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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.
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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 1*p*, 2*p*, ..., all the
way up to 59*p*. In fact, what they did was to give all the multiples
from 1*p* up to 20*p*, and then go up in steps of 10, so finishing
the table with 30*p*, 40*p* and 50*p*. If you wanted to
know, say, 47*p*, you added 40*p* and 7*p*. 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 |
2*p* |

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.

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Last modified: 6 June 2001
Duncan J. Melville

dmelville@stlawu.edu