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• Hypercomplex_number abstract "In mathematics, a hypercomplex number is a traditional term for an element of an algebra over a field where the field is the real numbers or the complex numbers. In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature. The concept of a hypercomplex number covered them all, and called for a science to explain and classify them.The cataloguing project began in 1872 when Benjamin Peirce first published his Linear Associative Algebra, and was carried forward by his son Charles Sanders Peirce. Most significantly, they identified the nilpotent and the idempotent elements as useful hypercomplex numbers for classifications. The Cayley–Dickson construction used involutions to generate complex numbers, quaternions, and octonions out of the real number system. Hurwitz and Frobenius proved theorems that put limits on hypercomplexity: Hurwitz's theorem (normed division algebras), and Frobenius theorem (real division algebras).It was matrix algebra that harnessed the hypercomplex systems. First, matrices contributed new hypercomplex numbers like 2 × 2 real matrices. Soon the matrix paradigm began to explain the others as they became represented by matrices and their operations. In 1907 Joseph Wedderburn showed that associative hypercomplex systems could be represented by matrices, or direct sums of systems of matrices. From that date the preferred term for a hypercomplex system became associative algebra as seen in the title of Wedderburn’s thesis at University of Edinburgh. Note however, that non-associative systems like octonions and hyperbolic quaternions represent another type of hypercomplex number.As Hawkins (1972) explains, the hypercomplex numbers are stepping stones to learning about Lie groups and group representation theory. For instance, in 1929 Emmy Noether at Bryn Mawr wrote on "hypercomplex quantities and representation theory".Review of the historic particulars gives body to the generalities of modern theory. In 1973 Kantor and Solodovnikov published a textbook on hypercomplex numbers which was translated in 1989; a reviewer says it has a "highly classical flavour". See Karen Parshall (1985) for a detailed exposition of the heyday of hypercomplex numbers, including the role of such luminaries as Theodor Molien and Eduard Study. For the transition to modern algebra, Bartel van der Waerden devotes thirty pages to hypercomplex numbers in his History of Algebra (1985).".
• Hypercomplex_number wikiPageExternalLink hypercomplex.
• Hypercomplex_number wikiPageExternalLink index.php?&lang=en.
• Hypercomplex_number wikiPageExternalLink octonions.html.
• Hypercomplex_number wikiPageExternalLink frobenius_-_hypercomplex_i.pdf.
• Hypercomplex_number wikiPageExternalLink study_-_complex_numbers_and_transformation_groups.pdf.
• Hypercomplex_number wikiPageExternalLink 1568981962.pdf.
• Hypercomplex_number wikiPageID "51438".
• Hypercomplex_number wikiPageRevisionID "605678609".
• Hypercomplex_number hasPhotoCollection Hypercomplex_number.
• Hypercomplex_number id "p/h048390".
• Hypercomplex_number title "Hypercomplex number".
• Hypercomplex_number urlname "HypercomplexNumber".
• Hypercomplex_number subject Category:History_of_mathematics.
• Hypercomplex_number subject Category:Hypercomplex_numbers.
• Hypercomplex_number type Abstraction100002137.
• Hypercomplex_number type Amount105107765.
• Hypercomplex_number type Attribute100024264.
• Hypercomplex_number type HypercomplexNumbers.
• Hypercomplex_number type Magnitude105090441.
• Hypercomplex_number type Number105121418.
• Hypercomplex_number type Property104916342.
• Hypercomplex_number comment "In mathematics, a hypercomplex number is a traditional term for an element of an algebra over a field where the field is the real numbers or the complex numbers. In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature.".
• Hypercomplex_number label "Hypercomplex getal".
• Hypercomplex_number label "Hypercomplex number".
• Hypercomplex_number label "Hyperkomplexe Zahl".
• Hypercomplex_number label "Liczby hiperzespolone".
• Hypercomplex_number label "Nombre hypercomplexe".
• Hypercomplex_number label "Numero ipercomplesso".
• Hypercomplex_number label "Número hipercomplejo".
• Hypercomplex_number label "Número hipercomplexo".
• Hypercomplex_number label "Гиперкомплексное число".
• Hypercomplex_number label "عدد عقدي فائق".
• Hypercomplex_number label "超复数".
• Hypercomplex_number sameAs Hyperkomplexní_číslo.
• Hypercomplex_number sameAs Hyperkomplexe_Zahl.
• Hypercomplex_number sameAs Número_hipercomplejo.
• Hypercomplex_number sameAs Nombre_hypercomplexe.
• Hypercomplex_number sameAs Numero_ipercomplesso.
• Hypercomplex_number sameAs Hypercomplex_getal.
• Hypercomplex_number sameAs Liczby_hiperzespolone.
• Hypercomplex_number sameAs Número_hipercomplexo.
• Hypercomplex_number sameAs m.0dkq8.
• Hypercomplex_number sameAs Q837414.
• Hypercomplex_number sameAs Q837414.
• Hypercomplex_number sameAs Hypercomplex_number.
• Hypercomplex_number wasDerivedFrom Hypercomplex_number?oldid=605678609.
• Hypercomplex_number isPrimaryTopicOf Hypercomplex_number.