In Defense of Octonions

Abstract

Various authors have observed that the unit of the imaginary numbers, i, has a special significance as a quantity whose existence predates our discovery of it.  It gives us the ability to treat degrees of freedom in the same way mathematically that we treat degrees of fixity.  Thus; we can go beyond the Real number system to create or describe Complex numbers, which have a real part and an imaginary part.  This allows us to simultaneously represent quantities like tension and stiffness with real numbers and aspects of vibration or variation with imaginary numbers, and thus to model something like a vibrating guitar string or other oscillatory systems.  But if we take away the constraint of commutativity, this allows us to add more degrees of freedom, and to construct Quaternions, and if we remove the constraint of associativity, what results are called Octonions.  We might have called them super-Complex and hyper-Complex numbers.  But we can go no further, to envision a yet more complicated numbering system without losing essential algebraic properties.  A recent Scientific American article by John Baez and John Huerta suggests that Octonions provide a basis for the extra dimensions required by String Theory and are generally useful for Physics, but others disagree.  We examine this matter.