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FQXI ARTICLE

September 16, 2019

The Crazy Old Uncle of Algebra

September 13, 2009

Octonions may be the key to understanding the universe, but what are they? There are four number systems in mathematics, each spanning a different number of dimensions, in which division is possible:

* real numbers (1 dimension);

* complex numbers (2 dimensions);

* quaternions (4 dimensions);

* octonions (8 dimensions).

- John Baez at the University of California, Riverside, has written that, "The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete."
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Quaternions are weirder still because they are

Octonions are worse still because they do not have the property of

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