Mesopotamian Sexagesimal Numbers, Part 3: Reciprocal Tables, Place-Value Notation, and Late Babylonian Mathematics (c. 2000 to 200 BC), History of Mathematics
Difficulty: Advanced | Prerequisites: Parts 1 and 2 of these study notes.
Source: Friberg, J. (2019) "Three thousand years of sexagesimal numbers in Mesopotamian mathematical texts," Archive for History of Exact Sciences 73:183–216. Sections 10–14.
Tags: place-value notation, floating sexagesimal, table of reciprocals, SM 2685, head numbers, multiplication tables, metrological tables, Old Babylonian, Neo-Sumerian, Late Babylonian, Seleucid, factorization algorithm, reciprocal compensation, CBS 1215, AO 6456, regular numbers, many-place reciprocals
This part covers the most transformative innovation in Mesopotamian mathematics: the invention of sexagesimal place-value notation around 2000 BC, and its consequences over the next 1,800 years. It begins with an atypical table of reciprocals (SM 2685) that may predate place-value notation, shows how Old Babylonian scribes used metrological tables to convert between real-world measures and abstract floating sexagesimal numbers, and ends with Late Babylonian mathematicians computing reciprocals of numbers with 17 sexagesimal places, purely for intellectual satisfaction.
Around 2000 BC, Mesopotamian mathematicians invented place-value notation for sexagesimal numbers by extending the factor-10-and-6 pattern indefinitely and abandoning quasi-integers. The atypical table of reciprocals SM 2685 shows how reciprocal pairs were systematically constructed using reciprocal compensation. Old Babylonian mathematics used metrological tables to bridge traditional measures and abstract place-value numbers. Late Babylonian scribes pushed factorization and reciprocal algorithms to extraordinary lengths, computing with numbers of 17 sexagesimal places.
Place-value notation (floating sexagesimal)
A notation where each position in a number represents a successive power of 60, extending indefinitely in both directions. "Floating" means there is no explicit marker for the ones place, so 1 30 could mean 90, or 1.5, or 5400. Context determined the magnitude. Think of scientific notation but in base 60 and without the exponent.
Table of reciprocals
A list of pairs (n, rec. n) such that n x rec. n = 60 (in the pre-place-value period) or = some power of 60 (in place-value notation). These were the foundation of all division: to divide by n, you multiply by rec. n.
Extended quasi-integer
A number of the form "quasi-integer" or "quasi-integer + quasi-integer times 1 gín." This is how reciprocals were expressed before place-value notation.
Sexagesimally regular twins
A pair of consecutive integers (n, n minus 1) that are both sexagesimally regular. Examples: (8, 9), (15, 16), (24, 25). These were used to construct additional reciprocal pairs.
Metrological table
A conversion table that translates a growing list of capacity, weight, area, or length measures into their corresponding sexagesimal place-value numbers. Old Babylonian students memorised these tables as a core part of their education.
Factorization algorithm
A method for computing reciprocals or square roots of many-place regular sexagesimal numbers by breaking them into their factors and computing the reciprocal (or root) of each factor separately.
Doubling and halving algorithm
A specific form of reciprocal compensation used to generate new reciprocal pairs from known ones: given (n, rec. n), produce (2n, 1/2 rec. n) or (1/2 n, 2 rec. n).
Head number
The first number (multiplier) in an Old Babylonian multiplication table. The set of 40 attested head numbers can be explained as reciprocal numbers from a table like SM 2685.
SM 2685 is a clay tablet from the Sulaymaniyah Museum in Kurdistan, probably a copy of a much older text from before the invention of place-value notation (c. 2000 BC). Every line contains the phrase "igi n gál.bi" ("the reciprocal of n"), confirming it is a table of reciprocals.
The table contains about 55 exact reciprocal pairs (n, rec. n) where n x rec. n = 60, plus two approximate pairs added for pedagogical reasons.
The pairs were constructed systematically in three stages:
Stage 1: Find reciprocals of all 25 sexagesimally regular integers between 2 and 60 (e.g. rec. 2 = 30, rec. 3 = 20, rec. 8 = 7 1/2, rec. 9 = 6 2/3).
Stage 2: Apply reciprocal compensation to generate 19 new pairs where n is a quasi-integer with a basic fraction. For instance, multiplying 8 by 2/3 gives 5 1/3, while multiplying 7 1/2 by 1 1/2 gives 11 15, so rec. 5 1/3 = 11 15.
Stage 3: Use "sexagesimally regular twins" (consecutive regular integers like 8 and 9, 15 and 16, 24 and 25) to construct three pairs with n greater than 33 1/3, such as 53 1/3 and its reciprocal 1 7 1/2.
The table also includes one exceptional pair: 1(60) 21 and its reciprocal 2/3 4 1/3 6 2/3, which is a "twice extended quasi-integer," demonstrating what happens with more complicated cases.
