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FQXI ARTICLE
July 7, 2022

The Crazy Old Uncle of Algebra
by Anil Ananthaswamy
September 13, 2009
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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."

Quaternions are weirder still because they are noncommutative. In mathematics, commutativity is the ability to change the order of terms in your mathematical operation without changing the end result. So, for example, using real numbers, multiplication is commutative because 1 x 2 is the same as 2 x 1. Baez describes quaternions as "the eccentric cousin who is shunned at important family gatherings."

Octonions are worse still because they do not have the property of associativity, which means that you get a different result if you change the order in which you carry out your mathematical operations. Multiplication is associative for the real numbers because, for example, multiplying 1 and 2 together first and then by 3, is the same as multiplying by 1 the product of 2 and 3 (that is, (1 x 2) x 3 = 1 x (2 x 3)). That isn’t true for octonions. "he octonions are the crazy old uncle nobody lets out of the attic," says Baez.

Return to the main article: "Taking on String Theory’s 10-D Universe with 8-D Math."

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