Hypercomplex number

In mathematics, a hypercomplex number is a traditional term for an element of a unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory. In mathematics, a hypercomplex number is a traditional term for an element of a unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory. 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 says finite-dimensional real composition algebras are the reals ℝ, the complexes ℂ, the quaternions ℍ, and the octonions ?, and the Frobenius theorem says the only real associative division algebras are ℝ, ℂ, and ℍ. In 1958 J. Frank Adams published a further generalization in terms of Hopf invariants on H-spaces which still limits the dimension to 1, 2, 4, or 8. 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 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'. In 1973 Kantor and Solodovnikov published a textbook on hypercomplex numbers which was translated in 1989. Karen Parshall has written 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. A definition of a hypercomplex number is given by Kantor & Solodovnikov (1989) as an element of a finite-dimensional algebra over the real numbers that is unital and distributive (but not necessarily associative). Elements are generated with real number coefficients ( a 0 , … , a n ) {displaystyle (a_{0},dots ,a_{n})} for a basis { 1 , i 1 , … , i n } {displaystyle {1,i_{1},dots ,i_{n}}} . Where possible, it is conventional to choose the basis so that i k 2 ∈ { − 1 , 0 , + 1 } {displaystyle i_{k}^{2}in {-1,0,+1}} . A technical approach to hypercomplex numbers directs attention first to those of dimension two. Theorem::14,15 Up to isomorphism, there are exactly three 2-dimensional unital algebras over the reals: the ordinary complex numbers, the split-complex numbers, and the dual numbers. In particular, every 2-dimensional unital algebra over the reals is associative. for some real numbers a0 and a1.Using the common method of completing the square by subtracting a1u and adding the quadratic complement a12 / 4 to both sides yields