The following is another reposting from my old blog, where it was the 11th entry. It is a quick summary of the number system I have rashly attempted to put together. It is rather (?) abrupt; for some background, combined with some false starts, see the other entries at the old blog.
A hyperexponential complex number (as I arbitrarily choose to use the term) refers to a number of the form Xn<x+iy>, where n, an integer, is the level, and x+iy, a complex number, is the locus.
The system uses the following basic definitions:
X0<x+iy> = x+iy
Xn<x+iy> = e^(Xn−1<x+iy>)
Thus,
X1<x+iy> = e^(x+iy), and
e^X−1<x+iy> = x+iy.
By extension,
Xm(Xn<x+iy>) = Xm+n<x+iy>.
Using X to represent the set of all hyperexponential complex numbers and taking the numbers of level 0 to correspond to C, the set of all complex numbers, we can write C ∈ X.
The numbers of each level can be represented on a Cartesian hyperexponential complex plane XPn where each number Xn<x+iy> is represented by the point (xn, yn).
Each point (xn, yn) on the plane XPn can be mapped down to a unique point (xn−1, yn−1) on the plane XPn−1 using the equalities
xn−1 = ƒ(xn, yn) = exp(xn) • cos(yn)
and
yn−1 = ƒ(xn, yn) = exp(xn) • sin(yn).
To express this mapping down in terms of hyperexponential numbers, let us use the sign
“\\\” (triple reverse solidus), read “maps down to”:
Xn<xn + iy n> \\\ Xn−1<xn−1 + iyn−1>
If (xn, yn) on XPn maps down to (xn−1, yn−1) on XPn−1, then we say that (xn−1, yn−1) on XPn−1 maps up to (xn, yn) on XPn. To express this mapping up in terms of hyperexponential numbers, let us use the sign “///” (triple solidus), read “maps up to”:
Xn−1<xn−1 + iy n−1>) /// Xn<xn + iyn>
If (xn, yn) on XPn maps down to (xn−1, yn−1) on XPn−1, then so do all the points (xn, yn + k•2π), where k is any integer. Conversely, if (xn−1, yn−1) on XPn−1 maps up to (xn, yn) on XPn, then it also maps up to all the points (xn, yn + k•2π), where k is any integer.
To restate the above in terms of hyperexponential numbers, if Xn<xn + iy n>) \\\ Xn−1<xn–1 + iyn–1>, then Xn<xn + i(y n + k•2π)> \\\ Xn−1<xn−1 + iyn−1> and Xn–1<xn−1 + iyn−1> /// Xn<xn + i
If two hyperexponential numbers of level n map down to the same number of level n–1, we will say they are congruent modulo <2πi>. The modulus expresses the fact that the loci of such numbers differ by an integer multiple of 2πi. We can express this with the usual congruence symbol ≡ as follows:
Xn<xn + i(yn + 2π)> ≡ Xn<xn + iyn> (mod <2πi>)
What I am attempting to propose is a number system where the relationship between e^0 and e^2πi, e^2πi and e^4πi, and similar pairs with exponents differing by an integer multiple of 2πi is viewed as a type of modular congruence rather than equality.
To offer a loose analogy, the proposed relationship between these numbers (of level 1) and the complex numbers (level 0) is like the relationship between a set of unbounded date/time values in measured in days and hours, where 24 hours equals 1 day, and a set of time values measured in hours repeating in a 24-hour cycle, ignoring the number of days. For example, if i corresponds to “6 am,” then e^i(π/2) corresponds to “0 days and 6 hours,” e^i(5π/2) to “1 day and 6 hours,” e^i(9π/2) to “2 days and 6 hours,” and so forth.
By this analogy, the fact that (e^i(π/2))^i is not equal or even congruent to (e^i(5π/2))^i even though e^i(π/2) and e^i(5π/2) are congruent, both mapping down to i, is no more surprising than the fact that, say, half of “0 days and 6 hours” is not equal to half of “1 day and 6 hours” even though both original time values come out to 6 am when measured in the modular 24-hour cycle.
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