6 MATHEMATICS IN MESOPOTAMIA Seleucid period a special separation symbol was introduced to indicate the absence of a number.Thus 1-60*+0-60 +4 =3604.However, even in this period no symbol was used to indicate the absence of a quan- tity at the right-hand end as in our 20.In both periods the content had to be relied upon to indicate the precise value of the entire number. The Babylonians used positional notation to represent fractions also.Thus intended as a fraction meant 20/60 and,intended as a fraction, could be 21/60 or 20/60 1/602.The ambiguity in their number system is therefore even greater than previously indicated. A few fractions had special symbols.Thus one finds: These special fractions,1/2,1/3,and 2/3,were to the Babylonians"wholes," in the sense of measures of quantities,and not a division of unity into parts, though they arose as measures of quantity having these respective relations to another quantity.Thus we might write 10 cents as 1/10 in relation to the dollar but think of the 1/10 as a unit in itself. Actually the Babylonians did not use base 60 exclusively.Sometimes the years were written 2 me 25,where me stood for hundred-in our symbols,225. Also lfmu was used for 1000,generally in nonmathematical texts,though it occurs even in mathematical texts of the Seleucid period.Sometimes 10 and 60 were mixed,as in 2 me 1,10,which means 2 x 100 +1 x 60 +10 or 270.Mixed systems involving units of 60,24,12,10,6,and 2 were used for dates,areas,measures of weight,and coinage,just as we use 12 for hours,60 for minutes and seconds,12 for inches,and 10 for ordinary counting.The Babylonian systems,like ours,were a composite of many historical and regional customs.However,in the mathematical and astronomical texts they did use base 60 quite consistently. We do not know definitely how base 60 came to be used.The suggestion may have come from the systems of weight measures.Suppose we have a system of weight measures with values in it having the ratios 1/2,1/3,2/3,1,10. Now suppose there is another system with a different unit but the same ratios,and political or social forces compel the fusing of the two systems.(We have meters and yards,for example.)If the larger unit were 60 times the smaller,then 1/2,1/3,and 2/3 of the larger unit would be integral multiples of the smaller one.Thus the larger unit might have been adopted because it was so convenient. As to the origin of positional notation,there are two likely explanations. In an older scheme of writing numbers I multiplied by 60 was written with a larger than the same symbol for 1.When the writing was simplified the larger was reduced in size but kept in the usual place for 60.Hence position