What are they and how are they used? Anyone know?
If you noodle around at John Baez's website, you will eventually come across a discussion on this topic. By dropping commutativity or associativity, you can move from complex numbers to hypercomplex numbers. The most familiar ones are the quaternions and the octonions. They have some properties that may be useful for modeling certain phenomena in particle physics, IIRC.
Here is a spiffy website.
Here's what good ol Mathworld has to say about quaternions:
Has anyone thought of doing calculus with quaternions? When going from real numbers to complex numbers you don't have to give up very much; you only have to give up ordering. When using quaternions you only have to give up commutativity, which isn't bad. Actually quaternions are still anticommutative... Nobody's done calculus/analysis with quaternions yet though, sounds like maybe it could be worth a PhD or two to prove some theorems about 'quaternion calculus' or something.
You should be careful about such statements. Hamilton did quite a bit of calculus/analysis with quaternions when he thought them up. They were at one point very popular for doing physics - a natural fit, with 1 real dimension for time and three imaginary dimensions for space. As time when on though, it became apparent that the real part of the quaternion field was always constant, and so it was dropped. Also, quaternion multiplication was short on use, except for when one multiplicand was real. So drop the real coordinate, and drop the multiplication except for by real numbers, and what do you have left? And lo, the vector was created, and it was good.
Today, quaternions are mostly used by game designers, who use them for calculating scene rotations, rather than rotation matrices. They say it works better, but I'm not sure why, since the calculations are entirely equivalent. I can only suppose the way it is calculated is more stable numerically.
And just a note; the quaternions are not anticommutative. For instance, 1 * i != -(i * 1), and i * i != -(i * i).
No matter what Hamilton did, I don't there's a quaternion version of integration or differentiation. That's the sort of thing I meant.
quaternionic integration is easy; just like the complex integral it can be defined simply by virtue that they form a vector space over the real numbers.
Differentiation has been studied for quaternions as well. The same definition that works for the Reals and Complex numbers still works for the Quaternions, and even for Cayley numbers (an 8-dimensional extension of the Reals which sacrifices full associativity - I'm guessing these are the "octonions" Janitor refered to, though I am unfamiliar with that term).
If anyone wants to read Hamilton's work on quaternions, here is a link to it
http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05230001&seq=9Over 700 pages of all the quaternions you can eat.
"Octonion" is clearly from the Latin word for "eight," just as "quaternion" is from the Latin word for "four." Icarus is correct about the octonions being Cayley numbers. This site says there is actually a different, less commonly-encountered, type of Cayley number as well.
Hypercomplex Numbers and Applications
A detailed description of communtative-associative hypercomplex numbers in user-defined dimensions and corresponding signal processing applications may be found on the web page at www.hypercomplex.us. You can download a white paper providing an excellent overview of this topic, and an evaluation copy of a toolkit providing hypercomplex numerical computing capability for the Matlab numerical computing environment. Because Gauss-Jordan elimination applies, the toolbox provides the unique capability to solve many types of inverse problems for multidimensional, multivariate signals - again, in any dimension selected by the user.
A definition of 4-dimensional, commutative-associative hypercomplex numbers as they relate to electromagnetic theory and special relativity at http://home.usit.net/~cmdaven/cmdaven1.htm.
Quanternian electromagnetic equations (4-dimensions, not-commutative) are given at http://www.hypercomplex.com.
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