# How does quater-imaginary (and other imaginary/complex bases) work?

So I've been working on a simple base-conversion program, and having given it the ability to convert from decimal to any base $> 1$ or $< 0$, as well as the $p$-adic (bijective, I think?) bases, I decided to tackle imaginary systems.

Unfortunately, I can't seem to understand any of the articles I find about them. The Wikipedia article on quater-imaginary is fairly comprehensible (I feel like I could write an algorithm for converting between decimal and it), but there are a couple of things I don't quite understand (why there are 4 digits, for example).

The article on complex bases is a bit more dense. This is mostly because I know very little of the notation used (and trying to Google it proves difficult, since they tend to be symbols more than words). I think I could extrapolate pure imaginary bases from quater-imaginary (once I've a fuller understanding of it), but I have no idea how to work complex bases at all.

So I guess my question is: Does anyone have any good resources on how complex number systems work? (That is to say, algorithms for converting between decimal and them, why some of the seemingly arbitrary decisions (like the number of digits) were made, et cetera). Not necessarily in simple terms, but at least presented in such a way that I could Google or otherwise find the meaning of any of the things I didn't immediately understand without too much trouble. (I'd rather be spending my time trying to learn about something than trying to search for information about it).

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You've tried reading Knuth's original article? –  Ｊ. Ｍ. Nov 5 '11 at 1:43
FWIW: conversion to decimal is always an easy task: you can use Horner's algorithm in general, and digit-grouping for bases that are prime powers. –  Ｊ. Ｍ. Nov 5 '11 at 1:45
I've wanted to look into his article, but it doesn't appear to be freely available, and I'd prefer not to have to spend anything if it can be helped. And yeah, decimal->base_n conversions are the things I've been having the most trouble with. –  Jef Nov 5 '11 at 2:07
You might have luck getting a copy at a nearby (university) library. –  Ｊ. Ｍ. Nov 5 '11 at 16:37
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