Why Unit Fractions?
Why did the ancient Egyptians (and others) persist in their use of Egyptian fractions for so many centuries? Was it a conceptual limitation, or simply a matter of notation? Some scholars contrast the _exactness_ of Egyptian expansions with the approximate nature of fixed-base expansions such as in decimal system and the Babylonian sexigesimal system. This contrast is interesting, although it takes some effort for modern readers (accustomed to fixed-base representations) to imagine the intellectual difficulties involved in this paradigm shift. The Egyptian preference for exact expansions reminds me of the Greek preference for geometry over symbolic arithmetic. When the Greeks discovered irrational numbers they realized that rational arithmetic can only approximate the values of most real numbers. As a result, not wanting to deal with approximations, they devoted themselves mainly to geometry. Even within geometry, their insistence on being able to give _exact_ constructions using straight-edge and compass is similar to the Egyptian insistence on giving exact expansions using unit fractions. There are at least two separate aspects to Egyptian fraction expansions that makes them puzzling to modern people. One is the variable "base", e.g., rather than expanding a fraction into a sum of fractions with the denominators equal to powers of a single base number (such as 10 or 60), they freely chose denominators to give an exact identity. Thus, while the Babylonians might have expressed 1/7 as (approximately) 1/7 = 8/60 + 34/60^2 + 17/60^3 the Egyptians would have preferred the exact expansion 1/7 = 1/14 + 1/21 + 1/42 This also highlights the other puzzling aspect of Egyptian fractions, namely, their preference for unit numerators. This might have derived from their "binary" approach to integer arithmetic, in which successive doublings of the operands were used to multiply numbers, so that effectively their numbers were expressed in the form N = c0*2^0 + c1*2^1 + c2*2^2 + ... where the coefficients ci are either 0 or 1. When they expanded their arithmetic to include fractions, they expressed all numbers in the form p/q = n1*A^-1 + n2*B^-2 + n3*C^-3 + ... where again the coefficients are either 0 or 1, but realizing that using A=B=C=...=2 would not allow exact expansions, they used independently variable denominators A,B,C.. Still, it isn't clear what purpose was served by the Egyptian unit fractions. Presumably one of the basic motivations for expanding rational fractions is to enable the comparison of different quantities. For example, if someone offers us 1/7 of a bushel of corn and someone else offers us 13/89 of a bushel, which should we take? In the Babylonian the two numbers would be expressed in sexigesimal as 1/7 = 8/60 + 34/60^2 + 17/60^3 13/89 = 8/60 + 45/60^2 + 50/60^3 Note that while the first terms are identical, the second term of 13/89 is larger than the second term of 1/7. In fact, it's clear that the fraction 13/89 exceeds 1/7 by about 6/60^2 + 33/60^3. This shows the value of expressing fractions in a fixed-base system: it enables us to immediately assess the relative magnitudes of different quantities by placing them on a common basis. However, the Egyptian approach doesn't seem to serve this purpose. One possible Egyptian expansion of 1/7 is 1/14 + 1/21 + 1/42, but how would they expand 13/89? Using a "binary" approach, they might have considered first expanding the numerator into powers of 2 as follows 13/89 = 8/89 + 4/89 + 1/89 Then from a table of 2/n expansions they would find 2/89 = 1/60 + 1/356 + 1/534 + 1/890 which immediately gives 4/89 = 1/30 + 1/178 + 1/267 + 1/445 and 8/89 = 1/15 + 1/89 + 2/267 + 2/445 The 1/89 term in the expansion of 8/89 could be combined with the 1/89 in the original expansion to give 2/89, for which we could substitute the 2/n table expression above. The terms 2/267 and 2/445 could be written as (1/3)(2/89) and (1/5)(2/89) respectively, so again we could substitute from the 2/n table expression. Adding up these terms would give 13/89 = 1/15 + 1/30 + 1/60 + 1/178 + + 1/180 + 1/267 + 1/300 + 1/356 + 1/445 + 1/534 + 1/890 + 1/1068 + 1/1335 + 1/1602 + 1/1780 + 1/4450 This gives a complete unit fraction expansion for 13/89, but it isn't obvious how this facilitates a comparison with 1/14 + 1/21 + 1/42. At some point, they would need to place the two numbers on a common denominator. Of course, we don't actually know how the ancient Egyptians would have expanded 13/89, since the tables that have survived don't include any general rules. Possibly they had some way of expanding the first few terms on specified denominators for purposes