Instead of looking at i as a square root of minus one, we view multiplication by i as a unary function that takes 4 applications before returning to the initial argument, or, if you like, we view i as the solution of i4-1=0 rather than i2+1 = 0.
This is nicely shown in this diagram. The red arrow is the unary function *i and makes a 4 element loop. We call i the rotational constant of order 4.
Similarly we can define a unary function *j that creates a 3 element loop. We call j the rotational constant of order 3. (Note that the name j has nothing to do with quaternions, but i is the familiar imaginary number.)
These “rotational numbers of order 3” are three-dimensional, and we can add them and multiply them with scalars and rotational constants. Here are some examples:
(a + jb + j2c) + (d + je + j2f) = (a+d) + j(b+e) + j2(c+f)
(a + jb + j2c) * 2 = 2a +j2b + j22c
(a + jb + j2c) * j = c + ja + j2b
We do not allow multiplication of rotational numbers with each other.
We will generalise this to define one rotational constant rn per loop order n, such that rnn=1.
Our system is conceptually similar to cyclotomic fields, expect that the unity roots are separate dimensions rather than complex numbers, which seems much more natural. All integers and rotational numbers are created with the same justification for their existence and free from contradictions. Defining i as the square root of -1 breaks the rules of the square root function and has delayed the acceptance of complex numbers by hundreds of years. In fact Descartes coined the name “imaginary” as a derogatory term.[[i]]
Mathematicians have tried to generalise the extremely useful complex numbers before. An extremely general approach is “Complex Numbers in n Dimensions” by Silviu Olariu [[ii]]. He offers several ways for some dimensions. His 3-dimensional case is exactly equivalent to our r3 generated numbers. He offers 4 different ways to arrive at four dimensions, one of which is equivalent to our r8 generated numbers. There are other cases because he allows different constants that have the same loop order.
The most famous example of an extension of the complex numbers are Hamilton’s quaternions, which also contain multiple loop orders. A quaternion contains 4 constants; one is 1 and the other three are all loop-4 numbers. All are defined as . However multiplication of quaternions is not commutative. Basically quaternions are three complex planes sharing the “1” dimension.
The popularity of quaternions is partly due to the property that they are closed under division. For algebras with commutative multiplication it has been proven that there are none other than the complex numbers that are closed under division [[iii]].
[[i]]Boyer and Merzbach, A history of mathematics
[[ii]] Silviu Olariu, Complex Numbers in n Dimensions, available at google books