A Hypothesis Explaining The Origin Of The Babylonian Number System

A Hypothesis Explaining the Origin of the Babylonian Number System James A. Montanye Abstract – The author argues that the ancient Babylonian number system emerged from the need of ordinary individuals to maximize the range of magnitudes that could be represented on the fingers of two hands.

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umerologists, anthropologists, archaeologists, historians, and mathematicians ponder the peculiarities of the ancient Babylonian number system, whose most primitive origin so far

has escaped positive explanation. The historian Daniel Boorstin, commenting on the origin of the twenty-four hour day, described the puzzle in this way: The Egyptians...apparently chose this number [twenty-four] because they used the sexagesimal system of numbers, based on multiples of six and powers of sixty, which had been developed by the Babylonians. This pushes the mystery back into earlier centuries, for we have no clear explanation of why the Babylonians built their arithmetic as they did. But their use of the number sixty seems to have had nothing to do with astronomy or the movement of heavenly bodies. (Boorstin 1983, 41) Pace Boorstein, math historians speculate that astronomy and geometry are likely sources for the sexagesimal number base: The assumption of Babylonian astronomers that the year had 360 days is very likely the origin of measuring angles in degrees; the fact that the angles of equilateral triangles are 60 degrees may explain, in part, the sexagesimal method of counting. (Gullberg 1997, 364) Other investigators consider anatomical explanations: Anthropological and archaeological studies of human culture have found the common number bases to be 2, 5, 10, and somewhat less frequently 20. These correlate nicely with features of the human anatomy. ... Put another way, people had bodily references in case their arithmetic failed them. But why 60? (Dunham 1994, 184)

Sumerologists assert that the sexagesimal (base-60, or “powers of 60”) system arose to meet the empirical needs of the Mesopotamian state, especially for public administration and astronomical calculations: Throughout the third millennium, counting and measuring systems were gradually revised in response to the demands of large-scale state bureaucracies. This development led in the end to the sexagesimal, or base 60, place value system ... (Robson 1999, Preface) Nothing in this explanation addresses the earlier “multiples of 6” (base-6) number system. Neither does it explain the sexagesimal system’s longevity and its many task-specific variations (Nissen and others 1993; Robson 2008), despite the system being at once awkward, prone to error, and employing separate number bases for computation (base-60) and record-keeping (base-10): [C]alculations in sexagesimal notation were made on temporary tablets which were then moistened and erased for reuse after the calculation had been transferred to an archival document in standard [i.e., decimal] notation. (qtd. in Robson 1999, 171) The sexagesimal place value system was used only for calculating with integers and simple fractions; the absolute decimal system was preferred for recording, and unit fractions were often used for metrologies. Reciprocals and division were avoided by using trial-and-error multiplication even where, in other periods, reciprocals of sexagesimally regular numbers were easily found. (Robson 2008, 212) Reconciling these disparate observations requires a positive explanation of the ancient system’s origin and evolution. A useful method for developing positive theory lies in the economist’s as if postulate that rationality gives rise to efficient social instruments of all sorts, including number systems. Subsequent institutionalization transforms these instruments, both for better and for worse, by adapting them for administrative convenience and corrupting them to confer private, entrepreneurial advantage (Quigley 1979; Scott 1998). This faceted perspective implies that the base-6 number system was the consequence of rational thought and pre-political action. The -2-

subsequent emergence of a Babylonian state bureaucracy necessitated a more powerful number system, which created, in turn, a class of professional mathematicians (Robson 2008) whose income and social status depended upon the arcana of sexagesimal calculation. Hence the system’s longevity and evolved complexity. The rational choice view fosters a positive explanatory hypothesis that fits the facts exactly. The result is not merely a “rational reconstruction” of ancient texts, an exercise disparaged by professional Sumerologists (Robson 2001). Rather, it represents a bootstrapped understanding of a math phenomenon that predates recorded history.

