Octonions and Aliens...

Octonions are an eight-dimensional number system that are non-commutative and non-associative, meaning the order of multiplication matters, and grouping of multiplications can change the result; however, they are considered "alternative" which means they satisfy a weaker form of associativity, and are also power associative; they are considered the largest normed division algebra over real numbers, making them an extension of complex numbers and quaternions.


Key properties of octonions:

Dimensionality:

Eight dimensions, making them a higher dimensional extension of quaternions.

Non-associativity:

Unlike real numbers, complex numbers, and quaternions, the order of multiplication in octonions can affect the result.

Non-commutativity:

Multiplication is not commutative, meaning a * b is not necessarily equal to b * a. 

Alternative property:

While not fully associative, octonions satisfy the alternative property, which means that for any three octonions a, b, and c, the following holds: a(ba) = (ab)a and a(aa) = (aa)a.

Power associative:

Raising an octonion to a power is associative.

Normed division algebra:

Octonions form a normed division algebra, meaning every non-zero octonion has a multiplicative inverse.

Cayley-Dickson construction:

Octonions can be constructed using the Cayley-Dickson process, which is also used to build quaternions from complex numbers. 

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