In the late Ur III / Neo-Sumerian era, Mesopotamian scholars developed the key innovation behind later Babylonian computation: sexagesimal (base‑60) place-value numbers usable for both large and fractional quantities (even though zero was not yet a fully standardized digit). This laid the foundation for Old Babylonian tabular calculation.
During the Old Babylonian period, scribal education produced large corpora of mathematical tablets featuring multiplication tables, reciprocal tables, roots/powers, and coefficient lists, alongside algorithmic problem texts. These tablets show sexagesimal place-value calculation as a learned scholarly technique.
A hallmark of the sexagesimal tradition was heavy use of reciprocals (especially for “regular” numbers with finite base‑60 expansions) to turn division into multiplication. This reciprocal-based computational style became central to both practical calculation and advanced school exercises.
The tablet known as Plimpton 322, from Larsa (Tell Senkereh), was produced in the early 2nd millennium BCE. It contains a 4‑column, 15‑row table written in sexagesimal place-value notation and is central evidence for sophisticated Old Babylonian numerical work (often discussed in connection with right triangles and reciprocal methods).
Modern analysis interprets Plimpton 322’s rows as encoding relationships equivalent to Pythagorean triples / right-triangle side relations in sexagesimal form, demonstrating structured numerical generation and tabulation techniques in Old Babylonian mathematics.
The student exercise tablet YBC 7289 records a remarkably accurate sexagesimal approximation to √2 (used as the diagonal-to-side ratio of a square), illustrating both geometric reasoning and high-precision computation in the Old Babylonian sexagesimal system.
Beyond Plimpton 322, multiple Old Babylonian mathematical tablets show procedures consistent with using the right-triangle relation in problem solving—evidence that such geometric-number techniques were part of the broader scribal mathematical repertoire.
Babylonian sexagesimal place-value writing used groups of wedge marks for 1–59, and for much of its history relied on context (and later a placeholder in internal positions) rather than a fully positional zero at the end of numerals—an important constraint shaping how computations were recorded and read.
Although the Old Babylonian period provides the largest surviving body of school mathematics, sexagesimal calculation and cuneiform scholarly numeracy continued through later Mesopotamian states, creating continuity between earlier mathematics and later astronomical computation.
In Babylonian mathematical astronomy, the sexagesimal computational tradition was applied to planetary prediction. Preserved System A tablets include Mercury ephemerides calculated for 424–401 BCE, showing mature numerical modeling techniques long after the Old Babylonian school texts.
During the early Hellenistic/Seleucid period, System A ephemerides continued to be produced; studies of preserved texts describe Mercury ephemerides spanning about 310–290 BCE, reflecting sustained high-level sexagesimal computation in astronomical contexts.
The transition into late Babylonian mathematical astronomy is also marked by lunar computations; summaries of the cuneiform record note that the oldest preserved lunar tablets date from 306 BCE, within the broader sexagesimal tradition that underpinned astronomical calculation.
Babylonian Mathematics and the Sexagesimal Tradition (c. 2000–300 BCE)
