But we’re so used to numbers being completely concrete that the extra degree of freedom seemed to make things more confusing. I kept telling my students that they were allowed to assume numbers had any order of magnitude they wanted.
We spent some time in class calculating Babylonian “reciprocals,” (pdf) and it was surprisingly difficult. I don't know how the people who used this system really thought of reciprocals. Those numbers also multiply together to make 1. Or we could interpret 5 as 5 sixties (300), and 12 as 12/3600. If we interpret 12 as 12/60 (1/5), then we can multiply it by 5 to make 1. Of course, there is a way to interpret the Babylonian symbols for 5 and 12 as reciprocals in the traditional sense, again because of the ambiguity that the lack of zero adds. It would make some sense only if we didn't have a zero, so 1 and 10 looked the same. It would be like us thinking of 4 and 25 as reciprocals because they multiply to 100. Why would this definition of reciprocal make sense? Because when you’re writing in base 60 without a zero, 60 looks just like 1! So do 1/60, 3600, and any other power of 60. But Babylonian reciprocal tables, which made calculations quicker, listed any two numbers that multiplied to a power of 60 as “reciprocals.” For example, 5 and 12 are “reciprocals” in this sense because they multiply to 60. We usually think of reciprocals as number pairs like 2 and 1/2 that multiply together to equal 1. One of the strange consequences of the lack of zero comes up in reciprocals. The oldest documented zero is surprisingly modern: it’s in a temple in India, and it dates from about 875 CE. They never made the leap to using a zero symbol at the end of a number to eliminate the ambiguity completely. But 60 and 1 would always be written identically. That way they could tell the number 3601, which would have been written 1,0,1, from 61, which would be written 1,1. Later, they added a symbol for zero, but it was only used for zeroes that were in the middle of the number, never on either end. In Plimpton 322, the tablet we studied in class, there are some gaps between numerals that represent zeros in the middle of a number, the way the 0 in 101 represents zero tens. I should say that it’s not quite true that the Babylonians didn’t have a symbol for zero. Without it, the numbers we write are inherently ambiguous, and we have to use context clues to figure out what order of magnitude is meant in a given situation. Why do we need a symbol that literally means nothing? But zero is what makes it possible for me to tell 1 from 10 from 1,000,000 from 0.1. The Babylonians didn’t have a symbol for zero.
No, the biggest issue we had was nothing. Image: public domain, via sugarfish and Wikimedia Commons. Unlike the Hindu-Arabic numerals we use today, Babylonian numerals “look like” the numbers they represent.īabylonian numerals are surprisingly easy to decipher. The biggest difficulty my students and I had deciphering these numbers was not the fact that there were so many extra numerals to keep track of. 25 means “two tens, five ones." 52 has the same symbols, but it means "five tens, two ones." Similarly, 1,3 in sexagesimal means “one sixty, 3 ones,” or 63, and 3,57 means “three sixties, fifty-seven ones,” or 237. To us, the digit 2 can mean 2, 20, 200, or 2/10, and so on, depending on where it appears in a number. Their notation is not terribly hard to decipher, partly because they use a positional notation system, just like we do. The Babylonian number system uses base 60 (sexagesimal) instead of 10. Image: Public domain, via Wikimedia Commons.Īs I told my class on Thursday, the theme of the first week of our math history course was “easy algebra is hard in base 60.” We started the semester in ancient Mesopotamia, trying to understand Babylonian* mathematical notation and decipher Plimpton 322, an enigmatic tablet from about 1800 BCE. Plimpton 322, an ancient Mesopotamian mathematical tablet whose purpose is still a mystery.
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