I recall sitting in (I think) the garden with Nanxi trying to solve through what these numbers could mean. Intelligently, she figured out how the Babylonians were doing this table of multiplication on base 60 - I'm kicking myself now that I didn't take a picture of her work because she also discovered how the decimals worked as well. As we go down the table from the beginning we see that 2 times 30 equals 60, 3 times 20 equals 60 and so on. When we get to 8, we find our first decimal which is interpreted as 8 times 7 equals 56 and 8 times 30/60 is 4; 56 plus 4 is equal to sixty. I believe there are certain numbers missing because they are prime or have these prime numbers as their factors. For example, 11 is missing and 22, 33, 44 and 55 are also missing. Although, I am not sure why they are missing but I think I just thought of an idea while writing this sentence out.
If we take the number 11, we could say that whole number wise, it fits into 60, 5 times (which would give 55). Now we need 5 more, the closest to which we could get by multiplying 11 by 27/60 (which gives 4.95). We are still left with 0.05 to make up and our current answer would look like 5, 27. We can already see that to get to 0.05 we would be doing another 27/60 at the third (or fourth?) place value which would yield another decimal value with 5 - we end up having a repeating decimal similar to the fraction 1/3. Babylonians I'm thinking, must have identified that there would be numbers with repeating decimals and thus excluded them from the table. What I think is fascinating is that they also identified that other numbers that are multiples of these numbers would also end up with repeating decimals and chose to exclude them as well.
Their fraction notation is similar to ours with the exception that they only simplify as far as base-60. Our 3/4 is their 45/60, our 1/2 is their 30/60 and so on. Wait, then there is a difference because our 3/4 is represented as 0.75 while theirs is represented as ", 45". Because we use a base-10 system, our decimals are over powers of 10 while Babylonians construct their decimals over powers of 60.
A table of 45 would look (I think) like this:
Interestingly enough, I'm not sure if I'm correct but I was noticing that with this odd-number base, the answers in the right side could only contain odd numbers. As soon as you encountered an even number, the decimal would most likely start repeating itself.
Ok -- very interesting comments! Some of the pairs don't seem to multiply to 45 (for example, 4 X14 is already 56), but at this point, I will let that part go as you have certainly thought about interesting aspects of rationality/ irrationality base 60!
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