Octonion

Octonions are a type of hypercomplex number, an eight-dimensional extension of the real numbers. They are also known as Cayley numbers or Cayley-Dickson algebras. Octonions are non-commutative and non-associative, which distinguishes them from real numbers, complex numbers, and quaternions.

Octonions can be represented as linear combinations of the basis elements 1, i, j, k, l, li, lj, lk. Multiplication of these basis elements follows specific rules, often visualized using mnemonic diagrams.

Unlike real numbers, complex numbers, and quaternions, the octonions lack the property of associativity. This means that for octonions a, b, and c, in general, (ab)c ≠ a(bc). However, they are alternative, meaning that the subalgebra generated by any two elements is associative.

Octonions are a normed division algebra, meaning that every non-zero element has a multiplicative inverse, and the norm satisfies |ab| = |a||b| for any octonions a and b. This property is crucial for many of their applications.

Applications of octonions include string theory, supergravity, and quantum logic. Their non-associative nature makes them less widely used than complex numbers or quaternions, but they offer unique properties applicable to certain areas of advanced physics and mathematics.