[JadeNB]

> 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.

[scentoni]

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

[JadeNB]

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).

[jjaredsimpson]

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.

[JadeNB]

> 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?