The Hypothesis Rationality and efficiency in the context of Babylonian antiquity is judged according to the needs of a pre-literate and pre-numerate civilization that almost certainly used fingers for counting, calculating, and displaying magnitudes. A base-6 number system would have been the most useful system in this context because it permits magnitudes in the range of 1-35 to be represented conveniently on two hands of five fingers each. A decimal (base-10) system, by comparison, limits finger arithmetic to magnitudes in the range of 1-10. A third possibility is a truly “digital” base-2 (or “powers of 2”) system that represents a broad range of magnitudes on ten fingers (i.e., 210 = 1,024), but is impractical because its application requires both extreme mental computing ability and the finger dexterity of a street mime. A simple proof confirms that a base-6 number system – which progresses naturally in “multiples of six” over the range encompassed by 10 fingers – maximizes the magnitude that can be represented using an x-multiple counting scheme. The magnitude m that can be represented on two hands of five fingers each is given by the equation m = (10 - x + 1)x + (x - 1). Differentiation gives the maximizing equation m'(x) = -2x + 12 = 0 proving that 6 is the value of x that maximizes m. Finger-based experimentation (as described below) yields the same result. Different maximizing values of x can result if finger orientation

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(e.g., the up/down scheme used by commodities pit-traders) is taken into account. The Babylonian system’s number bases of six and sixty imply that finger orientation was irrelevant. Finger counting in base-6 notation is straightforward. Whereas decimal arithmetic represents magnitudes by placing digits in the position of “units” and “tens,” base-6 arithmetic places them in the position of “units” and “sixes.” Some examples illustrate the mechanics. Let the fingers of one hand represent unit values 1-5, while fingers of the other hand represent multiples of 6 (or “carries”). The magnitude 5, for example, is represented by no (zero) fingers on the “sixes” hand and five fingers on the “units” hand ([0 x 6] + [5 x 1] = 5). The magnitude 6 is represented by one finger (or the thumb) on the “sixes” hand and no (zero) fingers on the “units” hand ([1 x 6] + [0 x 1] = 6). The magnitude 20 is represented by three fingers on the “sixes” hand and two fingers on the “units” hand ([3 x 6] + [2 x 1) = 20). And so on. A magnitude as large as 35 can be represented in this fashion ([5 x 6] + [5 x 1] = 35). The much later Roman custom of designating “units” and “tens” with the left hand, and “hundreds” and “thousands” with the right hand confirms the scheme’s practicability. A base-6 counting system cannot represent fractionally small values and values greater than 35, and so is unsuitable for advanced applications. Sexagesimal calculation (base-60, or 10 multiples of 6) increased the power of ancient arithmetic, as the Babylonian’s prowess with mathematical astronomy attests (Swerdlow 1998). The sexagesimal system nevertheless was prone to error and ambiguity because (i) notation was relatively cumbersome, (ii) computation and recording required extensive numerical ability, study, practice, and diligence (Dunham 1994, 183-4; Gullberg 1997, 56-7; Robson 2008), and (iii) the system embodied no concept of (or character for) zero as a place-holder and trailing digit. These difficulties pressured the evolution of a more tractable system. The first adaptation may have been to decimalize (i.e., to transform into “standard notation”) the results of sexagesimal calculations (Nissen and others, 1993; Robson 1999 and 2008). Decimal arithmetic eventually replaced the ancient system in its entirety.

Conclusion A rational and efficient number system arises and evolves within an environmental context. The ancient sexagesimal (base-60) number system developed from an older base-6 -4-

system of finger counting. Decimal (base-10) arithmetic replaced the sexagesimal system, and largely has been replaced in turn by digital (base-2) electronic calculations that are represented as decimal and hexadecimal (base-16) values.

References Boorstin, Daniel. 1983. The Discoverers. New York: Random House. Dunham, William. 1994. The Mathematical Universe. New York: John Wiley & Sons. Gullberg, Jan. 1997. Mathematics: From the Birth of Numbers. New York: Norton. Nissen, Hans, Peter Damarow, and Robert Englund. 1993. Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East. Chicago: University of Chicago Press. Quigley, Carroll. 1979. The Evolution of Civilizations: An Introduction to Historical Analysis, 2d. Indianapolis: Liberty Fund. Robson, Eleanor. 2008. Mathematics in Ancient Iraq: A Social History. Princeton: Princeton University Press. –––. 2001. ‘Neither Sherlock Holmes nor Babylon: A Reassessment of Plimpton 322.’ Historica Mathematica 28, pp. 167-206. –––. 1999. Mesopotamian Mathematics, 2100-1600 BC: Technical Constraints in Bureaucracy and Education. Oxford Editions of Cuneiform Texts, Vol. XIV. Oxford, UK: Clarendon Press. Scott, James. 1998. Seeing Like A State: How Certain Schemes to Improve the Human Condition Have Failed. New Haven: Yale University Press. Swerdlow, N.M. 1998. The Babylonian Theory of the Planets. Princeton: Princeton University Press.

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