In such cases, Babylonians were adept at approximating. For example, a seventh cannot be expressed exactly with a finite number of places. Some fractions are not so easy to express in sexagesimal. In our notation, the second place beyond the semicolon stands for three-thousand-six-hundredths.įor example, as one sixtieth is 0 1, a fifth of a sixtieth is 0 0, 12, which is 12 three-thousand-six-hundredths. They were also able to represent all multiples of such.įor example, as a thirtieth is 0 2 seven thirtieths is 0 14.īabylonians also routinely made use of fractional numbers with more than one fractional place. However, Babylonians didn't restrict themselves to pure reciprocals. We also have that a quarter is 0 15, a fifth is 0 12, a sixth is 0 10, a twelfth is 0 5, a fifteenth is 0 4, a twentieth is 0 3, a thirtieth is 0 2 and a sixtieth is 0 1. Similarly \(3\times 20 = 60\) and so a third is 0 20. Thus the sexagesimal representation for a half is 0 30. From this, we see that a half is the same as thirty sixtieths. Since \(2\times 30 = 60\) we have that half of sixty is thirty. The tables above make a number of pure reciprocals very easy to compute. For example 2, 13 45 will represent 2 sixties and 13 and a fractional part of 45 sixtieths. We will use a semicolon to separate the whole and fractional parts. To make life easier for ourselves, we will introduce a notation to indicate which part of a number is fractional. The Babylonians didn't have anything like our decimal point to separate whole and fractional parts of a number. That had to be determined from the context in which the number was used. There was no way to determine from the notation which digits in a number were being considered fractional. Either the minutes and seconds, or just the seconds or perhaps none of the three units may be considered fractional, depending on the context.īabylonian fractions were like this. Therefore, given a time 1:25:59, the part of the time that is considered to be fractional is dependent on context. In this case, a minute or an hour is a whole number of seconds. Again, each minute is divided into sixty equal parts (seconds).įinally, seconds can be taken as the unit of measurement. In this case, we are taking minutes as the unit of measurement with seconds being fractions of a minute. On the other hand, half a minute is thirty seconds. In such a calculation, we are taking hours to be the unit of measurement and minutes are fractions of an hour.Įach hour is divided equally into sixty fractional parts (minutes). Most people know by rote that half an hour is thirty minutes. Perhaps the notation above doesn't seem so unusual if we denote it 1:25:59, representing one hour, twenty-five minutes and fifty-nine seconds. Our system of hours, minutes and seconds is sexagesimal.Įach minute is sixty seconds and each hour is sixty minutes, for a total of three-thousand-six-hundred seconds in an hour. But we actually make regular use of a sexagesimal system in day-to-day life.
Thus, 1, 25, 59 represents 1 three-thousand-six-hundred, 25 sixties and 59.Īt first, this may appear unfamiliar. We'll also use a more familiar representation for each of the sexagesimal digits. Rather than work with these cuneiform numerals, we will use the notation that we introduced previously, with each digit separated by a comma.
In the page on Numerals we saw that the Babylonians had cuneiform numerals that denoted each of the numbers below sixty. In our page on Counting systems we saw that the Babylonians used a sexagesimal (base sixty) system of counting.