― Aldous Huxley

- Obfuscation.

- Storing signed numbers in unsigned fields (nah, not really).

So this may eventually find a use, likely with higher-dimensional math.

https://en.m.wikipedia.org/wiki/Fundamental_theorem_of_algeb...

The first use of negative roots was to find the (real) roots of certain polynomials. They were considered a mathemathical hack: useful, but specious on their own. And then it turned out that if you take them on their own terms, they're fantastically useful.

So there's no possible way this could be useful for "higher dimensional math" or anything, because there's no entity being added here. Higher dimensional math already uses numbers. At most there might be one or two concepts that might in some manner if you squint hard enough could be better represented by negative base numbers, but even then the positive base representation would probably still be how people actually understood it.

That's what I was thinking.

Highly unlikely, but fun to think about.

Actually, another neat example is analog vs digital computation. your models can get very different answers. I recall some Mandelbrot guy talking about that.

I think the ideal wikipedia page for this kind of thing has a 'Properties' section from which to start thinking about this

Different representations of the same thing are not useless; one might make an argument (which I can only back up off the top of my head with mathematical examples, but I suspect that there are also many in the physical sciences) that they are at the root of much progress. The canonical example is to try to do positive-integer arithmetic with Arabic versus Roman numerals; they represent exactly the same thing, but I'll bet you can compute 16 ⨉ 17, but not XVI ⨉ XXIII (without converting), in your head.

    XVI ⨉ XXIII
  = XVI ⨉ (XVI + I)
  = (XVI ⨉ XVI) + XVI
  = CCLVI + XVI
  = CCLXXII

Note that your calculation for some reason (EDIT: ah, maybe because my Arabic-numeral problem has 16 ⨉ 17?) replaces XXIII = 23 by XVII = 17, and then black-boxes the calculation XVI ⨉ XVI = CCLVI (which I at least wouldn't know without converting).

      1 6
    1 1 6
    7 7 42
    1, 6+7, 42 = 100 + 130 + 42 = 272

      X V I
    X C L X
    X C L X
    I X V I
    I X V I
    I X V I
    CC LL(=C) XXXXX(=L) VVV(=XV) III = CCCLXVIII = 16*23=368

There's probably a way to formalize this with an entropy argument: that roman numerals are inefficient encoding. Because given some n-length string of numerals, firstly many are invalid encodings, and secondly among the valid numerals there isn't a uniform distribution from strings to integers. Something like that.

Although it's in some sense the same algorithm, the Roman-numeral version requires far more memorisation. For example, once I know that 2 ⨉ 3 = 6, I know without further memorisation (or, rather, with only a meta-memorisation that generalises readily to other contexts) that 20 ⨉ 30 = 600; but, even once I know that II ⨉ III = VI, I have to memorise separately that XX ⨉ XXX = DC.

> This supports the arguments that Arabic numbers really are better suited for things like multiplying. They don't have the property that multiplication is convolution

I think "They" here is "Roman numerals", not (as the structure seems to suggest) "Arabic numerals", right?