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An ancient Mesopotamian tablet, Plimton 322, which has been the subject of many academic papers over the course of the last few decades, has columns of numbers related to right triangles, but we don’t know exactly how or why the table was created.
It uses the relative utility of base 60, or sexagesimal, versus the base 10, or decimal, system we use today.
To be clear, base 60 has a big advantage over base 10: 60 is divisible by 3, and 10 isn’t. It’s easy to write the fractions 1/2, 1/4, and 1/5 in base 10: they’re 0.5, 0.25, and 0.2, respectively. But 1/3 is 0.3333…. Its decimal representation doesn’t terminate. That really isn’t too much of a problem for us because we are comfortable representing numbers as either decimals or fractions. But the Babylonian number system did not represent fractions in terms of numerators and denominators the way we do. They only used the sexagesimal form, which would be like us only using decimals instead of writing numbers as fractions. In sexagesimal, 1/3 has an easy representation as. It’s 20/60, which could be written as .20 in a sexagesimal system. (It wasn’t written precisely that way by ancient Mesopotamians because they did not have an equivalent to a decimal point.)
The more prime factors, the better when it comes to representing numbers easily using a positional number system like base 10 or 60, but those extra factors come at a cost. In base 10, we only have to learn 10 digits. Base 30, the smallest base that is divisible by 2, 3, and 5 (60 has an extra factor of 2 that doesn’t make a huge difference in how easy it is to represent numbers), requires 30 distinct digits. If we wanted to write fractions like 1/7 using an analogous representation, we’d have to jump all the way up to base 210. Working with so many digits becomes cumbersome very quickly.
Fractions whose denominators only have factors of 2 and 5 have finite decimal representations. Base 12 would be fairly convenient as well. It has prime factors of 2 and 3, and it’s pretty easy to count to 12 on your fingers using the knuckles of one hand instead of the individual fingers.
With base 12, we’d lose the ability to represent 1/5 or 1/10 easily. But 30 or 60, the smallest bases that allow the prime factors 2, 3, and 5, are awfully big. It’s a trade-off. Personally, the idea of having to keep track of 30 or 60 different digits, even if they’re fairly self-explanatory, as the Babylonian digits were, is too much for me, so I’m sticking with 10 or 12. But go ahead and rock the sexagesimal if that's your thing.
Base 60 certainly has that prime advantage over base 10,
Bottom line: Pi, 3.14159265358979 in our base 10 has this for Base 60= 3.
The part that bothered me is that base 2 base 10 or base 60 are all of the domain integer, while Pi is of the domain irrational.
Changing the base of a number cannot transition it from one domain to the other. It's has a finite closed set fixed mapping which can't be altered.
Then I got lazy, and went to Wikipedia. https://en.wikipedia.org/wiki/Sexagesimal
