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Mathematics in Babylon - The reciprocals of regular numbers The Technique is a
procedure for finding the reciprocals of regular numbers. A
regular number is one which can be expressed as a multiple of 2, 3,
and 5 only, and so has a finite sexagesimal reciprocal. Old
Babylonian scribes went to a lot of trouble in school texts to make
sure that students only had to find reciprocals of regular numbers.
Indeed, most of the time, students were just expected to find
reciprocals that were in the standard table (this gave reciprocals of
regular numbers between 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 . Because Let us go through a
simple example, using the favorite Old Babylonian number Multiply As a step-by-step procedure, we proceed as
follows: Step 1: Break off the largest number in the standard reciprocal table. (5) Step 2: Find its reciprocal. ( Step 3: Multiply this number by the remainder of
the original number. ( Step 4: Add 1. ( Step 5: Find the reciprocal of this number
(repeat steps 1 to 4 if necessary) (reciprocal of Step 6: Multiply the original reciprocal by this
one. ( 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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