Mesopotamian Sexagesimal Numbers, Part 2: Development of Notations and Early Mathematical Texts (c. 2600 to 2250 BC), History of Mathematics
Difficulty: Intermediate to Advanced | Prerequisites: Part 1 of these study notes (preliterate tokens and proto-cuneiform systems).
Source: Friberg, J. (2019) "Three thousand years of sexagesimal numbers in Mesopotamian mathematical texts," Archive for History of Exact Sciences 73:183–216. Sections 5–9.
Tags: sexagesimal, basic fractions, quasi-integers, regular numbers, non-regular numbers, reciprocals, area numbers, factor diagram, Early Dynastic III, Old Akkadian, cuneiform mathematics, nindan, sar, division exercises, tables of squares
By the mid-third millennium BC, the early number systems had evolved but were still far from the place-value notation that would come later. This part covers the period from roughly 2600 to 2250 BC, when Sumerian and Akkadian scribes were performing division, multiplication, and area calculations using metrological units, not abstract numbers. "Basic fractions" (1/3, 1/2, 2/3) and "quasi-integers" (an integer plus a basic fraction) were central to computation. Understanding how scribes worked without place-value notation makes the later invention of that notation far more intelligible.
Between roughly 2600 and 2250 BC, Mesopotamian scribes developed basic fractions (1/3, 1/2, 2/3, 5/6) borrowed from the weight system and used "quasi-integers" (like 1 2/3 or 8 1/2) for division and multiplication. Division by non-regular numbers like 7 required approximate reciprocals and clever algorithmic workarounds. Tables of squares extended the area system downward using sexagesimal fractions (gín, gín.bi, gín.ba.gín).
Basic fractions
The fractions 1/3, 1/2, 2/3, and 5/6, for which special cuneiform signs existed. These were borrowed from the Sumerian system of weight measures and applied across all number systems. Think of them as the only fractions the system could write natively.
Quasi-integer
A number of the form "integer plus basic fraction," such as 1 2/3, 8 1/2, or 3 1/3. Before place-value notation, these were the primary way of expressing non-whole numbers in calculations.
Sexagesimally regular number
A number n for which there exists another number n' such that n x n' equals some power of 60 (or, in the pre-place-value period, such that n x rec. n = 60). In simple terms, a number you can divide into 60 evenly. Examples: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, etc.
Sexagesimally non-regular number
A number that has no exact reciprocal within the sexagesimal system. The number 7 is the classic example: 7 x 8 1/2 = 59 1/2, close to 60 but not exact.
Reciprocal (rec. n)
The number n' such that n x n' = 60 (in the pre-place-value period) or n x n' = some power of 60 (in place-value notation). Finding reciprocals was the Mesopotamian approach to division.
Reciprocal compensation
The technique of multiplying one factor up while dividing another down by the same amount, keeping the product constant. For example, halving 7 and doubling 8 1/2 gives 3 1/2 x 17 = 59 1/2. This was a core computational method.
Nindan
The basic Sumerian/Old Babylonian unit of length, approximately 6 metres.
Sar (square nindan)
The basic unit of area measure, equal to the square of one nindan. 1 gín = 1/60 sar.
Gín, gín.bi, gín.ba.gín
Sexagesimal fractions of the sar: gín = 1/60 sar, gín.bi = 1/60 gín (= 1/3600 sar), gín.ba.gín = 1/60 gín.bi. These represent an early attempt to extend the area system downward using sexagesimal fractions.
Metro-mathematical algorithmic computation
A step-by-step calculation in which each step builds on the previous result, carried out entirely in metrological units (not abstract numbers). Each line converts a unit into the next smaller unit and computes how many rations fit.
Four successive periods show how the number signs evolved while keeping the same underlying structure:
Proto-cuneiform/proto-Elamite (c. 3300 to 2900 BC): Impressed cup and disk shapes. The system ran from 1 up to 10 x 60, with no specific signs for fractions.
Old Sumerian/Old Akkadian (third millennium BC): Essentially the same notations, possibly rotated. Basic fractions (1/3, 1/2, 2/3, 5/6) were added, borrowed from the weight system.
Neo-Sumerian/Babylonian (late third millennium onward): "Round" forms replaced by "sharp" (wedge-shaped) forms. Basic fractions continued.
Place-value notation (c. 2000 BC onward): The system was extended indefinitely in both directions. The alternation of factors 10 and 6 became uniform: every place is simply worth 60 times the next.
The decimal system D disappeared when the proto-Elamite script ceased to exist. By the Early Dynastic III period, only Systems S (sexagesimal) and A (area numbers) remained from the early systems. Area numbers survived into the Old Babylonian period; sexagesimal numbers continued to the end of the cuneiform script.
TSS 50 is a metro-mathematical exercise from the Early Dynastic III period. The problem: divide the contents of a full granary into individual rations of 7 sìla each (the sìla was a capacity unit of roughly 1 litre). Find the number of rations.
