Cayley table

W Cayley Hamilton

David Cayley

In the News (Wed 30 May 12)

EXECUTIVE POWER

Iran

ANALYSIS

Pakistan

Family compact

	Cayley-Dickson construction - Wikipedia, the free encyclopedia
		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.
		This algebra was discovered by Graves in 1844, and is called the octonions or the "Cayley numbers".
	en.wikipedia.org /wiki/Cayley-Dickson_construction (737 words)

	Octonion - Wikipedia, the free encyclopedia
		Despite this, the octonions retain importance for being related to a number of exceptional structures in mathematics, among them the exceptional Lie groups.
		Addition of octonions is accomplished by adding corresponding coefficients, as with the complex numbers and quaternions.
		The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on.
	en.wikipedia.org /wiki/Octonions (836 words)

	[No title]
		Cayley's theorem +------------------------------------------------------------ Cayley's theorem assures that every finite group is isomorphic to a permutation group.
		number field +------------------------------------------------------------ A number field is a finite extension of Q, the field of rational numbers.
		It is a field extension of Q which is also a vector space of finite dimension over Q. Since the elements of a number field are algebraic numbers, roots of a fixed polyonomial a_0+a_1 z+...
	www.math.harvard.edu /~knill/sofia/data/algebra.txt (1599 words)

	Question Corner -- The Hypercomplex Numbers (Site not responding. Last check: 2007-10-13)
		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.
		In both the Quaternians and the Cayley numbers, multiplication is non-commutative.
	www.math.toronto.edu /mathnet/questionCorner/hypercomplex.html (637 words)

	Octonion (Site not responding. Last check: 2007-10-13)
		The octonions form an 8-dimensional (non-associative) division algebra over the real number s, and can therefore be thought of as octets (or 8-tuples) of real numbers.
		Addition of octonions is accomplished by adding corresponding coefficients, as with the complex number s and quaternions.
		The only finite-dimensional associative division algebras over the reals are the real number s, the complex number s, and the quaternions.
	www.serebella.com /encyclopedia/article-Octonion.html (234 words)

	A History of Hypercomplex Numbers (Site not responding. Last check: 2007-10-13)
		Cayley describes the 8-dimensional octonions, called the Cayley numbers, which are both noncommutative and nonassociative, in an article "On Jacobi's elliptic functions, in reply to the Rev. B.
		Cayley's "A memoir on the theory of matrices" published in Philosophical Transactions, introducing matrices, their addition, multiplication, zero and identity, and a new interpretation of determinants.
		Karl Weierstrass proves that complex numbers are the only finite dimensional extension of the reals which preserves all the laws of arithmetic.
	history.hyperjeff.net /hypercomplex.html (2046 words)

	Cayley-Dickson Construction of Quaternions, Octonions, etc.
		To understand this, you will need knowledge of the complex numbers, such as is often taught in a high school second-year algebra course.
		The Cayley-Dickson construction produces a sequence of higher-dimensional algebras that are like numbers inasmuch as they have a norm and a multiplicative inverse.
		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.zipcon.net /~swhite/docs/math/quaternions/Cayley-Dickson.html (625 words)

	New Scientist Archive \| Selected Article (Site not responding. Last check: 2007-10-13)
		The ultimate number - the humble 8 - lies at the heart of a mathematical system known as the octonions, and this system appears to be the key that will allow physicists to fit quantum theory and gravity together.
		Numbers in these systems are the only ones to have a "norm", effectively the number's distance from the origin (see Graphic).
		Interestingly, each of these numbers is 2 greater than that of a normed division algebra: subtract 2 from 3, 4, 6 and 10, and you get 1, 2, 4 and 8.
	www.incunabula.org /blog/articles/stewart.html (2511 words)

	sci.math FAQ: Fundamentals
		But if we restrict ourselves to odd numbers, their sum is not an odd number and so we know right off the bat that the set of odd numbers and addition cannot constitute a group.
		Now each of these is a rational number (it can be written as a fraction), and they are getting closer and closer to a number we've probably seen before (just take out your calculator and find the square root of two).
		Which was really too bad because they were given the name of imaginary numbers and now that the name stuck we realize that they are numbers just as good as any of the ones we have been using for centuries.
	www.faqs.org /faqs/sci-math-faq/numbers (3288 words)

	Quaternion References
		Cayley, Arthur: 1864 On the notion and boundaries of algebra.
		M.P. Cayley, Arthur: On the quaternion equation qQ — Qq’ = 0.
		Cayley, Arthur: On the matrical equation qQ — Qq’ = 0.
	home.att.net /~t.a.ell/QuatRef.htm (10852 words)

