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- Hypercomplex_number abstract "In mathematics, a hypercomplex number is a traditional term for an element of a unital algebra over the field of real numbers. In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature, added to the real and complex numbers. The concept of a hypercomplex number covered them all, and called for a discipline 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 (associative division algebras). Finally in 1958 J. Frank Adams used topological methods to prove that there exist only four finite-dimensional real division algebras: the reals ℝ, the complexes ℂ, the quaternions ℍ, and the octonions 𝕆.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 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).".
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- Hypercomplex_number wikiPageExternalLink octonions.html.
- Hypercomplex_number wikiPageExternalLink frobenius_-_hypercomplex_i.pdf.
- Hypercomplex_number wikiPageExternalLink study_-_complex_numbers_and_transformation_groups.pdf.
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- Hypercomplex_number wikiPageExternalLink 4V6F5ID5PHYD7UJ4F4ZDWHD2SBEXGCGQ.
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- Hypercomplex_number wikiPageWikiLink 2_×_2_real_matrices.
- Hypercomplex_number wikiPageWikiLink Abstract_algebra.
- Hypercomplex_number wikiPageWikiLink Addison-Wesley.
- Hypercomplex_number wikiPageWikiLink Algebra_over_a_field.
- Hypercomplex_number wikiPageWikiLink Archive_for_History_of_Exact_Sciences.
- Hypercomplex_number wikiPageWikiLink Associative.
- Hypercomplex_number wikiPageWikiLink Associative_algebra.
- Hypercomplex_number wikiPageWikiLink Associative_property.
- Hypercomplex_number wikiPageWikiLink B.L._van_der_Waerden.
- Hypercomplex_number wikiPageWikiLink Bartel_Leendert_van_der_Waerden.
- Hypercomplex_number wikiPageWikiLink Bartel_van_der_Waerden.
- Hypercomplex_number wikiPageWikiLink Benjamin_Peirce.
- Hypercomplex_number wikiPageWikiLink Bhama_Srinivasan.
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- Hypercomplex_number wikiPageWikiLink Biquaternion.
- Hypercomplex_number wikiPageWikiLink Bulletin_of_the_American_Mathematical_Society.
- Hypercomplex_number wikiPageWikiLink Category:History_of_mathematics.
- Hypercomplex_number wikiPageWikiLink Category:Hypercomplex_numbers.
- Hypercomplex_number wikiPageWikiLink Cayley–Dickson_construction.
- Hypercomplex_number wikiPageWikiLink Charles_Sanders_Peirce.
- Hypercomplex_number wikiPageWikiLink Classical_mechanics.
- Hypercomplex_number wikiPageWikiLink Classification_of_Clifford_algebras.
- Hypercomplex_number wikiPageWikiLink Clifford_algebra.
- Hypercomplex_number wikiPageWikiLink Closure_(mathematics).
- Hypercomplex_number wikiPageWikiLink Commutative.
- Hypercomplex_number wikiPageWikiLink Commutative_property.
- Hypercomplex_number wikiPageWikiLink Completing_the_square.
- Hypercomplex_number wikiPageWikiLink Complex_number.
- Hypercomplex_number wikiPageWikiLink Composition_algebra.
- Hypercomplex_number wikiPageWikiLink Coquaternion.
- Hypercomplex_number wikiPageWikiLink Cross-ratio.
- Hypercomplex_number wikiPageWikiLink Degenerate_bilinear_form.
- Hypercomplex_number wikiPageWikiLink Degenerate_form.
- Hypercomplex_number wikiPageWikiLink Dimension.
- Hypercomplex_number wikiPageWikiLink Distributive_property.
- Hypercomplex_number wikiPageWikiLink Division_algebra.
- Hypercomplex_number wikiPageWikiLink Dual_number.
- Hypercomplex_number wikiPageWikiLink Eduard_Study.
- Hypercomplex_number wikiPageWikiLink Electromagnetic_theory.
- Hypercomplex_number wikiPageWikiLink Element_(mathematics).
- Hypercomplex_number wikiPageWikiLink Elie_Cartan.
- Hypercomplex_number wikiPageWikiLink Elsevier.
- Hypercomplex_number wikiPageWikiLink Emil_Artin.
- Hypercomplex_number wikiPageWikiLink Emmy_Noether.
- Hypercomplex_number wikiPageWikiLink Field_(mathematics).
- Hypercomplex_number wikiPageWikiLink Frederick_William_University.
- Hypercomplex_number wikiPageWikiLink Frobenius_theorem_(real_division_algebras).
- Hypercomplex_number wikiPageWikiLink Geometric_algebra.
- Hypercomplex_number wikiPageWikiLink Georg_Scheffers.
- Hypercomplex_number wikiPageWikiLink Group_representation.
- Hypercomplex_number wikiPageWikiLink Hurwitzs_theorem_(composition_algebras).
- Hypercomplex_number wikiPageWikiLink Hurwitzs_theorem_(normed_division_algebras).
- Hypercomplex_number wikiPageWikiLink Hyperbolic_quaternion.
- Hypercomplex_number wikiPageWikiLink Hypercomplex_analysis.
- Hypercomplex_number wikiPageWikiLink Ian_R._Porteous.
- Hypercomplex_number wikiPageWikiLink Idempotent_element.
- Hypercomplex_number wikiPageWikiLink Involution_(mathematics).
- Hypercomplex_number wikiPageWikiLink John_T._Tate.
- Hypercomplex_number wikiPageWikiLink John_Tate.
- Hypercomplex_number wikiPageWikiLink Joseph_Wedderburn.
- Hypercomplex_number wikiPageWikiLink Karen_Parshall.
- Hypercomplex_number wikiPageWikiLink Kleins_encyclopedia.
- Hypercomplex_number wikiPageWikiLink Lie_group.
- Hypercomplex_number wikiPageWikiLink Lorentz_transformation.
- Hypercomplex_number wikiPageWikiLink Lorentz_transformations.
- Hypercomplex_number wikiPageWikiLink Mathematics.
- Hypercomplex_number wikiPageWikiLink Mathematics_Magazine.
- Hypercomplex_number wikiPageWikiLink Matrix_(mathematics).
- Hypercomplex_number wikiPageWikiLink Max_Herzberger.
- Hypercomplex_number wikiPageWikiLink Maximilian_Herzberger.
- Hypercomplex_number wikiPageWikiLink Maxwells_equations.
- Hypercomplex_number wikiPageWikiLink Multicomplex_number.
- Hypercomplex_number wikiPageWikiLink Nilpotent.
- Hypercomplex_number wikiPageWikiLink Norm_(mathematics).
- Hypercomplex_number wikiPageWikiLink Number.
- Hypercomplex_number wikiPageWikiLink Number_system.
- Hypercomplex_number wikiPageWikiLink Octonion.
- Hypercomplex_number wikiPageWikiLink Orthogonalization.
- Hypercomplex_number wikiPageWikiLink Pauli_algebra.
- Hypercomplex_number wikiPageWikiLink Pauli_matrices.
- Hypercomplex_number wikiPageWikiLink Phase_(waves).
- Hypercomplex_number wikiPageWikiLink Quadratic_form.
- Hypercomplex_number wikiPageWikiLink Quantum_mechanics.
- Hypercomplex_number wikiPageWikiLink Quaternion.
- Hypercomplex_number wikiPageWikiLink Real_number.
- Hypercomplex_number wikiPageWikiLink Richard_Brauer.
- Hypercomplex_number wikiPageWikiLink Rotation.
- Hypercomplex_number wikiPageWikiLink Sedenion.