Converting the Old Babylonian Tablet ‘Plimpton 322’ into the Decimal System as a Classroom Exercise: Historical Background of Plimpton 322

One of the most fascinating physical artifacts is a small Old Babylonian clay tablet labeled Plimpton 322, which is usually dated to around 1900–1600 BCE. The tablet contains a table of numbers that are written in the wedge-shaped cuneiform script typical of ancient Mesopotamian civilizations and that are expressed in sexagesimal (base 60) notation.

Figure 1: Plimpton 322, whose real size is approximately 13 cm × 9 cm × 2 cm. Wikimedia Commons.

As the photograph of the tablet suggests, table entries from the damaged areas of the tablet must be reconstructed, and historians are confident that we do not have the entire tablet because there is an obvious break on the left side. However, we do have table headings. (Some computational errors in the numbers have been corrected by math historians over the years.)

George Arthur Plimpton, who collected the tablet, purchased it from a dealer of antiquities, which means we do not know its provenance, or place of origin. (For other books and objects from Plimpton's collections, see Convergence's “Mathematical Treasures – The George Arthur Plimpton Collection.”) As a whole, the thousands of Mesopotamian tablets that have been analyzed indicate that those containing mathematics, but not obviously administrative or economic records, were created in schools that trained scribes to work in the bureaucracies that managed the royal dynastic empires that ruled in the region. Some were multiplication tables and lists of measurements that were copied and memorized by students; some were model examples prepared by teachers; some set out advanced mathematical problems; and the purpose for some remains unknown [Robson 2002; Bernard, Proust, and Ross 2014; Robson 2009; Collins 2008; Barrow-Green, Gray, and Wilson 2019].

Otto Neugebauer and Abraham Sachs were the first to look carefully at Plimpton 322 and realize its mathematics was interesting [Neugebauer and Sachs 1945]. In the past few decades, scholars have tended toward a consensus that the tablet's style and content resemble scribal school tables unearthed at Larsa, a major city-state in the south. Beyond those areas of agreement, though, debates continue over the exact reason why Plimpton 322 was inscribed. Some have theorized that it served as a resource for generating triangles that satisfy the Pythagorean Theorem; Robson and others argued that the values represent a descending list of reciprocal pairs that assisted with identifying the hypotenuses of right triangles whose short sides equal 1, given the long sides; Britton, Proust, and Shnider saw the tablet as setting out and solving a challenge problem on the sides and diagonals of rectangles; and Mansfield and Wildberger joined those who thought the tablet was used for trigonometry [Barrow-Green, Gray, and Wilson 2019; Robson 2002; Britton, Proust, and Shnider 2011; Mansfield and Wildberger 2017]. This ongoing scholarly discussion, of course, is one reason why mathematicians, educators, and students find this tablet so interesting.