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357 N o t e s o f M i d d l e A m e r i c a n A r c h a e o l o g y a n d E t h n o l o g y Carnegie Institution of Washington Division of Historical Research No. 90 April 8, 1948 Did the Maya Have a Zero? The Meanings of Our Zero and the Maya “Zero” Symbols Charles C. Fulton whole number. Some people still think of zero in this way. In this sense it meant “nothing” or “no value” in a purely negative way; the symbol simply had no meaning at all by itself. It was Stevinus who first held that in Arithmetic not unity, but zero, properly corresponds to the point in Geometry. From this has developed our most common modern concept of zero as the starting point of cardinal value. This is also a very restricted sort of “nothing”; just as a geometrical point has no dimensions but does have position. It should be noted carefully that this concept applies chiefly to our modern way of measuring continuous magnitudes . We do not begin with zero to number a file of soldiers; but we do begin with zero to measure lengths, times, and other quantities that we consider continuous or infinitely subdivisible. (Sometimes the zero is a purely conventional starting point of value, as for instance the 0 of the Fahrenheit thermometer .) This whole concept is fairly recent and I think it extremely unlikely that the Maya had any closely similar idea. This thought of zero as a point from which cardinal value takes its origin is, however , frequently read into the Maya symbols. We have also a modern interpretation of the positional zero, no longer claiming that it means nothing at all as compared with the other digits. In the case of measurements, the symbol 0 may be a digit of measurement, representing the closest value for that place just as truly as any other digit does when so used. Some zeros may be used merely to fix the position of the decimal point and are not “significant’”; Did the Maya have a zero? They had several symbols which have been translated by means of our zero, and the real question, naturally, is whether the translations are correct. Perhaps the answer might be either yes or no. Our symbol 0 represents several different concepts, according to the way it is used. The Maya also had at least two “zero” concepts, distinct if not entirely different, and represented by different symbols: these have both been translated “zero” or transliterated with our symbol 0. Spinden and other archaeologists have held that the Maya invented zero long before the Hindus, while at the same time saying that the Maya zero represents not “nothing” but completion. To state my own view briefly at the outset, I believe our modern mathematical concepts of zero do embrace the Maya concepts; but I think that the most common idea of our zero is not at all equivalent to either “zero” of the Maya. Ask anyone what our zero means, and he will probably answer that it means “nothing.” This is indeed the pervading idea, and moreover a meaning that the symbol has had ever since its invention. Yet seldom does the symbol 0 or the word zero represent an absolute nothingness. Except in such cases as the formal contrast between zero and infinity, it is nearly always a very restricted kind of “nothing” or “no value” that is denoted by 0 or the word zero. Our zero began about 1500 years ago as the representation of a mere blank space. It was just a meaningless cipher, used to hold the meaningful digits in their proper places, so as to represent the charlES c. FulTon 358 the others are “significant figures” of measurement with the same status as any of the other digits. The Maya cannot have had this concept except we change from a cardinal to an ordinal point of view. This may be a good place to point out, however, that the Maya zeros were assuredly “significant.” The Maya did not state any “round numbers” as approximations, but as dates to the exact day. Our public also accept, though largely unconsciously , the concept of zero as a number of counting preceding one; that is, an ordinal zero. In such cases the 0 does not necessarily mean “nothing” in any sense. As a pure ordinal it indicates...

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