The Egyptian 2/n table, the recto table of the Ahmes (Rhind) papyrus

The Egyptian concept of fraction requires that any fraction be represented as a sum of unit fractions without any repetitions, except 2/3 which was allowed. Thus, for example, our common fraction 2/5 would be treated as a problem, not as an answer. The problem is to divide 2 by 5; the answer would be any sum of unit fractions without repetition. One answer is 1/3 + 1/15, the preferred answer. Another possible answer would be 1/4 + 1/10 + 1/20, but that's a more complicated answer having both more terms and larger denominators. Note that 1/5 + 1/5 would not be an answer because 1/5 is repeated.

The Egyptian algorithms for multiplication and division are based on addition, subtraction, and doubling. Therefore, one ingredient necessary to compute products and quotients involving fractions is a table of doubles of unit fractions. It's also necessary for addition since when adding two sums of unit fractions, some particular unit fraction might occur twice.

The back (recto) of the most important Egyptian mathematical papyrus, the Ahmes, or Rhind, papyrus, includes a table of doubles of unit fractions. We can call it a 2/n table. Here it is, transcribed into modern numerals. Note that only the denominators are listed in this transcription. In one column appears the denominator of the unit fraction to be doubled, and in the next column appear the denominators of the unit fractions for that double.

5 3 15
7 4 28
9 6 18
11 6 66
13 8 52 104
15 10 30
17 12 51 68
19 12 76 114
21 14 42
23 12 276
25 15 75
27 18 54
29 24 58 174 232
31 20 124 155
33 22 66
35   30 42
37 24 111 296
39 26 78
41 24 246 328
43 42 86 129 301
45 30 60
47 30 141 470
49 28 196
51 34 102
53 30 318 795
55 30 330
57 38 114
59 36 236 531
61 40 244 488 610
63 42 126
65 39 195
67 40 355 536
69   46 138
71 40 568 710
73 60 219 292 365
75 50 150
77 44 308
79 60 237 316 790
81 54 162
83 60 332 415 498
85 51 255
87 58 174
89 60 356 534 890
91 70 130
93 62 186
95 60 380 570
97 56 679 776
99 66 198
101  101 202 303 606

At first glance, the only apparent regularity in the table occurs for denominators divisible by 3, and for those the rule is:

2
3n		  =  1
2n		  +  1
6n

Upon further analysis, you can perceive other principles used in constructing the table.





(taken from a table of Prof. David Joyce, Clark University)