	UC Davis Math: Glossary (Site not responding. Last check: 2007-10-13)
		Usually this means the maximum number of disjoint circles that can be drawn on the surface such that the complement is connected.
		In topology, a number, polynomial, or other quantity associated to a topological object such as a knot or 3-manifold which depends only on the underlying object and not on its specific description or presentation.
		The energy of a given state of the grid is given by the number of atoms which are spin up or down and by the number of pairs of neighboring atoms whose spins agree or disagree.
	www.math.ucdavis.edu /profiles/glossary.html (9932 words)

	4-dim HyperDiamond Lattice
		For a number of the form 2^p + 1 to be prime, p must be of the form 2^k.
		Today the usual definition of a perfect number is in terms of its divisors, but early definitions were in terms of the 'aliquot parts' of a number.
		Since energy levels are positive numbers, and so should correspond to a straight line in the complex plane, such a zeta function - quantum system correspondence could be used to verify the Riemann hypothesis, that all the nontrivial zeroes of the zeta function are on the straight line Re(z) = 1/2 in the complex plane.
	www.valdostamuseum.org /hamsmith/PrimeFC.html (5922 words)

	MAT 318 Projects
		For any number of sides, there is a regular polygon in the plane with that number of sides, so there are infinitely many.
		The octonians, or Cayley numbers, are another extension of this idea, with seven square roots of -1, in which multiplication is not even associative (so (ab)c is not always the same as a(bc)).
		Another possible direction is to study the geometry of the quaternions and Cayley numbers, in a manner similar to the Complex numbers and Plane Geometry project.
	www.math.sunysb.edu /~ksir/s00/mat318proj.html (2461 words)

	[No title]
		There are very few algebras over the real numbers for which every element has a multiplicative inverse: if we demand associativity and commutativity, just the reals themselves and the complex numbers.
		Well, few people doubt that the real numbers are fundamental to physics (though some advocates of the discrete might prefer the integers), and with emergence of quantum theory, if not sooner, the basic role of the complex numbers also became clear.
		For the complex numbers, you can check that (a,b)* (a,b) = (a,b) (a,b)* = K (1,0) where K is a real number called the "norm squared" of (a,b).
	www.math.niu.edu /~rusin/known-math/95/octonions.phys (2154 words)

	Amazon.com: The Book of Numbers: Books: John H. Conway,Richard Guy (Site not responding. Last check: 2007-10-13)
		Though number theory does not lend itself to fun and games, the authors take such joy in the order and patterns of numbers that you can't help being fascinated by what is actually a fairly difficult subject.
		Many of the numbers covered in the book are named after the person most responsible for making it famous, an aspiration that most mathematicians would no doubt confess to.
		The coverage is wide: primes, reals, Cayley numbers, Eisenstein numbers, polygonal numbers, catalan numbers, Stirling numbers of both types and of course Bell numbers.
	www.amazon.com /exec/obidos/tg/detail/-/038797993X?v=glance (1871 words)

	[No title]
		A complex number can be written in the form a+bi where i is the imaginary number satisfying i = EMBED Equation.3 and generally denoted byNow the quaternions are the natural extension of the complex numbers via something called the Cayley-Dickson process.
		A real number is just that, a complex number has a real part (the a) and an imaginary part (the bi).
		Use a function to read a quaternion number, a function to print a quaternion number, a function to add two quaternion numbers, a functions to multiply two quaterion numbers plus other appropriate functions as needed.
	www.mtsu.edu /~csal/spring.05/cs2170/olab2sp05.doc (424 words)

	Research Interests (Site not responding. Last check: 2007-10-13)
		We then seek the cutset that minimizes the ratio of the number of edges in the set to the number of vertices in the smaller of the two pieces.
		Since a graph is a sort of degenerate manifold, it is not surprising that the isoperimetric number is related to the eigenvalues of the graph.
		Buser developed the isoperimetric number as a tool for studying spectral geometry; the idea was to gain information regarding the spectrum of a manifold based on properties of certain associated graphs.
	www.math.ksu.edu /~jasonr/Research.html (1123 words)

	What's next? (Site not responding. Last check: 2007-10-13)
		Both are numbers in ZF set theory [Enderton77, Henle86, Hrbacek84] and so they are sets as well.
		Cardinals are numbers that represent the sizes of sets, and ordinals are numbers that represent well ordered sets.
		These numbers are constructed by means of ultrafilters [Henle86] and they are used in non-standard analysis.
	db.uwaterloo.ca /~alopez-o/math-faq/node20.html (226 words)

