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Coincidentally, I am reading a leading history of mathematics, and just read the section on Mesopotamian math (the source of our base 60 customs, such as time,

Coincidentally, I am reading a leading history of mathematics, and just read the section on Mesopotamian math (the source of our base 60 customs, such as time, degrees in circle, etc.). The Sumerians (of Mesopotamia, as was Babylon) not only used base 60, but also were the first known to use position or place value to indicate power of 60 the same way we do in base 10, a significant invention. For contrast, the Romans did not use position: IV is 4, not 15. Thus 11 (imagine Sumerian characters; sorry I don't know my Unicode well enough) in Sumerian is 61 in our contemporary system, and 563 in Sumerian would be 18,363 today. I'm impressed: Could people work with these numbers in their heads? Did someone doing Sumerian addition and multiplication have to memorize 60 x 60 tables, the way we memorized 10 x 10 tables in grade school? (The historian is silent on this matter.) > if you point just the joints, not the tips, you get base 12. / If you count multiples of 12 with the five fingers in the other hand, you get to 60. / And this is the reason we have the 12-hour ... The historian doesn't mention this (and how would anyone know today how the Mesopotamians counted on their hands?), but does say, "There is reason to believe that this choice of 60 rather than 10 as a unit occurred in an attempt to unify systems of measure, although the fact that 60 has many divisors may also have played a role." Other societies commonly have used base 20 (my speculation: maybe natural if you are barefoot or wear sandals) and base 5. I highly recommend the book, the leader in its time (1987) and maybe still today: A Concise History of Mathematics, 4th revised Ed., by Dirk Struik

thaumasiotes 10y ago
> Other societies commonly have used base 20 I've seen this claim several times about French. Apparently it's a standard description of the French number system, but it is totally and obviously wrong on the facts. Here are the characteristics you'd see in a base-20 number system: special words for 0/1/2/3/4/5/6/.../19 (the digits); special words for 20/400/8000/maybe 160000/... (the place values). No special word for 100, which is the not-obviously-significant value "50". No special word for 1000, which is the even-less-significant value "2A0". Here are the characteristics you'd see in a base-10 number system: special words for 0/1/2/3/.../9; special words for 10/100/1000/maybe 10000/... . Here's what French has: special words for 0/1/2/.../16; special words for 10/20/30/40/50/60/80/100/1000/... . It's quite plain that French uses a base-10 number system with some irregular number names. Is there a society that actually used base 20 numbers? I also note, in https://en.wikipedia.org/wiki/Sexagesimal#Usage https://en.wikipedia.org/wiki/Sexagesimal#Usage , that the Sumerians didn't have symbols for 0..59 . They represent large numbers as a series of coefficients of powers of 60, but their "digits" have a base-10 internal structure, precisely as we represent minutes taking one of 60 values today ("6:32"). That suggests to me that the language, as opposed to the number writing system, was probably base 10.
- hackuser 10y ago
  Note that I didn't say anything about France. You seem very doubtful that any society would use something besides base 10; why? > Is there a society that actually used base 20 numbers? From Struik, the book mentioned above, here is what I could find quickly: Of 307 number systems of primitive American peoples investigated by W.C. Eels, 146 were decimal, 106 quinary [base 5] and quinary decimal, vigesimal [base 20] and quinary vigesimal. The vigesimal system in its most characteristic form occurred among the Mayans of Mexico and the Celts in Europe. Give the book a try if you are interested in these things; he writes clearly and engagingly, with a personality and sophistication; the short factual passage above isn't representative. With due respect, I'm going to rely on the highly-regarded writing of arguably the leading historian in the field over Wikipedia. None of these number systems map directly to ours; I've omitted plenty of complexity (and Struik probably omitted more).
  - thaumasiotes 10y ago
    > Note that I didn't say anything about France. You seem very doubtful that any society would use something besides base 10; why? That's all you. I'm not doubtful that a society would use something besides base 10, but when you make the same claim that a lot of other people make, you risk being perceived as similar to those other people. The form of my comment is "I hear this same claim a lot from people who clearly don't know what they're talking about. Do you have something better than they have?" I bought the book. Struik doesn't bother to expand on the selection you quoted at all. His citation, however, does have a bit more ( https://archive.org/stream/jstor-2972526/2972526_djvu.txt https://archive.org/stream/jstor-2972526/2972526_djvu.txt ): > The vigesimal system usually occurs in connection with some other system. > Quinary-vigesimal. This is most frequent. The Greenland Eskimo says "other hand two" for 7, "first foot two" for 12, "other foot two" for 17 and similar combinations to 20, "man ended." The Unalit is also quinary to twenty which is "man completed." But 40 is "two sets of animals' paws," 60 "three sets of animals' paws" and so on regularly to 400 where there is an interesting change in the formation of this primary base (20 X 20) from animals back to man, for 400 is "20 sets of man's paws." A list of nahuatl numbers can be found here ( http://www.omniglot.com/language/numbers/nahuatl.htm http://www.omniglot.com/language/numbers/nahuatl.htm ), and it is quinary-vigesimal. Maya numerals, in their symbolic written form, are also quinary-vigesimal (in the same sense that written Sumerian numerals are decimal-sexagesimal -- that is to say, they are purely quinary, but the forms 20 to 24 aren't used), and Celtic numerals (the words) seem to be a weird quinary-decimal-vigesimal hybrid. This doesn't fill me with confidence in Struik. But Celtic aside, I am satisfied that several American number systems were vigesimal and not decimal. >> I also note, in https://en.wikipedia.org/wiki/Sexagesimal#Usage https://en.wikipedia.org/wiki/Sexagesimal#Usage , that the Sumerians didn't have symbols for 0..59 . They represent large numbers as a series of coefficients of powers of 60, but their "digits" have a base-10 internal structure > With due respect, I'm going to rely on the highly-regarded writing of arguably the leading historian in the field over Wikipedia If you looked at the link, you might notice that all I was relying on was pictures of the cuneiform glyphs. But hey, why read when you can just generally slam wikipedia?