The two approximate pairs (5 1/2 with 10 2/3 15, and 7 with 8 1/2 4) were probably added to teach students about the limits of exact reciprocals.
There are 40 documented head numbers in Old Babylonian multiplication tables: 50, 48, 45, 44 26 40, 40, 36, 30, 25, 24, 22 30, 20, 18, 16 40, 16, 15, 12 30, 12, 10, 9, 8 20, 8, 7 30, 7 12, 7, 6 40, 6, 5, 4 30, 4, 3 45, 3 20, 3, 2 30, 2 24, 2 15, 2, 1 40, 1 30, 1 20, 1 15, 1 12.
Previous attempts to explain this list from the standard table of reciprocals could only account for a subset (those that are straightforward reciprocals of small regular numbers). The discovery of SM 2685 provides a much better explanation: the head numbers can be derived from the reciprocal numbers in an atypical table of reciprocals very much like SM 2685.
This also suggests that the need for multiplication tables keyed to reciprocals may go back even further than the Old Babylonian period, possibly to the Early Dynastic III period when tables of squares (CUNES 50-08-001) required multiplying quasi-integer area values by successive square numbers.
The invention of place-value notation for sexagesimal numbers appears to have been Neo-Sumerian (c. 2000 BC). Metrological tables were introduced either at the same time or within a few hundred years, in the Old Babylonian period.
A metrological table converts a systematically arranged list of traditional measures (capacity, weight, area, or length numbers) into their sexagesimal place-value equivalents. Students memorised these tables as a core part of their education alongside tables of reciprocals and multiplication tables.
The workflow for solving a mathematical problem was:
Convert the given metrological data into floating sexagesimal place-value numbers using metrological tables
Perform the arithmetic in place-value notation
Convert the result back into metrological measures
Example from MLS 1842: "1 ma.na 7 1/2 gín" is silently converted to "1 07 30" in place-value notation. "30 gur" becomes "2 30" (since 1 gur = 5 x 60 sìla = "5 00" sìla, so 30 gur = 2 30 x 1 00 sìla = "2 30").
Some Old Babylonian texts contain no metrological data at all, working entirely with abstract sexagesimal place-value numbers. These include computations of square roots via factorization and construction of new reciprocal pairs using reciprocal compensation, such as CBS 1215, which generates increasingly complex pairs starting from (2 05, 28 48) and reaching (10 06 48 53 20, 5 55 57 25 18 45).
Late Babylonian texts from the second half of the first millennium BC reintroduced and extended the Old Babylonian methods.
W 23021 (Achaemenid period, c. 450 BC) is a school text containing algorithmic computations of reciprocals via factorization. For example, it shows that rec. 52 40 29 37 46 40 = 1 08 20 37 30. The method works by eliminating factors one by one in a left column, then multiplying together the reciprocals of those factors in a right column.
AO 6456 (Seleucid period, c. 200 BC) is an extended table of reciprocals constructed by reciprocal compensation with place-value numbers. The construction:
Compute all powers of 3 and 5 as sexagesimal numbers with at most 6 places, plus their reciprocals
Apply repeated doubling and halving to generate new pairs, as long as at least one number stays within 6 places
The second number of the pair was allowed to have many more places
The results are extraordinary. One entry: rec. 1 29 40 50 24 27 = 40 08 32 44 57 28 29 55 20 09 52 35 33 20 (the number 1 29 40 50 24 27 is the 19th power of 3). Another: a 17-place number paired with a 6-place reciprocal, where the 17-place number is the 22nd power of 20 (= rec. 3).
The table contains roughly 300 pairs, all numbers ordered from 1 to 2 (where the table apparently ended at a companion tablet going from 2 to 3). The same mathematicians who constructed these tables were, in some cases, the same people who built the many-place sexagesimal tables used in Late Babylonian mathematical astronomy.
Friberg identifies the key innovations that made place-value notation possible:
The cuneiform sign for "1(60)" was already hard to distinguish from the sign for "1," which made treating them as positional variants natural rather than confusing.
If the special sign for 10 x 60 was replaced by the ordinary sign for 10, then the repeating alternation of factors 10 and 6 could extend forever in both directions.
Quasi-integers were abandoned in favour of integral multiples of gín, gín.bi, and gín.ba.gín, clearing the way for a uniform fractional system.
The same alternation of 10 and 6 in the factor diagram for counting numbers could be imitated downward for sexagesimal fractions.
Metrological tables were invented to bridge traditional measurement systems and the new abstract place-value numbers.
Two features of the full-blown place-value system were present from the very beginning in the proto-cuneiform/proto-Elamite period: the repeating alternation between factors 10 and 6, and the fact that the sign for 60 was just a bigger version of the sign for 1. The invention of place-value notation was, in a sense, the recognition that these structural features could be exploited systematically.
