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Comment by empath75 | Hacker News Reader

I can’t imagine there ever would be.

I don't think anyone ever imagined a use for imaginary numbers either, but those turned out to be quite useful for reducing dimensionality. Towards the bottom of the article it states that Donald Knuth proposed imaginary base numerical systems.

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

Imaginary numbers were useful as soon as they were invented. They are algebraically closed, unlike the real numbers, and are required for the fundamental theorem of algebra.

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.

In addition to the other points, imaginary numbers immediately added to the representation of things we could represent. Simply rewriting existing numbers in other ways isn't all that interesting; it doesn't add anything. We already knew there's an infinite number of ways to serialize numbers to symbols. Negative base numbers don't behave differently than positive base numbers, because "negative base numbers" and "positive base numbers" are artifacts of English and don't actually exist mathematically. There are numbers, which are written in different representations, but are not actually different entities.

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.

> 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,

That's what I was thinking.

It will be really funny if there is some bit of new, undiscovered physics, that went undiscovered for so long just because it's best described using some esoteric maths like complex base numbers.

Highly unlikely, but fun to think about.

Are there any examples where the representation of a number matters in physics?

I recall some effort in moving from imperial measurement to metric. Reality itself does not care, but the math sure gets easier if you carefully select units.

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

See Gaussian (cgs) units for an example, compare Coulomb's law in cgs and SI: https://en.wikipedia.org/wiki/Gaussian_units#Unit_of_charge

Well, the number of digits tell you the sign of the number, right? It could save a sign bit

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

The bit saved for the sign is lost because you need more bits to represent a number.

True, and it seems you actually need exactly one extra bit for all positive numbers

So in your mind people are just coming up with these things for no reason, is that it?

It’s fun and interesting to figure out. There are lots of ways to represent numbers but they don’t change the thing they represent.

> There are lots of ways to represent numbers but they don’t change the thing they represent.

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

I didn't say that it wasn't computeable, only that I bet you couldn't do it in your head.

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).

I agree. You can use the same algorithm for computing both. But Arabic has better constants. Because it's a discrete convolution instead of grouping and aggregating.

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

Same process in Roman

      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

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, so you can't even do things like truncate your computation to get an approximation.

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.

> I agree. You can use the same algorithm for computing both.

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?

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