thaumasiotes 10y ago

> Other societies commonly have used base 20 I've seen this claim several times about French. Apparently it's a standard description of the French number system, but it is totally and obviously wrong on the facts. Here are the characteristics you'd see in a base-20 number system: special words for 0/1/2/3/4/5/6/.../19 (the digits); special words for 20/400/8000/maybe 160000/... (the place values). No special word for 100, which is the not-obviously-significant value "50". No special word for 1000, which is the even-less-significant value "2A0". Here are the characteristics you'd see in a base-10 number system: special words for 0/1/2/3/.../9; special words for 10/100/1000/maybe 10000/... . Here's what French has: special words for 0/1/2/.../16; special words for 10/20/30/40/50/60/80/100/1000/... . It's quite plain that French uses a base-10 number system with some irregular number names. Is there a society that actually used base 20 numbers? I also note, in https://en.wikipedia.org/wiki/Sexagesimal#Usage https://en.wikipedia.org/wiki/Sexagesimal#Usage , that the Sumerians didn't have symbols for 0..59 . They represent large numbers as a series of coefficients of powers of 60, but their "digits" have a base-10 internal structure, precisely as we represent minutes taking one of 60 values today ("6:32"). That suggests to me that the language, as opposed to the number writing system, was probably base 10.

hackuser 10y ago
Note that I didn't say anything about France. You seem very doubtful that any society would use something besides base 10; why? > Is there a society that actually used base 20 numbers? From Struik, the book mentioned above, here is what I could find quickly: Of 307 number systems of primitive American peoples investigated by W.C. Eels, 146 were decimal, 106 quinary [base 5] and quinary decimal, vigesimal [base 20] and quinary vigesimal. The vigesimal system in its most characteristic form occurred among the Mayans of Mexico and the Celts in Europe. Give the book a try if you are interested in these things; he writes clearly and engagingly, with a personality and sophistication; the short factual passage above isn't representative. With due respect, I'm going to rely on the highly-regarded writing of arguably the leading historian in the field over Wikipedia. None of these number systems map directly to ours; I've omitted plenty of complexity (and Struik probably omitted more).
- thaumasiotes 10y ago
  > Note that I didn't say anything about France. You seem very doubtful that any society would use something besides base 10; why? That's all you. I'm not doubtful that a society would use something besides base 10, but when you make the same claim that a lot of other people make, you risk being perceived as similar to those other people. The form of my comment is "I hear this same claim a lot from people who clearly don't know what they're talking about. Do you have something better than they have?" I bought the book. Struik doesn't bother to expand on the selection you quoted at all. His citation, however, does have a bit more ( https://archive.org/stream/jstor-2972526/2972526_djvu.txt https://archive.org/stream/jstor-2972526/2972526_djvu.txt ): > The vigesimal system usually occurs in connection with some other system. > Quinary-vigesimal. This is most frequent. The Greenland Eskimo says "other hand two" for 7, "first foot two" for 12, "other foot two" for 17 and similar combinations to 20, "man ended." The Unalit is also quinary to twenty which is "man completed." But 40 is "two sets of animals' paws," 60 "three sets of animals' paws" and so on regularly to 400 where there is an interesting change in the formation of this primary base (20 X 20) from animals back to man, for 400 is "20 sets of man's paws." A list of nahuatl numbers can be found here ( http://www.omniglot.com/language/numbers/nahuatl.htm http://www.omniglot.com/language/numbers/nahuatl.htm ), and it is quinary-vigesimal. Maya numerals, in their symbolic written form, are also quinary-vigesimal (in the same sense that written Sumerian numerals are decimal-sexagesimal -- that is to say, they are purely quinary, but the forms 20 to 24 aren't used), and Celtic numerals (the words) seem to be a weird quinary-decimal-vigesimal hybrid. This doesn't fill me with confidence in Struik. But Celtic aside, I am satisfied that several American number systems were vigesimal and not decimal. >> I also note, in https://en.wikipedia.org/wiki/Sexagesimal#Usage https://en.wikipedia.org/wiki/Sexagesimal#Usage , that the Sumerians didn't have symbols for 0..59 . They represent large numbers as a series of coefficients of powers of 60, but their "digits" have a base-10 internal structure > With due respect, I'm going to rely on the highly-regarded writing of arguably the leading historian in the field over Wikipedia If you looked at the link, you might notice that all I was relying on was pictures of the cuneiform glyphs. But hey, why read when you can just generally slam wikipedia?