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

Plimpton 322 attracted interest from scholars because the listed values appear to represent primitive Pythagorean triples (i.e., triples of strictly positive integers \((a,b,c)\) satisfying \(a^2+b^2=c^2\), and for which the three numbers have no common factors). The tablet contains four columns, representing (from right to left):

• the row number in the table;
• the hypotenuse of a right triangle;
• the shortest side of that triangle;
• either the square of the hypotenuse divided by the square of the longest side, or the square of the shortest side divided by the square of the longest side [Robson 2002].

Notice that the ambiguity in the leftmost column is minimal: the two proposed values differ by 1, and the question is whether we have 0 or 1 before the comma.

The degree measure of smallest angle in the right triangles associated with the table’s 15 rows is distributed quite well between 30 and 45 degrees (the following values are rounded):

44.8 ; 44.3 ; 43.8 ; 43.3 ; 42.1 ; 41.5 ; 40.3 ; 39.8 ; 38.7 ; 37.4 ; 36.9 ; 35.0 ; 33.9 ; 33.3 ; 31.9.

One explanation proposed for this feature of the table is that it derives from the fact that the regular sexagesimal integers (i.e., integers whose reciprocal has a terminating sexagesimal representation) are uniformly distributed among the integers [see, for example, de Solla Price 1964, p. 3].

The idea for the student activity that we describe in the next section was inspired by the theory proposed by Mansfield and Wildberger [2017], who argued that Plimpton 322 is an exact ratio-based trigonometric table that abandons the notion of angle. Indeed, working with sines and cosines corresponds to working with side lengths of right triangles, and the numbers given in the first column of Plimpton 322 are not approximations of the ratio recorded in that column. The main difficulty in appreciating this “exactness” aspect of Plimpton 322 comes from the fact that the mathematics underlying it is related to the base 60. The numbers in Plimpton 322 are rational numbers which have terminating sexagesimal representation. If we translate these numbers into base 10, most of them have non-terminating decimal representations because they are not decimal fractions. Thus simply getting a decimal translation (i.e., writing the given numbers in base 10) is not particularly illuminating.

For didactic purposes we present instead a construction method for a decimal analogue of Plimpton 322, produced by performing operations in base 10 (rather than in base 60), and where all rational numbers are decimal fractions. This construction provides a nice classroom activity for prospective K–12 mathematics teachers, as well as high school students. The activity guides students, working together in small groups, through an interesting arithmetic algorithm that produces primitive Pythagorean triples in a fashion consistent with the list of numbers on Plimpton 322. It also offers students opportunities to make and test mathematical conjectures as they work together through the various steps of the algorithm. Moreover, the activity consolidates several notions related to numbers and numeration systems (e.g., fractional versus decimal representations, different bases). Last but not least, the activity allows the students to better understand the famous tablet Plimpton 322 by removing the difficulty related to the base 60.  

The complete student-ready activity sheet for constructing a decimal analogue of Plimpton 322 can be downloaded as a pdf in the next section of this article. Instructors who wish to first introduce the sexagesimal system to their students may also be interested in using the student activity “Babylonian Astronomy and Sexagesimal Numeration,” designed by Daniel E. Otero.