For finding the reciprocals of regular numbers not in the table, the students used a standard procedure, first explained by A. Sachs and called The Technique. Here we briefly describe the algorithm and give some examples.

For the convenience of the modern reader, we give an utterly ahistorical
justification of the Technique in modern algebraic terms. The basic idea
is to write the reciprocal of *r *as a product of two terms, one of
which we already know because it is in the table. In order to do
this, we first write *r* as a sum *r = x + y*, where *x*
is a number from the standard reciprocal table. Then we notice that

Let us go through a simple example, using the favorite Old Babylonian
number `2,5` (= 125). This number is called the *igum*.
Its reciprocal, which we want to find, is called the *igibum*.
According to the technique, we want to write `2,5 `as the sum of
two numbers, one of them from the standard table. The Babylonians
'broke off' the largest number that was in the table, in this case 5.
The reciprocal
of `5` is `12`. Multiply `12` into (the remaining)
`2`
to get `24`. Add `1`, you will see `25`. The
reciprocal of `25` is `2,24`. Multiply `2,24`
by `12`. You will see `28,48`. The *igibum
*is
`28,48`.

As a step-by-step procedure, we proceed as follows:

Step 0: Given a regular number. (`2,5`)

Step 1: Break off the largest number in the standard reciprocal table.
(5)

Step 2: Find its reciprocal. (`12`)

Step 3: Multiply this number by the remainder of the original number.
(`12` times `2` is `24`)

Step 4: Add 1. (`1` plus `24` is `25` )

Step 5: Find the reciprocal of this number (repeat steps 1 to 4 if
necessary) (reciprocal of `25` is `2,24`)

Step 6: Multiply the original reciprocal by this one. (`2,24`
times `12` is `28,48`)

The Technique is well-described in the tablet VAT 6505, published by
Neugebauer in MKT 1, 270ff. The tablet is somewhat broken and not
all the problems can be restored. Here are the numbers from two of
the problems, written in cuneiform so it is easy to see the 'breaking off':
has the
broken off, and
has the
broken off.

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Last modified: 6 June 2001

Duncan J. Melville

dmelville@stlawu.edu