of comparisons, but there is no evidence of this. Based on the examples they gave, we would expect them to expand 13/89 into something like 13/89 = 1/8 + 1/48 + 1/4272 or perhaps, to minimize the largest denominator, the might have used 13/89 = 1/12 + 1/20 + 1/90 + 1/801 + 1/2670 but this still gives no easy basis of comparison with some other fraction, such as 1/7 = 1/14 + 1/21 + 1/42. The most expedient way of comparing the magnitudes would be to simply cross-multiply to clear the fractions, finding that 13*7 = 91 exceeds 1*89 = 89, but again there is no evidence the ancient Egyptians looked at it this way. Although, there are undeniably several interesting algebraic patterns in the historical Egyptian expansions, it nevertheless remains unclear (to me) what the purpose of those expansions really was, i.e., what function they served. Were they just exercises in manipulation or did they serve some useful purpose? How did the Egyptians compare the sizes of two general fractions? How did they add, subtract, multiuply, and divide general fractions? Did they USE the 2/n table for anything? Some have suggested that the partitioning of estates might have been one motivation, and I can see that this might have given the Egyptians a special interest in unit fractions, although I'm not sure what benefit they got from expressing unit fractions as sums of other unit fractions. It's interesting to consider other possibilities, such as gambling. The modern theory of probability originated in a series of letters between Fermat and Pascal on the subject of partitioning the stakes of an unfinished game of chance. I'm no scholar of ancient cultures, but I'd be willing to bet that the ancient Egyptians practiced some forms of gambling. Maybe some forgotten predecessors of Fermat and Pascal were concerned about the same thing, and worked out a set of mathematical techniques for dealing with these kinds of partitions. Still, I can't quite see how to make use of Egyptian unit fractions for any of these purposes. One possible reason the practice of expressing numbers as unit fractions endured for so long is the limitations of notation. Darrah Chavey points out that the ancient Egyptians wrote a number 1/n as the number n with an oval above it. This is just a single-variable symbol, and doesn't readily accommodate the two variables needed to express the ratio of an arbitrary numerator and denominator. It is necessary to devise a completely new notation. Moreover, not only is a new notation required, it may have been difficult for them to imagine a single quantity with TWO variable and independent arguments. They could adjoin unit fractions by addition, but couldn't conceptually consolidate them ito a single entity. Searching for clues to explain the motives behind the use of unit fractions, we might examine the Rhind Papyrus itself. Recall that Ahmes poses the problem of dividing 3 loaves of bread equally between 5 people. Naturally each person get's 3/5 of a loaf, but there are multiple distinct ways of partitioning the loaves to accomplish this. One way would be to cut each loaf into five equal parts and give each person three parts. This would require 12 cuts. Another way would be to make one cut in each loaf, dividing it into 3/5 and 2/5 parts, and give each of three people one of the 3/5 parts. This leaves three parts of size 2/5. One of these could be cut in half, and each of the remaining two people could be given a 2/5 and and 1/5 slice. This would require only 4 cuts. In the book "Ancient Puzzles", Dominic Alivastro suggests that Ahmes might have wanted to solve this problem by cutting one loaf into five equal slices, and the other two loaves each into three equal slices. Then take one of the 1/3 loaf slices and cut it into five equal slices. Each person could then be given his share in the form 3/5 = 1/3 + 1/5 + 1/15 Alivastro suggests that this might be more readily perceived as equable than the partition 3/5, 3/5, 3/5, (1/5 + 2/5), (1/5 + 2/5), although it must be said that the uniform partition into congruent shares (1/5 + 1/5 + 1/5) would presumably be even more obviously equable. Both of these partitions requires 12 cuts, so we cannot prefer one over the other based on economy of cuts. Overall the most plausible explanation for the ancient fixation on unit fractions seems to be that they had difficulty conceiving of a single quantity in terms of two variables (numerator and denominator), and were looking for simple "whole" fractional quantities. Just as the "whole" natural numbers are those of the form n/1, it was natural to imagine that the "whole" fractional numbers are of the form 1/n.