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 mulitplication and division are based on addition, subtraction, and doubling. Therefore, one ingrediant 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:
Upon further analysis, you can perceive other principles used in constructing the table.
Back to course page