Search FQXi


Malcolm Riddoch: "JBW: “The ‘measurement’ of a property does not ‘exist’ until a..." in Alternative Models of...

Al Schneider: "I have had this idea for some time and worked on it a lot over the years...." in Alternative Models of...

Peter Morgan: "Note that the author link,,..." in An algebraic approach to...

Zeeya Merali: "This is a place to discuss the article, "An algebraic approach to Koopman..." in An algebraic approach to...

Joe Fisher: "A visible person could dig or bore a fixed hole with visible sides in a..." in The Quantum Agent

Robert McEachern: "Eckard, I would argue that all symbolic representation (and thus all of..." in The Quantum Agent

Joe Fisher: "The most compelling evidence that the ability of white men of only being..." in Undecidability,...

Jack James: ""That is, instead of a world or physical laws, it is the local state of the..." in Could Consciousness Forge...

click titles to read articles

The Quantum Agent
Investigating how the quantum measurement process might be related to the emergence of intelligence, agency and free will.

First Things First: The Physics of Causality
Why do we remember the past and not the future? Untangling the connections between cause and effect, choice, and entropy.

Can Time Be Saved From Physics?
Philosophers, physicists and neuroscientists discuss how our sense of time’s flow might arise through our interactions with external stimuli—despite suggestions from Einstein's relativity that our perception of the passage of time is an illusion.

A devilish new framework of thermodynamics that focuses on how we observe information could help illuminate our understanding of probability and rewrite quantum theory.

Gravity's Residue
An unusual approach to unifying the laws of physics could solve Hawking's black-hole information paradox—and its predicted gravitational "memory effect" could be picked up by LIGO.

February 22, 2020

The Crazy Old Uncle of Algebra
by Anil Ananthaswamy
September 13, 2009
Bookmark and Share

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."

Comment on this Article

Please read the important Introduction that governs your participation in this community. Inappropriate language will not be tolerated and posts containing such language will be deleted. Otherwise, this is a free speech Forum and all are welcome!
  • Please enter the text of your post, then click the "Submit New Post" button below. You may also optionally add file attachments below before submitting your edits.

  • HTML tags are not permitted in posts, and will automatically be stripped out. Links to other web sites are permitted. For instructions on how to add links, please read the link help page.

  • You may use superscript (10100) and subscript (A2) using [sup]...[/sup] and [sub]...[/sub] tags.

  • You may use bold (important) and italics (emphasize) using [b]...[/b] and [i]...[/i] tags.

  • You may also include LateX equations into your post.

Insert LaTeX Equation [hide]

LaTeX equations may be displayed in FQXi Forum posts by including them within [equation]...[/equation] tags. You may type your equation directly into your post, or use the LaTeX Equation Preview feature below to see how your equation will render (this is recommended).

For more help on LaTeX, please see the LaTeX Project Home Page.

LaTeX Equation Preview

preview equation
clear equation
insert equation into post at cursor

Your name: (optional)

Please enter your e-mail address:
Note: Joining the FQXi mailing list does not give you a login account or constitute membership in the organization.