Mesopotamian Sexagesimal Numbers, Part 1: Preliterate Tokens and Proto-Cuneiform Number Systems, History of Mathematics
Difficulty: Intermediate | Prerequisites: Basic understanding of number bases (e.g. decimal vs binary), familiarity with ancient Mesopotamian chronology helpful but not required.
Source: Friberg, J. (2019) "Three thousand years of sexagesimal numbers in Mesopotamian mathematical texts," Archive for History of Exact Sciences 73:183–216. Sections 1–4.
Tags: sexagesimal, base-60, Mesopotamia, proto-cuneiform, proto-Elamite, clay tokens, bullae, number systems, Uruk, Susa, cuneiform mathematics, capacity numbers, bisexagesimal, metrological systems, history of counting
Before anyone in Mesopotamia could write, they were already counting in base 60. Small clay tokens, shaped as cones, lenses, and disks, stood for quantities like 1, 10, and 60, and were sealed inside hollow clay balls (bullae) for record-keeping. When writing appeared around 3300 BC, these tokens became impressed signs on clay tablets, forming the proto-cuneiform and proto-Elamite scripts. This part of the study notes covers how those earliest number systems worked, what their different variants were used for (counting bread rations vs measuring barley volumes vs tracking trade), and how even the most basic accounting exercises from 3000 BC already involved surprisingly sophisticated mathematics.
Mesopotamians used a base-60 (sexagesimal) counting system from before writing (c. 3300 BC) onward. Preliterate clay tokens mapped directly onto later written number signs. Multiple parallel number systems existed simultaneously for different purposes: sexagesimal for general counting, bisexagesimal for bread rations, decimal for proto-Elamite contexts, and separate systems for capacity, weight, area, and time.
Sexagesimal (base-60) number system
A positional number system built on the progressive series of units 1, 10, 60, 600 (10 x 60), 3600 (60 x 60), and so on, with factors alternating between 10 and 6. In simple terms, this is counting in groups of 60 rather than groups of 10.
Tokens
Small clay figurines (cones, spheres, disks, lenses, rods) used for counting and record-keeping in Mesopotamia for roughly 5,000 years before the invention of writing. Think of them as physical digits you could hold in your hand and seal inside a clay envelope.
Bullae
Hollow clay balls in which groups of tokens were enclosed, sometimes with impressions on the outside indicating the contents. These were the "sealed envelopes" of preliterate accounting.
Proto-cuneiform script
The earliest writing system used on clay tablets in Uruk (southern Mesopotamia), dating from around 3300 BC. Not yet the wedge-shaped cuneiform of later periods, but an ancestor of it.
Proto-Elamite script
A largely undeciphered writing system used in Susa (in modern-day Iran), roughly contemporary with proto-cuneiform. It shared the same sexagesimal counting system but also used a separate decimal system.
System S (sexagesimal counting numbers)
The standard system for general counting, shared by both proto-cuneiform and proto-Elamite scripts. Units progress as 1 > 10 > 60 > 600 > 3600, with factors alternating 10, 6, 10, 6.
System B (bisexagesimal counting numbers)
A variant counting system used exclusively for counting bread rations in proto-cuneiform texts. It differs from System S by having special notations for 2 x 60, 10 x 2 x 60, and 60 x 2 x 60.
System C (capacity numbers)
A metrological number system for measuring volumes of barley and similar products. It used its own set of units (c, d, D, C) with conversion factors of 6, 10, and 3, and included fractional sub-units.
System D (decimal counting numbers)
A base-10 counting system found only in the proto-Elamite script, used for specific purposes alongside the sexagesimal system. It disappeared when the proto-Elamite script ceased to exist.
Factor diagram
A schematic showing the relationships between successive units of a number system, with arrows indicating the multiplication factor needed to move from one unit to the next. This is the standard way historians of mathematics represent these ancient number systems.
Metro-mathematical exercise
A mathematical problem that works with metrological (measurement) data rather than abstract numbers. In simple terms, a word problem where you have to keep track of units of measure (like barley volumes or silver weights) throughout the calculation.
"Almost round" number
A number that can be expressed as a round number minus or plus a small simple fraction of that round number. For example, 13.5 x 60 = (1 minus 1/10) x 15 x 60. These appear frequently in early mathematical texts and are unlikely to be coincidental.
For roughly five millennia before writing appeared (c. 3300 BC), people in Mesopotamia and neighbouring regions used small clay figurines called tokens for communication and archiving. Shortly before writing was invented, groups of tokens began to be sealed inside hollow clay balls (bullae), sometimes with marks on the outside describing the contents.
Bullae have been found at Uruk (southern Mesopotamia) and Susa (in modern Iran), which tells us these accounting practices were geographically widespread.
