hypercomplex

Hypercomplex is a broad term used to describe number systems that extend the complex numbers. In mathematics, hypercomplex number systems are finite-dimensional algebras over the real or complex numbers, and they often feature multiplication that is not commutative or not associative. The term encompasses a range of constructions, from well-known examples to more general algebraic frameworks such as Clifford algebras, split-complex numbers, and dual numbers. While some hypercomplex systems form fields, many do not, and may include zero divisors or lack associativity.

Among the most studied hypercomplex systems are quaternions and octonions. Quaternions, discovered by William Rowan Hamilton in 1843, are a four-dimensional real division algebra with basis 1, i, j, k and multiplication rules where i^2 = j^2 = k^2 = ijk = -1. They provide a robust representation of three-dimensional rotations and are widely used in computer graphics, robotics, and navigation. Octonions, eight-dimensional and non-associative but alternative, extend quaternions and have applications in certain areas of theoretical physics and geometry. Higher-dimensional Cayley-Dickson constructions lead to sedenions (16-dimensional), which fail to be division algebras due to the presence of zero divisors.

Other related systems include split-complex numbers, where a squared unit equals one, and dual numbers, where the square of the unit is zero. Clifford algebras unify many of these ideas and play a central role in geometry and mathematical physics. Hypercomplex analysis is the study of function theories on these algebras, generalizing aspects of complex analysis to higher dimensions.

In contemporary usage, hypercomplex serves as an umbrella term for finite-dimensional real or complex algebras that extend the complex numbers, reflecting a wide spectrum of algebraic and geometric structures.