Around the 1:10 mark, Mansfield says, “We count in base 10, which only has two exact fractions: 1/2, which is 0.5, and 1/5.” My first objection is that any fraction is exact. Perhaps the utility of different types of trig tables is a matter of opinion, but the UNSW video also has some outright falsehoods about accuracy in base 60 versus the base 10 system we now use. But go ahead and rock the sexagesimal if that's your thing.īase 60 certainly has that prime advantage over base 10, but I was annoyed by the way Mansfield overstated that advantage in the promotional video they made to accompany the paper. 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 30 or 60, the smallest bases that allow the prime factors 2, 3, and 5, are awfully big. (One of my math history students wrote a post arguing for a base 12, or dozenal, number system.) With base 12, we’d lose the ability to represent 1/5 or 1/10 easily. 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. Base 12 would be fairly convenient as well. Working with so many digits becomes cumbersome very quickly.įractions whose denominators only have factors of 2 and 5 have finite decimal representations. 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. 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. In base 10, we only have to learn 10 digits. 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. (It wasn’t written precisely that way by ancient Mesopotamians because they did not have an equivalent to a decimal point. In sexagesimal, 1/3 has an easy representation as. They only used the sexagesimal form, which would be like us only using decimals instead of writing numbers as fractions. But the Babylonian number system did not represent fractions in terms of numerators and denominators the way we do. That really isn’t too much of a problem for us because we are comfortable representing numbers as either decimals or fractions. Its decimal representation doesn’t terminate. 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. To be clear, base 60 has a big advantage over base 10: 60 is divisible by 3, and 10 isn’t. Specifically, I was irritated at the strange remarks one of the researchers made about the relative utility of base 60, or sexagesimal, versus the base 10, or decimal, system we use today. In my post, I criticized the publicity video the researchers made to accompany the release of the paper. This ancient Mesopotamian tablet, 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. Last month, I wrote about the hype surrounding a new paper about the much-studied tablet Plimpton 322.
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