Students often think place-value notation was "invented" in a single moment of insight. It was not. It emerged from a series of incremental innovations over centuries: the visual similarity of the signs for 1 and 60, the extension of sexagesimal fractions via gín and gín.bi, and the development of metrological tables.
Students sometimes assume that floating place-value notation (with no indication of magnitude) was a limitation the Babylonians wished they could fix. The system worked because context, not notation, supplied the magnitude, and metrological tables handled the conversion. The "floating" quality was a feature, not a bug, for computational purposes.
Students sometimes think the Late Babylonian many-place reciprocal computations were useful for astronomy. They were not directly. Friberg describes them as having "no practical applications whatsoever." They were intellectual exercises, albeit performed by the same people who did the astronomical work.
The five innovations that enabled place-value notation (Section 14) are a likely essay or short-answer topic. Be able to list and explain each one.
The three-stage construction of reciprocal pairs in SM 2685 (Stage 1: integer reciprocals; Stage 2: reciprocal compensation with basic fractions; Stage 3: regular twins) is worth understanding as a process.
The Old Babylonian workflow (metrological data to place-value to arithmetic to metrological answer) is central to understanding how the system was actually used.
The factorization method for computing reciprocals (eliminate factors, multiply their reciprocals) is the ancestor of algorithms still recognisable today.
True or false: In floating sexagesimal place-value notation, the number "1 30" always means 90. (False: it could mean 90, or 1.5, or 5400, depending on context.)
Fill in the blank: To divide by n in the Babylonian system, you multiply by ______. (the reciprocal of n)
True or false: The table of reciprocals SM 2685 uses sexagesimal place-value notation. (False: it predates place-value notation and uses quasi-integers.)
Fill in the blank: The doubling and halving algorithm generates a new reciprocal pair (2n, ______) from a known pair (n, rec. n). (1/2 rec. n)
True or false: Late Babylonian many-place reciprocal computations had direct practical applications. (False: Friberg describes them as having no practical applications whatsoever.)
Q: Describe the three stages by which the reciprocal pairs in SM 2685 were likely constructed.
A: Stage 1 found reciprocals of all 25 regular integers between 2 and 60 (e.g. rec. 8 = 7 1/2). Stage 2 applied reciprocal compensation with basic fractions: multiplying n by 2/3, 1/2, or 1/3 while multiplying rec. n by 1 1/2, 2, or 3, generating 19 new pairs where n is a quasi-integer. Stage 3 used sexagesimally regular twins (consecutive regular integers like 8 and 9) to construct three pairs with n greater than 33 1/3.
Q: What was the Old Babylonian workflow for solving a metrological mathematical problem?
A: Convert the given measures into floating sexagesimal place-value numbers using memorised metrological tables. Perform the arithmetic (multiplication, division via reciprocals, etc.) in place-value notation. Convert the result back into traditional metrological units.
Q: List the five innovations Friberg identifies as necessary for the invention of sexagesimal place-value notation.
A: (1) The visual similarity of signs for 1 and 60. (2) Replacing the special sign for 10 x 60 with the ordinary sign for 10, allowing the 10-and-6 alternation to continue indefinitely. (3) Abandoning quasi-integers in favour of integral multiples of gín, gín.bi, and gín.ba.gín. (4) Recognising that the 10-and-6 alternation could extend downward for fractions. (5) Inventing metrological tables to link traditional measures to abstract place-value numbers.
Q: What is the factorization method for computing reciprocals, and how does it work in CBS 1215?
A: The method breaks a number into its factors, computes the reciprocal of each factor separately, and multiplies the individual reciprocals together. CBS 1215 starts from the pair (2 05, 28 48), where 2 05 = 5 x 5 x 5, verifies it via factorization, then applies repeated doubling and halving to build increasingly complex pairs, ending with (10 06 48 53 20, 5 55 57 25 18 45).
The place-value system covered here is the direct ancestor of the sexagesimal notation still used for degrees, minutes, and seconds of arc in astronomy and navigation, and for hours, minutes, and seconds of time.
The reciprocal compensation algorithm connects to the broader history of algorithms and computational methods, and can be compared with similar techniques in Indian and Chinese mathematics.
The Late Babylonian reciprocal tables connect to the development of mathematical astronomy, where the same scribes used many-place sexagesimal numbers to compute planetary positions with remarkable accuracy.
Place-value notation, floating sexagesimal, table of reciprocals, SM 2685, SM 2574, CBS 1215, W 23021, AO 6456, head numbers, multiplication tables, metrological tables, Old Babylonian mathematics, Neo-Sumerian, Late Babylonian, Seleucid, Achaemenid, factorization algorithm, doubling and halving, reciprocal compensation, regular twins, extended quasi-integers, many-place reciprocals, mathematical astronomy, Friberg, Al-Rawi, Sulaymaniyah Museum, Larsa