Sexagesimal reciprocals
The Babylonians used several tables for computation. One basic table was
a reciprocal table. This table listed "regular" numbers and their reciprocals. A regular number is one whose reciprocal
can be expressed in a finite number of digits in base 60, in other words, a number which is a product of 2's, 3's, and 5's. A
few nonregular numbers are 7, 11, 13, 14, 17, 19, and 21.
This reciprocal table was used for division. To divide by a number, just look up its reciprocal, and then multiply by the
reciprocal. Thus, a separate division algorithm is not needed.
Below are two tables of reciprocals of regular numbers. The first includes reciprocals through 60,
and it's short enough to quickly construct by hand. The second goes through 3600, but would take much
more time to complete by hand.
Note that
there is no decimal point. Each line gives several reciprocals. Consider the line with 1:20 and 45. This says that to divide
by 1:20 (which is 80 decimally), instead multiply by 45. Also, to divide by 45, instead multiply by 1:20. Of course, decimal
points, actually se agesimal points, were not recorded by the Babylonians, so this line was interpreted several ways based on
where the implicit sexagesimal point was located. For example,
Number Reciprocal Interpretation
1.20 .45 1 + 20/60 and 45/60 are reciprocals
1:20. .00:45 60 + 20 and 45/3600 are reciprocals
.01:20 45. 1/60 + 20/3600 and 45 are reciprocals
Table of reciprocals of regular numbers through 60
1  1 
1:06:40  54 
1:12  50 
1:15  48 
1:20  45 
1:30  40 
1:40  36 
1:52:30  32 
2  30 
2:13:20  27 

2:24  25 
2:30  24 
3  20 
3:45  16 
4  15 
5  12 
6  10 
6:40  9 
7:30  8 

Table of reciprocals of regular numbers through 3600
1  1  1:02:30  57:36  1:04  56:15 
1:06:40  54  1:07:30  53:20  1:09:07:12  52:05 
1:10:18:45  51:12  1:12  50  1:14:04:26:40  48:36 
1:15  48  1:20  45  1:21  44:26:40 
1:23:20  43:12  1:24:22:30  42:40  1:26:24  41:40 
1:28:53:20  40:30  1:30  40  1:33:45  38:24 
1:36  37:30  1:38:45:55:33:20  36:27 
1:40  36  1:45:28:07:30  34:08  1:46:40  33:45 
1:48  33:20  1:51:06:40  32:24  1:52:30  32 
1:55:12  31:15  2  30  2:05  28:48 
2:08  28:07:30  2:13:20  27  2:15  26:40 
2:20:37:30  25:36  2:24  25  2:28:08:53:20  24:18 
2:30  24  2:40  22:30 

2:42  22:13:20  2:46:40  21:36  2:48:45  21:20 
2:52:48  20:50  2:57:46:40  20:15  3  20 
3:07:30  19:12  3:12  18:45  3:20  18 
3:30:56:15  17:04  3:36  16:40  3:42:13:20  16:12 
3:45  16  4  15  4:03  14:48:53:20 
4:10  14:24  4:16  14:03:45  4:26:40  13:30 
4:30  13:20  4:41:15  12:48  4:48  12:30 
4:56:17:46:40  12:09  5  12 
5:20  11:15  5:24  11:06:40  5:33:20  10:48 
5:37:30  10:40  5:45:36  10:25  6  10 
6:15  9:36  6:24  9:22:30  6:40  9 
6:45  8:53:20  7:01:52:30  8:32  7:12  8:20 
7:24:26:40  8:06  7:30  8 

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