## Egyptian Fractions

Ed Pegg Jr., June 21, 2004

Imagine that you are overseeing a farm.  There are 29 equal sacks of grain to be distributed evenly, to 45 people. There is only one table, so you can't divide up more than one sack at a time.  How do you make sure each person gets a fair share?

Consider this:  1/3  + 1/5  + 1/9 = 29/45.  So, 15 of the sacks can be divided into 3 parts, 9 of them into 5 parts, and 5 of them into nine parts.  Then, each of the 45 people gets one of each type of part.  In the diagram below, each person gets a red, yellow, and green part.  Problem solved!

Figure 1.  Dividing 29 sacks equally between 45 people.  Each person gets .

This problem dates back over 4000 years. There was no algebra or geometry.  No decimals.  Whole numbers existed, but not fractions like 29/45.  The ancient egyptians would indicate a "part" number by a symbol that looked like a mouth, somewhat.  Thus, "part 7" would indicate 1/7. An egyptian fraction would be something like part 3 + part 5 + part 9. In other words, 1/3  + 1/5  + 1/9 = 29/45. As seen above, this system was useful in dividing sacks of grain.

How smart were the egyptians, 4000 years ago? You can see if you can solve one of their problems. Find three different whole numbers, all less than 600, so that their reciprocals (the "parts") add up to 2/95. Consider 48 and 4560.  As reciprocals, 1/48 + 1/4560 = 2/95.  That doesn't solve the problem, since 4560 exceeds 600.

As mentioned earlier, this problem was solved 4000 years ago, in around 2000 BC, without algebra, number theory, or computers.  A large table of similar solutions was recorded for posterity.  Later, in 1600 BC, a scribe by the name Ahmes found the document, and recorded the "ancient" results. He also wrote down a number of math problems to be solved just for fun, which makes him the earlier known recreational mathematician. Much later, in 1858 AD, Ahmes' papyrus scroll was found in Thebes, where it was bought by Henry Rhind. Eventually, the scroll went to the British Museum. The first two columns of the table below are the numbers scribed by Ahmes.

 Fraction Ahmes (Rhind) Papyrus,  1/a + 1/b as a + b 2/p = 1/A + (2A - p)/Ap,  with A = (p+1)/2 Other representations  of interest 2/3 2 + 6 2 + 6 2/5 3 + 15 3 + 15 2/7 4 + 28 4 + 28 6 + 14 + 21 2/9 6 + 18 5 + 45 2/11 6 + 66 6 + 66 2/13 8 + 52 + 104 7 + 91 10 + 26 + 65 2/15 10 + 30 8 + 120 12 + 20 2/17 12 + 51 + 68 9 + 153 2/19 12 + 76 + 114 10 + 190 2/21 14 + 42 11 + 231 15 + 35 2/23 12 + 276 12 + 276 2/25 15 + 75 13 + 325 2/27 18 + 54 14 + 378 2/29 24 + 58 + 174 + 232 15 + 435 2/31 20 + 124 + 155 16 + 496 2/33 22 + 66 17 + 561 2/35 30 + 42 18 + 630 20 + 140 2/37 24 + 111 + 296 19 + 703 2/39 26 + 78 20 + 780 52 + 60 + 65 2/41 24 + 246 + 328 21 + 861 2/43 42 + 86 + 129 + 301 22 + 946 2/45 30 + 90 23 + 1035 36 + 60 2/47 30 + 141 + 470 24 + 1128 2/49 28 + 196 25 + 1225 42 + 98 + 147 2/51 34 + 102 26 + 1326 2/53 30 + 318 + 795 27 + 1431 2/55 30 + 330 28 + 1540 40 + 88 2/57 38 + 114 29 + 1653 2/59 36 + 236 + 531 30 + 1770 2/61 40 + 244 + 488 + 610 31 + 1891 2/63 42 + 126 32 + 2016 56 + 72 2/65 39 + 195 33 + 2145 45 + 117 2/67 40 + 335 + 536 34 + 2278 2/69 46 + 138 35 + 2415 2/71 40 + 568 + 710 36 + 2556 2/73 60 + 219 + 292 + 365 37 + 2701 2/75 50 + 150 38 + 2850 60 + 100 2/77 44 + 308 39 + 3003 63 + 99 2/79 60 + 237 + 316 + 790 40 + 3160 2/81 54 + 162 41 + 3321 2/83 60 + 332 + 415 + 498 42 + 3486 2/85 51 + 255 43 + 3655 55 + 187 2/87 58 + 174 44 + 3828 2/89 60 + 356 + 534 + 890 45 + 4005 2/91 70 + 130 46 + 4186 2/93 62 + 186 47 + 4371 2/95 60 + 380 + 570 48 + 4560 2/97 56 + 679 + 776 49 + 4753 2/99 66 + 198 50 + 4950 90 + 110 2/101 101 + 202 + 303 + 606 51 + 5151

Accurate reckoning: the entrance into knowledge of all existing things and all obscure secrets. -- Ahmes, 1600 BC

What method did the ancient egyptians use to come up with these? In answer to the earlier question, 2/95 = 1/60 + 1/380 + 1/570. The Ahmes table has many "best" solutions, with either the lowest largest denominator, or other optimizations.

One method, suggested by Akhmim P. Eves, uses n/pq = 1/(p(p + q)/n) + 1/(q(p + q)/n). For example:

2/35, p = 5, q = 7, or 2/(5 7) = 1/(5(5+7)/2) + 1/(7(5+7)/2) = 1/30 + 1/42
2/91, p = 7, q = 13, or 2/(7 13) = 1/(7(7+13)/2) + 1/(13(7+13)/2) = 1/70 + 1/130

Note, as well, the solution for 2/101, namely 2/p = 1/p + 1/2p + 1/3p + 1/6p. Ahmes likely knew of some kind of proto-number theory, and seemed well-versed in it. Every composite entry in the table is based on a decomposition of n into its prime factors.

Mathematica Code:

Code used for this column can be found in the Mathematica Information Center, at the Ten Algorithms for Egyptian Fractions entry, by David Eppstein.

References:

David Eppstein, Ten Algorithms for Egyptian Fractions, Mathematica in Education and Research, 1995, p5-15.

Bruce Friedman, mathorigins.com, http://www.mathorigins.com/image grid/BRUCE OLD_005.htm.

J J O'Connor and E F Robertson, Egyptian numerals, http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Egyptian_numerals.html.