Hypercomplex

Part of speech: adjective

Definitions

1. Relating to or describing mathematical systems that extend beyond complex numbers with multiple imaginary units and noncommutative multiplication
2. Pertaining to algebraic structures that generalize complex numbers into higher dimensions with intricate arithmetic properties
3. Concerning mathematical entities involving advanced number systems beyond complex numbers, often featuring multidimensional components and complex operations

Etymology: "Hypercomplex" is a plain compound: the prefix "hyper-", from Greek "hyper" meaning "over" or "beyond", attached to "complex." The "complex" here carries its mathematical sense, a number of the form a + bi where i is the imaginary unit whose square is -1. "Hypercomplex" therefore names systems that go past that familiar two-part arrangement. The logic of the prefix matches the mathematics. Complex numbers add one imaginary unit to the real numbers, and hypercomplex systems add further units, so that a single number can have several independent components. The term covers structures such as the quaternions, which have three imaginary units, and it allows that multiplication may not commute, meaning the order of factors can change the result.