Tokens found inside bullae can be matched to later proto-cuneiform and proto-Elamite number signs
Rods corresponded to the unit 1, lenses to 10, cones to 60, and punched cones to 10 x 60
This mapping is confirmed by comparing bulla contents with the well-documented System S
A key example is bulla Sb 1927 from proto-Elamite Susa. Its contents (one punched cone, three smaller cones, three lenses) correspond to the sexagesimal number 1(10 x 60) + 3(60) + 3(10) = 13 x 60 + 30 = 810. This turns out to be an "almost round" sexagesimal number: 13.5 x 60 = (1 minus 1/10) x 15 x 60.
The inscription on the outside of Sb 1927 was imprinted right to left, consistent with the proto-Elamite script's direction. The three round holes on the outside resulted from one of the smaller cones being pushed point-first into the clay.
The same method of matching tokens to later number signs works for other number systems beyond sexagesimal counting.
Disks, balls, large balls, and high disks found in bullae can be matched to the units of System C (capacity numbers), used for measuring volumes of barley and similar products
The capacity system C used conversion factors of 6, 10, and 3 between its units (c, d, D, C)
A separate proto-Elamite decimal system D(pE) also existed. Bulla Sb 1967 from Susa appears to contain two distinct sets of tokens: tetrahedral tokens for the decimal system and ball-shaped tokens for the capacity system. The two larger balls and four small balls represent the capacity number 2D 4d = 24 d, while the tetrahedra represent the decimal number 3(100) + 2(10) + 4 = 324.
The decimal number 324 is itself an "almost round" number: 324 = 27 x 12 = (1 minus 1/10) x 6 x 60. The interpretation suggests that 24 d of barley were accounted for as wages for (1 minus 1/10) x 6 x 60 work days, or approximately 12 work months.
The factor diagram for System S shows the characteristic alternation between factors 10 and 6 that would eventually become central to the invention of sexagesimal place-value notation a thousand years later.
MSVO 4, 66 is a well-known proto-cuneiform "bread-and-beer text" from the Jemdet Nasr period (c. 3000 BC). It is a school text, not an administrative document, identified as such by the absence of names and titles in the space where an official would normally be listed.
The text is remarkable because it uses several different number systems simultaneously:
Bisexagesimal numbers (System B) for counting bread rations
Sexagesimal numbers (System S) for counting jars of beer
Capacity numbers (System C) for measuring barley expenditure, including fractional sub-units
Bisexagesimal numbers differ from ordinary sexagesimal numbers by having special signs for 2 x 60, 10 x 2 x 60, and 60 x 2 x 60. They appear to have been used exclusively for counting rations of bread and similar products.
Capacity fractions in System C start with the "minor" unit M, equal to 1/5 of the basic capacity unit c. Fractions of M (from m2 = 1/2 M down to m6 = 1/6 M) were written as wreaths of small marks.
The calculations in MSVO 4, 66 are sophisticated. For example, the text computes that 1(60) rations of size M require 12 c = 2d of barley, and that 5(10 x 2 x 60) rations of size m6 require 5 x 40 c = 1C 3d 2c of barley.
The grand totals on the reverse are suspiciously tidy:
Total bread rations = 1(60 x 2 x 60), a perfectly round bisexagesimal number
Total beer jars = 1(10 x 60), a perfectly round sexagesimal number
Total barley expenditure = (1 minus 1/200) x 2C, an "almost round" capacity number
This pattern strongly suggests the text was designed as a teaching exercise, not a real administrative record.
CUSAS 1, 143 is a proto-cuneiform text from the Uruk III period (c. 3200 to 3000 BC) that has been identified as dealing with loan and interest. Like all known mathematical cuneiform texts from before the Neo-Sumerian period, it is a metro-mathematical exercise.
The text states that the daily ration of a non-fluid product is m4 (one-quarter of the minor capacity unit M). The corresponding monthly cost in barley is explicitly broken down into:
The "loan" amount: 1c 2M m2 = 30 x m4
The "interest": m2 m5 dxs
The interest is precisely 1/10 of the loan. The verification: 1c 2M m2 = 7M m2 = 15 m2, and 1/10 of 15 is 1.5, which equals (10 + 4 + 1) dxs = m2 m5 dxs.
So the monthly expense is computed as (1 + 1/10) x 30 x m2, a 10% interest rate applied to the base cost.
The text then computes the expense for five months, arriving at 12c 1M m2 (though the correct answer should be 12c 1M m4, indicating a small scribal error).
Two notable features: the text uses proto-cuneiform number signs for time measures (1 month and 5 months), and no cuneiform sign for "1/10" is used, only a sign meaning "interest." The concept of a fixed fractional interest rate was already embedded in the administrative vocabulary.
Students often think sexagesimal means "base 60 like our base 10," with each place simply worth 60 times the next. It does not work that cleanly in the early period. The factor diagram alternates between 10 and 6, so the progression is 1, 10, 60, 600, 3600, not 1, 60, 3600, 216000.
