OCTONIONS Meaning and
Definition
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Octonions, also known as Cayley's octaves or the eight-dimensional division algebra, are a type of mathematical object that extends and generalizes the concepts of real numbers, complex numbers, and quaternions. They form a non-associative, non-commutative algebraic structure that possesses eight dimensions.
Unlike real numbers, where each element is solely expressed as a scalar, octonions introduce the concept of imaginary units. In this eight-dimensional system, the multiplication of these imaginary units follows specific rules defined by their multiplication table. Each octonion can be represented as a linear combination of these units, where the coefficients are real numbers.
The multiplication of octonions is non-commutative, meaning that changing the order of the multiplication affects the result. Octonions also violate associativity, meaning that the order of operations matters during multiplication. Consequently, octonions do not possess fundamental properties like division or a well-defined modulus.
Mathematicians have found various applications for octonions, particularly in theoretical physics, string theory, and some areas of mathematics. Researchers have also discovered connections between octonions and other mathematical structures, such as exceptional Lie groups and Jordan algebras. Octonions offer a unique and intriguing algebraic system that provides insights into complex mathematical phenomena and continues to be explored in both theoretical and applied research.
Etymology of OCTONIONS
The word octonions is derived from the Latin word octo meaning eight and the suffix -onion, which refers to a type of mathematical object. The term octonions was coined by mathematician John T. Graves in 1843, who generalized the concept of quaternions (which were discovered by mathematician William Rowan Hamilton) to eight dimensions. Therefore, the word octonions literally translates to eight quaternions.
