Comparing bases of arithmetic with a pair of calipers

Looking at the links in the description of this YouTube video about measuring tools in machining, good calipers typically can measure between 0 and 8 inches/200 mm to a precision of 0.001 inch/0.02 mm. So, over the course of the full travel, a caliper has about 8000-10 000 possible measurements, more or less, which takes four or five decimal figures to display.

So, let's use 8000 as our benchmark. Imagine for some reason that we were converting to a new numbering system and needed to build new instruments. How many figures will our calipers need to display in any given base?

• At the bottom end of the scale is binary, naturally. These calipers would have 1 1111 0100 0000 divisions and therefore 13 figures.
• Ternary calipers are already an enormous improvement - 101 222 022 divisions, 9 figures.
• Quaternary brings us to 1 331 000, 7 figures.
• Quinary and senary (a.k.a. seximal), 224 000 and 101 012, are one step better and use 6 figures.
• Septimal (32 216) through nonary (11 868) use 5 figures.
• Decimal (8000) through vigesimal (base twenty, 1000) use 4 figures.
• ...and then it's 3 figures until you get to base ninety.

So, what does this say to me?

I would argue that, for most terrestrial purposes, this degree of precision is a good proxy for how many figures a person doing manual calculations would need to be able to process to complete their tasks. Some calculations are more precise than this, naturally, but outside of fields like accounting or astronomy, they are unlikely to be grossly more precise than this. Therefore, a pragmatic comparison of how difficult manual calculations are should be focused on calculations using this many figures - comparing four figures of long division in decimal to seven in quaternary, or four in decimal to six in senary, or four in decimal to ... four in dozenal. Or hexadecimal. Or vigesimal.

Look, numbers being longer might be a good reason to not go for a small base, but numbers being shorter is a poor reason to go for a big base because the numbers are barely shorter. Returns diminish after decimal, and I think that's really the most important takeaway.

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Edit 2021-02-12: Because it feels a little bit unfair to choose a number so ideally suited to decimal, if we instead bump it up to 15 000 divisions (as would be for a 300mm (~12") metric caliper), this ... adds one figure each to binary, decimal, and bases 21 and 22. And the most significant digit is 1 in all of those cases.

So yes, there's not nothing in it between decimal and vigesimal, but in this range there's still not much.