IANAM...but I do dabble. I was reading a good explanation of imaginary numbers, and the question occurred to me. The phrasing of my question insinuates not that I'm the first to come up with this question, but I do lack the math vocabulary to search for this topic efficiently. Thanks.
Posted Jul 15, 2011, 12:31 AM UTC
The z-axis?
maybe I'm just thinking about this too much, or maybe it's just my limited grasp of imaginary numbers. They are used to solve problems outside of the euclidean geometry and trig/physics space. They're used for signal analysis, electromagnetics, etc. I'm just wondering if the third dimension provides any assistance in non-geometric problem solving...does that make sense?
Only works in dimensions a power of 2. See Hurwitz's theorem and the Cayley-Dickson construction.
I'm going to take an aspirin and try to read those again in about an hour...but it does seem to be what I was looking for. Thanks for the links!
In layman's terms: you can make a 1-dimensional number system (the real numbers), a 2-dimensional number system (the complex numbers), a 4-dimensional number system (the hamiltonions), or an 8-dimensional number system (the octonions). There are no others[.](http://a.b edit: no others that are finite dimensional, at least.)
If you try to add on a third dimension to the complex numbers, you won't be able to make the result into a number system.
That is, you can't have addition, subtraction, multiplication, and division.
(To this humble mathematician, it seems that the universe is biased towards small powers of 2...)
I'm not sure if they pop up anywhere else in maths. But I know in physics etc, that's all there is. It's mainly to solve the dilemma with the square root of negative numbers. Once that is done, the whole number line is covered.
If you are talking about quaternions, then there are two more, namely 'j' and 'k'...