The answer given: 45(60 x 60) 42(60) 51 rations, with a remainder of 3 sìla.
The text does not show how this was computed. Friberg reconstructs the likely method by assuming the granary held 40(60) gur.mah, using the conversion factors:
1 gur.mah = 8 barig
1 barig = 6 bán
1 bán = 10 sìla
The division proceeds algorithmically, building each step on the previous:
1 bán (= 10 sìla) = 1 ration, remainder 3 sìla
1 barig (= 6 bán) = 8 rations, remainder 4 sìla
1 gur.mah (= 8 barig) = 1(60) 8 rations, remainder 4 sìla
10 gur.mah = 11(60) 25 rations, remainder 5 sìla
1(60) gur.mah = 1(60 x 60) 8(60) 34 rations, remainder 2 sìla
10(60) gur.mah = 11(60 x 60) 25(60) 42 rations, remainder 6 sìla
40(60) gur.mah = 45(60 x 60) 42(60) 51 rations, remainder 3 sìla
This is division by 7, a non-regular number, carried out entirely in metrological units without any abstract number notation. The method is confirmed by a roughly contemporary text from Ebla (TM.75.G.2346) that uses the same algorithmic approach explicitly.
This text contains eight connected exercises focused on multiplication and division by quasi-integers, specifically 1 2/3 5 (gín) and 1 2/3.
The notation "1 2/3 5" is shorthand for 1 2/3 + 5/60, or equivalently 1 + 40/60 + 5/60 = 105/60. Here "gín" means 1/60, borrowed from the weight system where 1 ma.na = 60 gín.
Exercise 1 is a division problem: the selling price for beads is 1 2/3 5 gín times the buying price. Beads sold for 1 ma.na minus 1/2 gín = 59 1/2 gín. Find the buying price.
The solution uses a "false solution" method:
Try n* = 4 gín. Then 1 2/3 5 x 4 = 6 2/3 + 1/3 = 7 gín.
The desired product is 59 1/2 gín = 8 1/2 x 7 gín.
So the true answer is n = 4 x 8 1/2 = 34 gín = 1/2 ma.na 4 gín.
The hidden point: this exercise is really about dividing by the non-regular number 7, since 1 2/3 5 gín x 34 = 59 1/2, which is 60 minus 1/2.
The relationship 7 x 8 1/2 = 59 1/2 = 60 minus 1/2 gives an approximate reciprocal. Reciprocal compensation ("halving and doubling") produces further such pairs: 3 1/2 x 17 = 59 1/2, and 1 2/3 5 x 34 = 59 1/2.
Exercises 2 to 8 involve multiplication or division by 1 2/3, a sexagesimally regular quasi-integer with reciprocal 36 (since 1 2/3 x 36 = 60). The questions are brief and cryptically formulated, dealing with market rates for commodities like potash.
CUNES 50-08-001 is a large Early Dynastic III tablet containing a sequence of "tables of areas of squares." The basic unit of length is the nindan (c. 6 m), and the basic unit of area is the sar = 1 square nindan.
The area system used larger units built from the sar:
100 sar = 1 iku
6 iku = 1 èshe
3 èshe = 1 bùr
Above the bùr: sexagesimal multiples (bùr'u, shár, shár.gal, shár.kid)
The largest units are purely theoretical: 1 shár.kid = 60 x 60 x 60 x 3 x 6 x 100 sar, roughly 14,000 square kilometres.
Sub-table A lists squares of sexagesimal multiples of the nindan. For example: sq. 10(gésh) = 3(shár) 20(bùr), sq. 1(shár) = 2(shár).gal.
Sub-tables B, C, D, E list squares of fractions of the nindan:
B: multiples of 1 nikkas = 1/4 nindan
C: multiples of 1 kùsh.numun = 1/6 nindan
D: multiples of 1 gish.bad = 1/12 nindan
E: multiples of 1 shu.bad = 1/24 nindan
The critical observation is that sub-tables C and D express their results as quasi-integer multiples of the sar and gín respectively. For example, sq. 1 kùsh.numun = 1 2/3 gín. All entries in sub-table C follow the pattern: a quasi-integer times 1 sar, then a quasi-integer times 1 gín.
This is an attempt to extend the area system downward sexagesimally, using gín = 1/60 sar, gín.bi = 1/60 gín, and gín.ba.gín = 1/60 gín.bi, a precursor to genuine sexagesimal fractions.
HS 815 is a school text from the Old Akkadian period. The problem: given that the long side of a rectangle is 1(60) 7 1/2 nindan and the area is 1 iku (= 1(60) 40 sar), find the short side.
In modern notation: 67.5 x s = 100, so s = 100/67.5.
