packbat | Summarizing our impressions of eleven bases of arithmetic

Here are the conditions of the experiment:

Starting with F₀ = 0 and F₁ = 1, calculate all Fibonacci numbers up to F_[ten] via addition.
Using long division, calcuate the ratio of F_[ten] and F_[nine] to estimate ϕ (the golden ratio).
Repeat for every positional system base that has a Wikipedia page: 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, and 60.

And here are some thoughts.

First of all: 2 is a tiny base - palpably tinier than 3. F₁₀₀ = 1021 and F₁₀₁ = 2001 in ternary; F₁₀₀₁ = 100010 and F₁₀₁₀ = 110111 in binary. That's so many digits. It didn't even matter to us that the individual operations were easier; we had to deal with so many individual operations - both because the divisor was longer and because we needed that many more decimal places - that it was straight-up not worth it to us. For hand calculations, 3 was way better.

Second: 20 is a large base. We didn't expect that because 16 was doable, but ... well, probably in part because we were using Maya numerals instead of extending Hindu-Arabic numerals with Latin letters, it felt enormously bigger. I think if we were fluent, then the Maya numerals would be worth it, and I don't think using Hindu-Arabic numerals + Latin letters would have helped much because fully half the numerals are non-Hindu-Arabic, but at least as we experienced it, it felt like it'd crossed a threshold from "like decimal but bigger" to "like whatever-base-60-is-called-we-never-remember but smaller". Except it wasn't big enough to let us get three decimal significant figures with only two significant figures; we still needed three.

(For base 60 we tried to fake our way through not exactly Babylonian cuneiform numerals. It was a struggle but it wasn't impossible.)

Otherwise: within that range from 3 to 16 where bases were not onerously small nor onerously large, we'd rather go down from ten than up but, as far as this particular problem went, there wasn't a lot between them. Going from 10 to 16 was harder than going from 10 to 12, but it could be done; 6 and down were probably easier than 8 and up but there's not a lot between them.

Interestingly, for the kind of task we did? The whole "which fractions are simple?" question didn't really matter. We were always dividing by large numbers rather than small numbers, so if there was even a repeating fraction, much less a terminating one, it was a surprise. The divisors might have been nice for the large bases - when we were doing the base-60 calculation, we could quickly calculate the divisor times 30 and then step the multiplier up one at a time to get to our target - but for the addition and subtraction we needed for this, it did not matter.

I think as far as practical utility of bases of arithmetic, there is a lot we didn't test by doing this operation, but the stuff we did test was very informative.

Edit 2021-02-12: Here is a Twitch highlight of the stream where we did the experiment, for people who want to watch nearly two and a half hours of arithmetic by hand.