# OB Multiplication Tables

Old Babylonian multiplication tables come in two varieties. There are the so-called single tables, listing multiples of a single number, called the principal number, and there are combined tables which have a number of single tables on one tablet. Often, multiplication tables occur on the same tablet as other, unrelated, information and it is clear that many of the tablets we have were school practice copies made by students learning their multiplication tables.

### Single Multiplication Tables

Single multiplication tables list the multiples of a single number, known as the principal number, p. Because the Mesopotamians used a sexagesimal (base 60) number system, you might expect that a multiplication table would have to list all the multiples from 1p, 2p, ..., all the way up to 59p. In fact, what they did was to give all the multiples from 1p up to 20p, and then go up in steps of 10, so finishing the table with 30p, 40p and 50p. If you wanted to know, say, 47p, you added 40p and 7p. Sometimes the tables finished by giving the square of the principal number, too.

Most of the principal numbers are regular sexagesimal numbers that also appear on the standard table of reciprocals. Part of the reason for this is that the Mesopotamians treated division as "multiplication by the reciprocal." Instead of computing 19 (divided by) 12, they would compute 19 (times the reciprocal of) 12.

The numbers in the standard reciprocal table for which there are no single multiplication tables known are:

1,04 1,06,40 1,12 1,21 1,52,30 2,13,20 27 32 54
Among these missing numbers there are three pairs of reciprocals, each pair having one three-digit member. Perhaps the oddest omission is the number 1,21 (= 81 decimal), whose reciprocal 44,26,40 is the only three-place sexagesimal number which does occur as the principal number for a single multiplication table. We might dismiss this as an accident of discovery were it not for the story Neugebauer tells in Exact Sciences in Antiquity (note 18 to Chapter 1). If a number does not have a finite sexagesimal reciprocal, that is, it is not regular, then the Mesopotamian scribes would say, "it does not divide." Referring to a tablet in the British Museum (BM 85210), which he had published in MKT, Neugebauer says:
'That a scribe was sometimes not sure when a number was regular or irregular is shown by a statement ... to the effect that "4,3 does not divide." This is wrong because 4,3 = 3 x 1,21 and both 3 and 1,21 are regular numbers whose reciprocals can be found in the standard table.'
In order to write "4,3 does not divide," the scribe clearly had to be unaware of the factorization of 4,3 as 1,21 times 3. This suggests the scribe did not know his 1,21 times table, which implies that he didn't have one. That is, it is possible that the Babylonians never did write a single multiplication table with principal number 1,21

The numbers that are not in the standard reciprocal table, but do have single multiplication tables are:

7 (the only number under 10 that is not regular sexagesimal);
22,30 (probably should be thought of as 3/8 or reciprocal of 2,40 = 160 decimal);
16,40 (1000 in decimal);
12,30 (750 in decimal);
8,20 (500 in decimal);
4,30 (reciprocal of 13,20 = 800 decimal);
1,15 (75 in decimal).
There are some later tablets (i.e., not Old Babylonian) that have other numbers as principal numbers of single multiplication tables.

It is worth noting that although the Babylonian 'scientific' system of computation was sexagesimal, they did go to the trouble of writing multiplication tables for the decimal numbers 100 (1,40), 200 (3,20), 400 (6,40), 500 (8,20), 750 (12,30), and 1000 (16,40) as well as the numbers 300 (5), 600 (10) and 900 (15) which occurred naturally. This indicates the strength of a numerical substrate based on 100.

Over 160 single multiplication tables are known; they come in a number of different forms, all of which are slight variants of each other. In the most common type, the table is written
 p a-rá 1 p a-rá 2 2p and so on.
The word 'a-rá,' of course, means 'times.'

### Combined Multiplication Tables

Combined multiplication tables have a number of single multiplication tables written on one tablet. Almost all of the principal numbers appearing in combined tables also appear as the principal number on a single table. One of the most interesting facts about OB combined multiplication tables is that the individual sub-tables are nearly always written in the same order of descending principal number. This implies that many of the known tablets are practice copies of portions of a complete or 'canonical' list of tables. Most tablets start either at the beginning or in the middle of the canonical list.

Among the 80 or so known combined multiplication tables one of them (A 7897) is a large cylinder containing an almost complete set of tables written in 13 columns. There is a hole through the center of the cylinder so that it could be turned on some kind of peg. The cylinder is quite fragmentary, but helps to construct the canonical list. The others survive in greater or lesser fragments and each included less sub-tables, but together they allow us to reconstruct the overall list.

The canonical table begins with a standard table of reciprocals and follows it with multiplication tables for the principal numbers

p = 50 48 45 44,26,40 40 36 30 25 24 22,30 20 18 16,40 16 15 12,30 12 10 9 8,20 8 7,30 7,12 7 6,40 6 5 4,30 4 3,45 3,20 3 2,30 2,24 2,15 2 1,40 1,30 1,20 1,15.
Of these, only 48 2,15 and 1,20 are not known as principal numbers for single multiplication tables, and 48 was often omitted from combined tables, too. There are no single multiplication tables (from the OB period) with principal numbers not on the canonical table.