Hypercomplex algebras are formed by doubling previously formed algebras.  For example, the Cayley-Dickson construction, (a,b)(c,d)=(ac-bd,ad-bc), forms the sequence of algebras real, complex, quaternion, octonion, sedenion.

Hierarchy of hypergeometric algebras

 . . . . . . the following outline structure might be better represented as a picture (although we should leave the outline structure here, and explain how to derive each algebra from its predecessor).

Reals

Complex

Quaternion

Octonion

Sedenion

Split-quaternion

Split-complex

Double

Dual

Cayley-Dickson construction

 . . . . . .

Kronecker product construction

 . . . . . .

Division algebra

 . . . . . . A division algebra is . . . . . .

Clifford algebras

 . . . . . . related to division algebras

Internet references

Euclidean Space: hypercomplex

Related pages in this website

Complex, Quaternion, Octonion, and Sedenion numbers are n-tuples of real numbers, where n=2,4,8,16, respectively.

 

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