There are some algebraic problems that can't be solved without sqrt(-1). The existence of a+b*i numbers creates a plane. The magnitude of such numbers is often useful. But this is a reduction in the specificity of a+b*i which included a direction, so you know the distance from a centerpoint, but not the direction. pi is the statistical measure of the amount of information you lost in reducing 2 dimensions a and b to a single magnitude. So there should be a binary-like statistical derivation starting from the amount of information lost that uses only numbers of the form 1+i that can derive both e and pi to show that log base i of e^n = 2/(i*pi), the entropy of e^n possible states.
pi/4 = 1-1/3+1/5-1/7...
-pi/4 = sum[ (-1)^n / (2n-1) ]
pi/4 =sum[ i^(2+2n) / (2n-1) = i^2n / (1-2n) ]
1=e^(i*pi*(2N+1/2)) for N=0, 1, 2....
logi(e^(i*(2N+1/2))=1/pi = 1/4*(1-2n)/(i^(2n)) = 1/4 * 1 / (1-1/3+1/5-1/7...)
move i over right and pi over left, and i^ both sides:
e^(pi*(2N+1/2)) = i^(1/i) = i^(-i)
There must be an error above because:
i^i = e^(-pi*(N+1/2)) = 0.20788
i^-i = e^(pi*(N+1/2)) = 1/0.20788
pi = -2*i*ln(i) for N=0
2N instead of N is used next to "i*pi" as the number of "pi's" needed to get back around 360 degrees.
-i/N = pi/2*logi(e) where -i/N is "entropy" per bit location, e is number of states per "i" bit location, and pi was used in finding entropy of black hole surface. So a large number of locations carry less entropy per location because there is really "0" net information in any physical system.
The two i's might be polar coordinates like a photon coming from an origin affected by mass (meters in Maxwell's equations) and charges (some type of i*meters, or meters=i*charge?).