Nega-binary (base -2) existed in the mid-1960s, and maybe earlier, as a candidate for arithmetic circuits. Knuth discuses radix i-1 in TAoCP, as a way to represent complex numbers; the virtue is uniform arithemtic circuits, with no need to separately consider real & imaginary parts. He also mentions Fibonacci base; I don't recall if he treats radix phi. Nega-binary can be viewed in concert with other representations such as sign-magnitude w/wo ones-complement, and the various ways of doing floating point. The world seems to have settled on two's complement, with some churn remaining for floating point. Radix pi was at least mentioned c. 1965. Not good for computing with. Polynomials mod P are sometimes extended with sqrt(x). Rich --------- Quoting Henry Baker <hbaker1@pipeline.com>:
Well, I'm not 100% certain, but *someone* must have written a paper *sometime* about positional number systems using an *algebraic* and/or *algebraic integer* radix and integer numerals.
Knuth? Knuth? Anyone? Anyone?
Several interesting things:
If p(r) is the minimal polynomial for r, and deg(p)=n, then we can express r^n in terms of lower powers of r, and thus there is some possible redundancy in the representations.
Also, if n>1, then there are multiple r's satisfying p(r)=0, so we have to relate representations using r and r', s.t. p(r)=p(r')=0.
Clearly, complex number systems of the 1+i type qualify, but I don't recall any such systems with n>2.
Also, cyclotomic polynomials have the same unfortunate property that base-(e^i) numbers have -- namely, it is a lot more difficult to represent large numbers.
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