The answer: s = 1 nindan 5 kùsh 2 shu.dù.a 3 shu.si 1/3 shu.si, using the Old Akkadian length conversions:
1 nindan = 12 kùsh (cubits)
1 cubit = 3 shu.dù.a
1 shu.dù.a = 10 shu.si (fingers)
The problem has a simple solution because 1(60) 7 1/2 = 27 x 2 1/2, and both 27 and 2 1/2 are sexagesimally regular. The solution uses repeated "reciprocal compensation" (tripling and trisecting):
1(60) 40 sq. nindan = 2 1/2 nindan x 40 nindan
= 7 1/2 nindan x 13 nindan 4 cubits
= 22 1/2 nindan x 4 nindan 5 1/3 cubits
= 1(60) 7 1/2 nindan x 1 nindan 5 2/3 cubits 3 1/3 fingers
The exercise had a clean solution precisely because the given number was sexagesimally regular, and because the conversion factors 12, 3, and 10 in the Old Akkadian length system were also regular.
Students often think division in the ancient world was simply "harder long division." It was not. Without place-value notation, division was carried out in metrological units, and each step required knowing the conversion factors between those units. The algorithm is fundamentally different from anything in modern arithmetic.
Students sometimes assume that "regular" and "non-regular" numbers are a modern classification imposed on Mesopotamian mathematics. They are not. The distinction was operationally real for scribes: regular numbers could be divided exactly, non-regular numbers required approximation. The scribes designed exercises around this distinction.
Students sometimes think quasi-integers (like 1 2/3) were just a clumsy notation for what we would write as fractions. In context, quasi-integers were the computational standard, not a workaround. They were as natural to a Sumerian scribe as decimals are to us.
The distinction between sexagesimally regular and non-regular numbers is foundational. Be able to explain what makes a number regular and give examples of each.
The "false solution" method (try a convenient value, compute the correction factor) is worth knowing as a technique. It reappears in Old Babylonian algebra.
The extension of area numbers downward via gín, gín.bi, and gín.ba.gín (CUNES 50-08-001) is the direct precursor of sexagesimal place-value fractions. Understand how this worked.
Reciprocal compensation is the single most important computational technique in this material. Be able to demonstrate it with a numerical example.
True or false: 7 is a sexagesimally regular number. (False: 7 x 8 1/2 = 59 1/2, not 60.)
Fill in the blank: A number of the form "integer plus basic fraction" is called a ______. (quasi-integer)
True or false: The basic fractions 1/3, 1/2, and 2/3 were originally borrowed from the weight system. (True.)
Fill in the blank: 1 gín = ______ of 1 sar. (1/60)
True or false: The reciprocal of 1 2/3 is 36. (True: 1 2/3 x 36 = 100/60 x 36 = 60.)
Q: Walk through the "false solution" method for dividing 59 1/2 by 1 2/3 5 gín, as in exercise 1 of CUNES 52-18-035.
A: Try n* = 4 gín. Compute 1 2/3 x 4 = 6 2/3, plus 5/60 x 4 = 1/3, giving 7 gín total. The desired answer is 59 1/2 gín = 8 1/2 x 7 gín. Multiply the trial value by the correction factor: n = 4 x 8 1/2 = 34 gín = 1/2 ma.na 4 gín.
Q: Why does the problem in HS 815 have a clean solution?
A: Because 1(60) 7 1/2 = 27 x 2 1/2, and both 27 and 2 1/2 are sexagesimally regular (their reciprocals exist within the system). Also, the conversion factors between length units (12, 3, 10) are all regular. This means repeated reciprocal compensation can reduce the problem step by step without introducing remainders.
Q: What are the basic fractions, and where do they come from?
A: The basic fractions are 1/3, 1/2, 2/3, and 5/6. They originated as cuneiform signs for fractions of the weight unit ma.na in the Sumerian weight system, and were then borrowed into all other number systems.
Q: Explain how CUNES 50-08-001 extends the area system downward using sexagesimal fractions.
A: The tablet defines gín = 1/60 sar, gín.bi = 1/60 gín, and gín.ba.gín = 1/60 gín.bi. Sub-tables C, D, and E express squares of sub-nindan lengths as quasi-integer multiples of these fractional area units, creating a limited but genuine system of sexagesimal fractions for areas.
The concept of sexagesimally regular numbers connects directly to Part 3 (reciprocal tables and place-value notation), where the entire system of tables of reciprocals is built on identifying and pairing regular numbers.
The "false solution" method in CUNES 52-18-035 is an early instance of what becomes standard Old Babylonian algebraic technique, used extensively in quadratic and other equation types.
The area tables in CUNES 50-08-001, with their downward sexagesimal extension, are a stepping-stone to the full invention of place-value fractions around 2000 BC.
Basic fractions, quasi-integers, regular numbers, non-regular numbers, reciprocals, reciprocal compensation, halving and doubling, false solution method, area numbers, factor diagram, nindan, sar, gín, gín.bi, gín.ba.gín, Early Dynastic III, Old Akkadian, TSS 50, CUNES 52-18-035, CUNES 50-08-001, HS 815, metro-mathematical, algorithmic computation, division by 7, tables of squares, Friberg, Ebla, cuneiform mathematics