# Converting the Old Babylonian Tablet ‘Plimpton 322’ into the Decimal System as a Classroom Exercise: Constructing the Decimal Analogue of Plimpton 322

Author(s):
Antonella Perucca (University of Luxembourg) and Deborah Stranen (University of Luxembourg)

As noted earlier, the purpose of Plimpton 322 has long been debated. It is also not certain how the displayed numbers were constructed. Our decimal analogue activity leads students through a construction method that is generally referred to as the “Reciprocal Pairs Algorithm.” In this section, we outline and comment upon the steps of that algorithm, as they are presented in the student activity sheet. To download the activity sheet itself as a pdf, click here.

1. We need to work with decimal fractions. We call a natural number regular for base 10, or a regular decimal, if its reciprocal has a terminating decimal representation. The regular decimal numbers up to 100 are:

1 , 2 , 4 , 5 , 8 , 10 , 16 , 20 , 25 , 32 , 40 , 50 , 64 , 80 , 100.

In the classroom activity, the pupils first ﬁnd these numbers by themselves (as products of a power of 2 and a power of 5).

1. We take certain pairs of regular decimal numbers $$(r,R)$$ from the above list and construct the rational numbers $L=\frac{1}{2}\left(\frac{R}{r}-\frac{r}{R}\right) \,\,\,\,;\,\,\,\, H=\frac{1}{2}\left(\frac{R}{r}+\frac{r}{R}\right).$

We want L to be positive and smaller than 1, for reasons that we explain below; this requirement implies that we must take $$r<R<(1+\sqrt{2})r$$. Since we are only interested in the ratio $$\frac{r}{R}$$, however, we get repetitions by varying $$r$$ and $$R$$. For these reasons, the table we provide in the classroom activity indicates ﬁve admissible pairs $$(r,R)$$ that give all the possible ratios for the list of regular decimals from step 1.

1. We are also interested in the numerators of the reduced fractional forms of $$L$$ and $$H$$, which we refer to in the activity as their “least numerators” and denote by  and  respectively. We further make use of the least denominator $$d$$ of $$H$$, which is always equal to the least denominator of $$L$$ (for reasons we discuss below). Furthermore, $$d$$ is a regular decimal because it divides $$2rR$$. These numbers (and also $$L$$, $$H$$, and their squares) are displayed in Figures 2 and 3, in fractional and decimal form respectively.
 $$L$$ $$H$$ $$L^2$$ $$H^2$$ $$n_L$$ $$d$$ $$n_H$$ row # $$\frac{3}{4}$$ $$\frac{5}{4}$$ $$\frac{9}{16}$$ $$\frac{25}{16}$$ 3 4 5 1 $$\frac{39}{80}$$ $$\frac{89}{80}$$ $$\frac{1521}{6400}$$ $$\frac{7921}{6400}$$ 39 80 89 2 $$\frac{369}{800}$$ $$\frac{881}{800}$$ $$\frac{136161}{640000}$$ $$\frac{776161}{640000}$$ 369 800 881 3 $$\frac{399}{1600}$$ $$\frac{1649}{11600}$$ $$\frac{159201}{2560000}$$ $$\frac{2719201}{2560000}$$ 399 1600 1649 4 $$\frac{9}{40}$$ $$\frac{41}{40}$$ $$\frac{81}{1600}$$ $$\frac{1681}{1600}$$ 9 40 41 5

Figure 2: Values for the decimal analogue of Plimpton 322 (in reduced fractional form).

 $$L$$ $$H$$ $$L^2$$ $$H^2$$ $$n_L$$ $$d$$ $$n_H$$ row # 0.75 1.25 0.5625 1.6625 3 4 5 1 0.2875 1.1125 0.23765625 1.23785825 39 80 89 2 0.46125 1.10125 0.2127515625 1.2127515625 369 800 881 3 0.249375 1.030625 0.062187890625 1.062187890625 399 1600 1649 4 0.225 1.025 0.50625 1.0506625 9 40 41 5

Figure 3: Values for the decimal analogue of Plimpton 322 (in decimal form).

What are the numbers $$L$$ and $$H$$?
The rational numbers $$L$$and $$H$$ can be interpreted as the short leg and the hypotenuse of a right triangle in which the length of the long leg equals 1. This means that $$L$$and $$H$$ are strictly positive rational numbers satisfying $$L<1$$ and the identity $$L^2+1=H^2$$. Notice that $$L$$ and $$H$$ necessarily have the same least denominator because of this identity; to see this, one can first show that the least denominator of $$L$$ is a possible denominator for  $$H$$ and then show that the same property holds by swapping $$L$$ and $$H$$.

How do we get Pythagorean triples?
Each row of the table can be interpreted as representative of a right triangle whose rational side lengths are $$(L, 1,H)$$. If we rescale such a triangle so that the side lengths are coprime integers, then we get the Pythagorean triple

$(n_L,d,n_H).$

The number $$n_L$$ is the least numerator of L, d is the least denominator of $$L$$ and of $$H$$, and $$n_H$$ is the least numerator of $$H$$. A Pythagorean triple is called primitive if the three numbers have no common factor: by construction, $$(n_L,d,n_H)$$ is a primitive Pythagorean triple.

For example, the right triangle corresponding to the ﬁrst row has side lengths ($$\frac{3}{4}$$, 1, $$\frac{5}{4}$$), and we get the primitive Pythagorean triple (3, 4, 5). The distinctive aspect of the Pythagorean triples that we ﬁnd is that, as observed above, the middle number (which is the least denominator of $$L$$ and of  $$H$$ is a regular integer.

 This algorithm is used in the reconstruction method proposed in [Mansfield and Wildberger 2017]. It was originally proposed as a possible method of construction behind Plimpton 322 in [Bruins 1949] and also endorsed by a number of later scholars (e.g.,  [Buck 1980], [Schmidt 1980], [Friberg 1981], [Robson 2002]). Simoson  presents an explanation of the algorithm via a “proof without words” figure, based on Old Babylonian cut-and-paste geometric constructions.