Students often assume the Mesopotamians used only one number system at a time. They did not. Multiple systems ran in parallel, each for a specific purpose (counting bread, measuring barley, weighing metals, measuring land). Choosing the wrong system was a real source of error.
Students sometimes confuse the proto-cuneiform script with the later cuneiform script. Proto-cuneiform used impressed cup and disk shapes; the familiar wedge-shaped signs came later.
Students sometimes assume that because Susa (proto-Elamite) and Uruk (proto-cuneiform) shared the sexagesimal system, they shared a single script. They did not. The proto-Elamite script remains largely undeciphered and is unrelated to proto-cuneiform, despite the number-system overlap.
The alternation of factors 10 and 6 in the factor diagram is the structural signature of the sexagesimal system and the key to understanding why place-value notation eventually worked. Expect questions asking you to explain or reproduce the factor diagram.
"Almost round" numbers appearing in school texts are strong evidence that a text is a pedagogical exercise rather than an administrative record. Be prepared to identify them.
The coexistence of multiple number systems (S, B, C, D, A, E, T) is a major theme. Know which system was used for which purpose.
The proto-cuneiform loan-and-interest text (CUSAS 1, 143) demonstrates that a 10% interest rate was standard and that the concept was built into the writing system's vocabulary, not expressed numerically.
True or false: The sexagesimal system uses a consistent factor of 60 between each successive unit in its early form. (False: it alternates between factors of 10 and 6.)
Fill in the blank: Small clay figurines used for counting before writing are called ______. (tokens)
True or false: The proto-Elamite and proto-cuneiform scripts are the same writing system. (False: they are unrelated scripts that happen to share the sexagesimal counting system.)
Fill in the blank: Hollow clay balls used to enclose groups of tokens are called ______. (bullae)
True or false: Bisexagesimal numbers were used for counting all types of goods. (False: they were used exclusively for counting bread rations and related products.)
Q: What is the progressive series of units in System S of sexagesimal counting numbers, and what are the factors between each successive pair?
A: The units are 1, 10, 60, 600, 3600, and so on. The factors alternate: x10, x6, x10, x6. Moving from 1 to 10 is a factor of 10; from 10 to 60 is a factor of 6; from 60 to 600 is a factor of 10; from 600 to 3600 is a factor of 6.
Q: What evidence suggests that bulla Sb 1927 from Susa contained sexagesimal number tokens?
A: The tokens inside (one punched cone, three smaller cones, three lenses) can be mapped to the proto-cuneiform/proto-Elamite signs for 10 x 60, 60, and 10 respectively. The resulting number, 13 x 60 + 30 = 810, is an "almost round" sexagesimal number, which fits the pattern seen in other early mathematical texts.
Q: How does the bisexagesimal system (System B) differ from the ordinary sexagesimal system (System S)?
A: System B has special notations for 2 x 60, 10 x 2 x 60, and 60 x 2 x 60, whereas System S has notations for 60, 10 x 60, and 60 x 60. System B was used specifically for counting bread rations, while System S was the general-purpose counting system.
Q: In the proto-cuneiform loan text CUSAS 1, 143, how is the interest rate expressed, and what is that rate?
A: The interest rate is 1/10 (10%) of the loan amount. It is not expressed with a numeral for "1/10" but rather through a cuneiform sign meaning "interest." The total monthly cost is computed as (1 + 1/10) times the base loan.
Q: What features of MSVO 4, 66 identify it as a school text rather than an administrative document?
A: The absence of names and titles in the space where responsible officials would normally be listed, the unusually large and round numbers in the grand totals, the variety of ration sizes used, and the fact that the grand total of barley is an "almost round" number, all point to a designed pedagogical exercise.
This material connects directly to the development of place-value notation (covered in Part 2), because the factor-diagram structure of the early sexagesimal system is what made place-value notation possible. The alternation of 10 and 6 became the basis for extending the system indefinitely in both directions.
The coexistence of multiple metrological systems also foreshadows the Old Babylonian invention of metrological tables (Part 3), which were designed to convert between these traditional systems and the new abstract place-value numbers.
The concept of "almost round" numbers and the deliberate design of school exercises around tidy totals connects to the broader history of mathematical pedagogy and the question of how mathematical knowledge was transmitted in ancient cultures.
Base-60, sexagesimal counting, Mesopotamian mathematics, Sumerian mathematics, proto-cuneiform numbers, proto-Elamite numbers, clay tokens, bullae, Uruk period, Jemdet Nasr period, Early Dynastic, bisexagesimal, System S, System B, System C, System D, capacity numbers, metrological systems, factor diagram, metro-mathematical, bread-and-beer text, MSVO 4 66, CUSAS 1 143, Sb 1927, Susa, barley rations, interest rate, ancient accounting, cuneiform number signs, almost round numbers, Friberg