Sexagesimal Place-Value (Babylonian floating-point notation)

Definition

Sexagesimal place-value is the specific base-sixty implementation of the place-value principle used throughout the Babylonian mathematical and astronomical corpus: sixty units of one rank are written as one unit of the next higher rank, and a small set of cuneiform wedge-signs takes different values according to position. Hunger-Steele characterise it as a floating-point system: a written numeral such as 23,5,56 can be read as (23×60²) + (5×60) + 56, or (23×60) + 5 + (56×60⁻¹), or any further power-of-sixty multiple, with no inherent indication of absolute magnitude.

In Tradition

Neugebauer, with Hunger-Steele and Hunger-Pingree concurring, treats sexagesimal place-value as the Babylonian arithmetical-computational medium proper. Neugebauer reads its 1,10 = 70 pattern as the same principle later employed in Ptolemy's table of chords (cuneiform or Greek alphabetic); Hunger-Steele record that in MUL.APIN it appears only inside procedure-internal calculations, never in the presentation of data or final results, which carry explicit units.

In Practice

For the reader of an ACT procedure text, ephemeris, or MUL.APIN-style calculation, the sexagesimal place-value is the medium that records every computed parameter. Modern editorial convention separates places by a comma (23,5,56) and marks the implied radix-point by a semicolon between integer and fractional places (12;47,30 = 12 + 47/60 + 30/3600). The floating-point ambiguity — the absence of any sign for missing zero-digits or absolute magnitude — is both a difficulty and a decipherment key: Neugebauer shows an apparently impossible Old-Babylonian problem-text relation 2f + x = 1 becomes consistent when the recorded "1" reads as 1,0 = 60, yielding x = 30 in agreement with surviving traces. Within the ACT corpus the medium carries System-A step-function zone constants, System-B zigzag M/m/Δ/d parameters, synodic-arc and length-of-daylight values, lunar-latitude Column E, and the mean synodic month 29;31,50,8,20 days — all expressed as sexagesimal sequences read by position.

Historical Origin

Attested across the Old Babylonian mathematical problem texts (c. 1900-1600 BCE), the MUL.APIN compilation (~1000 BCE composition), and the Seleucid-period ACT lunar and planetary ephemerides (c. 250-50 BCE). Modern critical treatments: Otto Neugebauer, *The Exact Sciences in Antiquity* (1957/1969), Chapter I §§12-13, pp. 16-18 and Chapter III §33, p. 65; Hermann Hunger & John Steele, *The Babylonian Astronomical Compendium MUL.APIN* (Routledge 2019), p. 21.

Further Reading

• Otto Neugebauer, The Exact Sciences in Antiquity
• Hermann Hunger & John Steele, The Babylonian Astronomical Compendium MUL.APIN