Old Babylonian Multiplication and Reciprocal Tables
These "table texts" are the first known examples of a tradition of mathematical tablemaking that is still alive and useful today....
Tony Phillips
Stony Brook University
Email Tony Phillips
An online survey of Old Babylonian mathematics is available on Duncan Melville's website.
Introduction
"Old Babylonian" refers here to the civilization that flourished some four thousand years ago (around 2000 B.C.) in what is presentday Iraq. Documents at the time were written in cuneiform ("wedgeshaped") characters on clay tablets, most of them rectangular and of a size to be held comfortably in the hand (about $5\times 8$cm or $2\times 3$ inches). Hundreds of thousands of tablets have been excavated from the ruins of this civilization; perhaps a thousand or so of these concern mathematics. For example, the collection of the University of Pennsylvania Museum of Archaeology and Anthropology, catalogued, with photographs, on the website of the Cuneiform Digital Library Initiative, includes 458 Old Babylonian mathematical tablets (of which about 110 have been published).
For Old Babylonians, mathematics was practical. Many of the mathematical tablets are "problem texts:" they contain problems or sets of problems, sometimes with solutions. Many of the problems involve geometry; the rest are almost always "word problems" where the context is the calculation of the area of an irregular field, the volume of a ditch, the number of bricks to build a ramp, ... . Understanding word problems requires, besides a knowledge of the language used to record them, a thorough understanding of the culture (in particular the different systems of units and their relation). Even the geometry problems require a specialized vocabulary and familiarity with the Babylonian way of explaining a solution. But along with the problem texts are "table texts," which can be understood by anyone who has mastered the Babylonian number system; they are the first known examples of a tradition of mathematical tablemaking that is still alive and useful today.
Multiplication tables from my Grade 3B notebook. The format is the same as the Old Babylonian (see the 12times table below). A significant difference is the listing of $0$ as a number; the Old Babylonians had no such symbol. 
There are tables of squares, of square roots, of cube roots and of similar computations; here we will concentrate on the two most basic: multiplication tables and tables of reciprocals (for Old Babylonians, division by $n$ amounted to multiplication by $\frac{1}{n}$). The study of these texts allows us a real intellectual connection with users of mathematics who preceded the Romans by as many years as the Romans preceded us.
My thanks to Bill Casselman and to Christian Siebeneicher for furnishing me with images (and to the Yale Balylonian Collection for permission to reproduce NBC 7344); and above all to Jöran Friberg who made me a set of beautiful images of tablets from the Martin Schøyen Collection. References to "Friberg" in this text lead to his book "A Remarkable Collection of Babylonian Mathematical Texts" (Springer, 2007).
The Old Babylonian number system
The Old Babylonian number system is characterized by three features:

Numbers are represented in a base60 ("sexagesimal") placevalue system. This was a local, and very significant, invention; we still retain their notation for the measurement of angles and time: degrees (or hours), minutes and seconds.

Numbers from 1 to 59 (there is no symbol for zero) are represented by combinations of a symbol for 1 and a symbol for 10. Like Roman numerals using only I and X. The Babylonian numbers from 1 to 20 (plus 30, 40, 50) appear in the central columns of each of the four multiplication tables shown below.

There is no "decimal point;" as Friberg puts it: "All Babylonian sexagesimal numbers are written as if they were integers, but the intended value of a sexagesimal number in a Babyonian cuneiform text can be its 'nominal' value multiplied by any positive or negative power of 60." The scale of the intended value must be determined by the context.
A tentimes table
In decimal notation a 10times table is not very interesting, but in base60 things are less trivial.



This 10times table, tablet VAT7858 in the collection of the Vorderasiatisches Museum, Berlin, exhibits one of the common formats for multiplication tables: the first line (partially illegible here) reads
10 arà(times) 1 10.
From line 2 on, in this format, the multiplier is left out:
arà 2 20
arà 3 30,
stepping up the multiplicand by 1 each time. VAT7858 gives us a nice introduction to Old Babylonian base60 placevalue notation:
arà 6 1
arà 7 1 10
...
arà 13 2 10 



The reverse reads
arà 14 2 20
...
arà 19 3 10
arà 20 3 20.
After 20, as usual, come the multiplicands 30, 40 and 50:
arà 30 5
and (partially illegible)
arà 40 6 40
arà 50 8 20.
The table ends with a colophon, or subscript, usually the scribe's name and the date.
Photographs by the late Peter Damerow (Damerow was active in the Cuneiform Digital Library Initiative), courtesy of Christian Siebeneicher.

A fivetimes table
A twelvetimes table



On this 12times table (tablet MS2184/3 in the Schøyen collection), the format is different from the more standard NBC7344: the multiplier is repeated on each line:
12 arà 1 12
12 arà 2 24
...
12 arà 10 2. 



On the reverse:
12 arà 11 2 12
...
12 arà 19 3 48
12 arà 20 4
12 arà 30 6
(the last three lines on the back of the tablet are partially illegible, but note again a special "subtractive" symbol for 19). 



The last two lines wrap around the bottom/top of the tablet:
12 arà 40 8
12 arà 50 10.
Photographs courtesy of Jöran Friberg. 
A $1\frac{1}{5}$times table



This table (MS3866 in the Schøyen collection) has "headnumber" 1 12, so it is a $1\frac{1}{5}$times table; it follows the standard format:
1 12 arà 1 1 12
arà 2 2 24
...
arà 16 19 12. 



On the reverse the first three lines are partially missing, but they would read:
arà 17 20 24
arà 18 21 36
arà 19 22 48 (note another "subtractive" symbol for 19).
Then:
arà 20 24
arà 30 36
arà 40 48
arà 50 1.
MS3866 ends with the square of the headnumber:
1 12 arà 1 12 1 26 24.
Photographs courtesy of Jöran Friberg. 
A table of reciprocals
The standard OldBabylonian table gives the reciprocals of the numbers ${\bf \frac{3}{2}}$, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 1, 1 04, 1 21; reciprocals are the sexagesimal numbers with which they multiply to give 1. Some tables omit the ${\bf \frac{3}{2}}$, some also give 1 12, 1 15, 1 20. Since 1 represents any power of $60 = 2^2\cdot 3\cdot 5$, these numbers must all be regular: they can have only 2, 3, and 5 as prime factors.
The numbers in the standard table have the property that they and their reciprocals require no more than three sexagesimal places. If we were to try a similar construction with our decimals, the list could only include $1, 2\leftrightarrow 5, 4\leftrightarrow 25, 8\leftrightarrow 125$ and $16\leftrightarrow 625$.
The tablet shown here is MS3874 in the Schøyen collection. The transcription and (translation) are copied from Friberg.
For mathematical uniformity, the first two lines can be understood as: "the reciprocal of 3/2 is 40," and "the reciprocal of 2 is 30."
Tony Phillips
Stony Brook University
Email Tony Phillips