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Also, octaves are 8-dimensional numbers, just as complex numbers are 2-dimensional and quaternions 4-dimensional, and more dimensions seem not to make any sense anymore, in number theory.

AM: I think you will find that octonions are 8-dimensional numbers, they are also called biquaternions.

Their properties or better the lack of familiar properties is rather curious:

  • Complex numbers can not be ordered: "i < 1" makes no sense
  • Quaternions do not commute when multiplied: a * b != b * a
  • Octonions do not have the associative property: a * (b * c) != (a * b) * c

(This latter lack of properties seems to be responsible for there not being a system of 16-dimensional numbers ...)

The existence of octonions does mean that in 7-dimensional space, just as in 3-dimensional space, but in no others, there exists an out-product for vectors.

(Okay, there are all rather impractical faits divers - but I thought you might like to know :)

AMG: Actually there is a system of 16-dimensional numbers. They're called sedenions. Read about them here: [L1 ]. Octonions aren't associative, but they are alternative: a * (a * b) == (a * a) * b. Sedenions aren't even alternative, but they do have power associativity: a * (a * a) == (a * a) * a. I'm having a hard time imagining an algebra that's not power associative. ;^)

AMG: Terrific quote about octonions:

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. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative. — John C. Baez

AMG: again, with a question for AM or whomever else: When you say "out-product", do you mean "outer product", a.k.a. "cross product"? Or am I confusing my terms?

Lars H: Your interpretation seems correct, AMG, but I'm not so sure about whether AM's claim is...

The thing that is special about the octonions is that nonzero elements have multiplicative inverses, but I can't see a reason why that would be important for making the vector component of multiplication an "outer product"; the scalar (real) component is needed for inverses to exist.

Moreover, the outer product on 3-space is simply a Lie algebra [L2 ] bracket, and Lie algebra structures exist on vector spaces of arbitrarily large dimension. More precisely, the Lie algebra so(3) (the space of infinitesimal rotations of 3-space) has the cross product of 3-vectors as bracket, and in general so(n) is a Lie algebra with n*(n-1)/2 dimensions.

AMG: I won't even pretend that I understand any of that. :^) Maybe someday, though, maybe with the help of the URL you give. Thanks!

AMG, years later: Is a multiplicative inverse like a reciprocal? x * mult_inv(x) = 1. What's the interpretation of "1" in a number system with more than one dimension? Does that mean the sum of the squares of the real components is 1?

AMG: If interested in this stuff, see Cayley-Dickson construction to multiply complex, quaternion, octonion and so forth.

slebetman: Actually, an octave is one "order of magnitude" for base 8 numbers (octals) just like decade is one order of magnitude for base 10 numbers. For example, the octal number 021 is two octaves above zero.

AMG: How about musical octaves? Halving or doubling a note's pitch produces another note that is an octave interval from the first. In Western music at least, the octave is divided into twelve semitones, of which seven are used in any given major or minor key scale. Therefore, if a "root" note is called 1, its octave would be 8.