Hypercomplex numbers

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 (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 ...
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