Mathematics for accountants
Sumerian mathematics grew out of practical needs. Cities had to count sheep, measure fields, allocate rations and calculate the volume of brickwork, and scribes were trained to do it. The mathematics that survives is therefore mostly arithmetic and measurement, not abstract proof. It was nonetheless sophisticated, and its number system is one of the most influential from the ancient world.
One limit should be stated at the start. Many of the most advanced mathematical tablets, such as those with elaborate problems and tables, come from the Old Babylonian period, roughly 1900 to 1600 BCE, when Sumerian was no longer a spoken language but was still used in schools. The earlier Sumerian evidence, from the third millennium BCE, is more modest. It is often hard to separate what Sumerian scribes could do from what later Babylonian scribes developed.
Counting in base 60
The sexagesimal number system uses 60 as its base. Sumerian scribes did not simply count in sixties, though. Their notation mixed bases 10 and 6, so that a sequence of units went 1, 10, 60, 600, 3,600 and so on, with each step alternating a factor of 10 and 6. A tablet writer pressed a stylus into the clay to form wedge shapes: a vertical wedge for 1 and a corner-shaped (chevron) wedge for 10.
Why 60 was chosen is uncertain. A common explanation is that 60 is divisible by many small numbers, namely 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30, which makes fractions easy to handle in a world of shared rations and divided land. Other explanations, such as the merging of two earlier counting systems, are not proven.
Traces of base 60 remain in the 60 minutes of an hour and 60 seconds of a minute. The 360 degrees of a circle are also often linked to Mesopotamian influence, though that link runs through later Babylonian astronomy and is not a simple inheritance from Sumer.
Place value
A major achievement was positional notation, in which the value of a sign depends on its position, like the digits in 305. In the mature system the same wedge signs could mean 1, 60 or 3,600 depending on place. This allowed a short written line to express large numbers and made multi-step calculations workable on clay.
Early versions had no sign for an empty place and no marker like a decimal point, so the reader had to work out the intended magnitude from context. A zero-like empty-place sign appears only much later, in the last centuries BCE. Context was part of the notation.
Multiplication tables and reciprocals
Scribal students learned from standard tables. Surviving school tablets include multiplication tables for numbers such as 2 to 20 and for key numbers like 25 and 50, along with lists of reciprocals. Division was handled as multiplication by a reciprocal. Because 60 has many divisors, numbers like 2, 3, 4, 5, 6, 8, 9, 10, 12 and 15 have clean reciprocals in base 60, while 7 and 11 do not, so tables skipped them or handled them in special ways. For example, dividing by 4 meant multiplying by 15, since 15/60 equals one quarter.
Most surviving tables are Old Babylonian, but their forms descend from earlier scribal teaching. Teaching relied on memorization and repeated copying.
Measures for grain, land and labour
Sumerian measurement was practical and varied. Separate systems existed for length, area, capacity and weight, each with its own units, and many were not decimal. Scribes learned lists that gave the units and their relations. Capacity units for grain, such as the sila and gur, allowed rations to be calculated. Area was measured with units tied to the size of fields, including the iku, a unit often said to be a little over a third of a hectare, though modern equivalents are estimates and varied by period.
Because these systems were not uniform, converting between them was an important skill. A field’s area could be calculated from its sides, the seed needed could be estimated from the area, and the expected harvest from the seed. These links between land, seed and crop appear in school problems and administrative practice, and they are part of why mathematics and accounting grew together in the early cities, alongside wider traditions of ancient weights measures.
Practical geometry
Geometry in Mesopotamia was applied. Scribes calculated the areas of rectangles and triangles, and they approximated the areas of irregular fields by dividing them into simpler shapes. They could work out the volume of a brick structure or an excavation, a common task for labour managers estimating how many workers and days a building project required. The best-known geometric tablets again belong to the Old Babylonian period and show an understanding of relationships between sides in right triangles, though this is a later Babylonian development rather than something proven for Sumerian scribes.
Circles were treated approximately, often by taking the circumference as three times the diameter, which was good enough for practical work.
How to weigh the legacy
The Sumerians did not invent mathematics, and other early civilizations such as Egypt developed their own methods, as in Rhind papyrus mathematics. What survives from Sumer is a clearly documented tradition of base-60 numeration, place value, tables and applied measurement, taught in schools and used daily for accounts. The strongest claims about its legacy, for time and angle measurement, rely on Babylonian developments that came afterwards. Even so, the Sumerian scribes who first recorded numbers on clay laid the groundwork for a mathematical culture that lasted for well over two thousand years.