Hyperkomplex

Hyperkomplex refers to a family of number systems that extend the concept of complex numbers. These systems are constructed by successively augmenting the number of basis elements. The real numbers form the first level. Augmenting with one imaginary unit yields the complex numbers. Further augmentation leads to quaternions, which have three imaginary units, and then octonions, with seven imaginary units. Each successive system is derived by a doubling process.

While complex numbers are commutative and associative, quaternions are associative but not commutative. Octonions, however, are neither commutative nor associative. This loss of properties makes higher-level hypercomplex systems more challenging to work with algebraically. The structure of these systems is related to division algebras, which are algebraic structures where every nonzero element has a multiplicative inverse. The real numbers, complex numbers, quaternions, and octonions are the only finite-dimensional real division algebras.

The study of hypercomplex numbers has applications in various fields, including physics, computer graphics, and engineering. For instance, quaternions are widely used for representing rotations in three-dimensional space, which is crucial in robotics and animation. Octonions, despite their non-associativity, appear in certain areas of theoretical physics, such as string theory and M-theory. The concept of hypercomplex numbers provides a framework for exploring algebraic structures beyond the familiar real and complex number systems.