	Archive
		Here Dave derives a recurrence relation for the number of domino tilings of Aztec rectangles with squares removed along one or both of the long edges.
		A theorem of Lindstrom-Gessel-Viennot is used to express this number in terms of determinants.
		This is a rough draft of a review paper summarizing some of Dave's work on the mathematics of quantum mechanics and, in particular, the Lagrangian/path integral formalism.
	www.math.columbia.edu /~ums/Archive.html (570 words)

	Octonions and sedenions
		There are three approaches to constructing the Cayley numbers.
		From the quaternions, it produces the Cayley numbers.
		The third approach to constructing the Cayley numbers is more combinatorial.
	orion.math.iastate.edu /jdhsmith/math/JS26jan4.htm (861 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		Every normed algebra with an identity is isomorphic to one of the following four algebras: the real numbers, the complex numbers, the quaternions, and the Cayley numbers.
		Every associative division algebra is isomorphic to one of the following: the algebra of real numbers, the algebra of complex numbers, and the algebra of quaternions.
		Every alternative division algebra is isomorphic to one of the following four algebras: the real numbers, the complex numbers, the quaternions, and the Cayley numbers.
	www.math.niu.edu /~rusin/known-math/96/divisalg.bok (193 words)

	Hypercomplex numbers
		Addition of such numbers is defined as with the complex numbers (elementwise addition), and multiplication with a real is also defined in a elementwise fashion.
		In the complex numbers we have defined a multiplication that makes the complex numbers a field (and could identify the reals with specific complex numbers).
		So the product of a number with its conjugate (or the other way around, this multiplication commutes) is the sum of such products in the parent ring, and so is the sum of such products in the ultimate base ring.
	homepages.cwi.nl /~dik/english/mathematics/numc.html (976 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		For example, their algebraic structure is the basis for considerable study in algebra, since they form a \emph on field \emph default (a set with two operations, known as addition and multiplication, with the usual properties).
		But, for us, studying analysis, the algebraic properties of the real numbers are (more or less) assumed, and it is the \emph on ordering \emph default of the numbers that is more important.
		There is only one way in which two different sequences of digits after the decimal point can represent the same number, and that is that, for example, \begin_inset Formula \[ 3.2871818999999...=3.2871819000000...,\] \end_inset where in the first case the string of 9's is infinite, and in the second case the string of 0's is infinite.
	www.lehigh.edu /dlj0/yesterday/courses/301f04-11-12.lyx (1299 words)

	ipedia.com: Cayley-Dickson construction Article (Site not responding. Last check: 2007-10-13)
		The complex numbers can be written as ordered pairs of real numbers and, with the addition operator being component-by-component and with multiplication defined by
		A complex number whose second component is zero is associated with a real number: the complex number is the real number.
		These operators are direct extensions of their complex analogs: if and 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.ipedia.com /cayley_dickson_construction.html (715 words)

	Maths - Octonion - Martin Baker
		Octonions are a superset of quaternions in the same way that quaternions are a superset of complex numbers.
		We might expect this sequence to continue with an element consisting of 16 numbers, but such an algebra does not exist, and the sequence ends with octonions.
		When we go from Complex numbers to Quaternions we loose commutatively and when we go from Quaternions to Octonions we loose associatively.
	www.euclideanspace.com /maths/algebra/realNormedAlgebra/octonion (649 words)

	Letter C
		A v-cage is a v-regular graph with maximum possible number of nodes.
		Then the British mathematician, Arthur Cayley (1821-1895), generalized this to an array of arbitrary number of rows and columns, known as a "matrix", adopting as the multiplicative rule for matrices the operation correct for determinants.
		(the triangular numbers, PL); and its topological genus, PL is
	members.fortunecity.com /jonhays/letterC.htm (7206 words)

	Sam Sirlin's aplc Page (Site not responding. Last check: 2007-10-13)
		Hamilton's quaternions include the complex numbers, plus two more square roots of -1, 1i2j3k4.
		Cayley's octionions include the quaternions, plus four more roots of -1, 1i2j3k4U5I6J7K8.
		I've been meaning to have a language with all the numbers for a while.
	home.earthlink.net /~swsirlin/aplcc.html (202 words)

	Degen-Graves-Cayley Eight-Square Identity
		Is there a connection or is it just the law of small numbers?) Anyway, Abel and Galois gave the reason for the latter while Hurwitz dealt with the former.
		Complex numbers, one step up, do not have a natural ordering; they can, after all, be seen as points in the complex plane.
		Thus, the octonions are sometimes referred to as Cayley numbers.
	www.geocities.com /titus_piezas/DegenGraves1.htm (2204 words)

Try your search on: Qwika (all wikis)