The ancient Egyptians had notation to represent the fraction 1/n, but not for every fraction of the form a/b. Because of this, they typically represented fractions as Egyptian fractions, which are finite sums of fractions of the form 1/n, all with different denominators. For example, 5/121 can be expressed in the form of an Egyptian fraction as 1/33 + 1/121 + 1/363.
The theory of Egyptian fractions is discussed in the Rhind mathematical papyrus, shown below. The manuscript was written by the scribe Ahmes in about 1550 BC, although Ahmes did not invent the technique, and the idea is thought to date back to around 2000 BC. Even though the idea of Egyptian fractions is no longer necessary, it has attracted the attention of other mathematicians over the centuries, and is still a current area of research.
In 1202, Fibonacci (c.1170–1245) described an algorithm for calculating Egyptian fraction representations which is now called the greedy algorithm. For example, to calculate an Egyptian fraction representation of 7/15, one starts with the largest possible fraction (hence the name “greedy”) that is less than 7/15 and that has the form 1/n. The number ½ is too big, because it exceeds 7/15, but ⅓ will work. We then write 7/15 as ⅓ + 2/15, and run the algorithm on the new term, 2/15. In this case, 1/7 is bigger than 2/15, but ⅛ is smaller than 2/15, so we write 2/15 = ⅛ + 1/120. The procedure now terminates because all the numerators are 1, and we have 7/15=⅓ + ⅛ + 1/120.
The greedy algorithm always terminates successfully, and it can be proved that the numerator of the fraction gets smaller each time the algorithm is applied. Unfortunately, the greedy algorithm sometimes produces very inefficient results. For example, running the algorithm on the fraction 5/121 produces five terms, one of which has a 25 digit number as its denominator. It is possible to do much better than this: there is a 3-term Egyptian fraction for 5/121 with reasonably sized denominators. The result of running the greedy algorithm on 31/311 is even worse: it produces ten terms, the last of which has a denominator with over 500 digits.
Much of the modern interest in Egyptian fractions can be attributed to J.J. Sylvester (1814–1897). Sylvester was interested in representing the number 1 as an infinite sum of Egyptian fractions, as shown above. The rule for generating the denominators of this series (2, 3, 7, 43, 1807, and so on) is that each successive term is given by 1 plus the product of the previous terms. It turns out that this series solves the problem of how best to underapproximate the number 1 using a sum of k Egyptian fractions.
Expressing 1 as a sum of Egyptian fractions is not difficult, because 1=½ + ⅓ + ⅙. What the underapproximation result says is that the first three terms of Sylvester’s series, ½ + ⅓ + 1/7, are the best way to underapproximate the number 1 using a 3-term Egyptian fraction. This means that if we pick any other three distinct denominators, such as 3, 4, and 5, with the property that ⅓ + ¼ + ⅕ < 1, then it is guaranteed that ⅓ + ¼ + ⅕ ≤ ½ + ⅓ + 1/7=41/42. Another way to say this is that any fraction bigger than 41/42 but less than 1 will require at least four terms in its Egyptian fraction representation.
It turns out that the best way to underapproximate 1 using a 4-term Egyptian fraction is ½ + ⅓ + 1/7 + 1/43, and this can be obtained simply by taking the best underapproximation with a 3-term Egyptian fraction (½ + ⅓ + 1/7) and adding one more new term (1/43). If a number x has the property that, for large n, the best way to underapproximate x with an n-term Egyptian fraction is to add one more term to the best underapproximation with n–1 terms, then the number x is said to have eventually greedy best Egyptian approximations. The previous discussion shows that the number x=1 has this property.
Two recent prominent mathematicians who did research on Egyptian fractions were Paul Erdős (1913–1996) and Ronald Graham (1935–2020), pictured above. In their 1980 monograph Old and new problems and results in combinatorial number theory, Erdős and Graham stated without proof that every positive rational number has eventually greedy best Egyptian approximations. They also speculated that this property holds for almost all real numbers. However, the recent paper On eventually greedy best underapproximations by Egyptian fractions by Vjekoslav Kovač proves that, on the contrary, the set of numbers with the “eventually greedy” property has Lebesgue measure zero. What this means is that it is vanishingly unlikely that a real number chosen at random will have eventually greedy best Egyptian approximations.
Kovač’s paper includes a brief survey of several other papers from the last few years on the topic of Egyptian fractions. I find it remarkable that this is still an active area of research, given that the idea is 4000 years old!
Picture credits and relevant links
The photograph of the Rhind Mathematical Papyrus is in the public domain. It appears on the Wikipedia page on Egyptian fractions.
The photograph of Paul Erdős is by Kmhkmh and the photograph of Ronald Graham is by Cheryl Graham. Both pictures appear on Wikipedia.
The complicated expression for 5/121 appears on the Wikipedia page on the greedy algorithm.
The other graphics are my own work.
Wikipedia also has entries on Fibonacci, James Joseph Sylvester, Paul Erdős, and Ronald Graham.
The sequence of denominators in Sylvester’s series is sometimes known as Sylvester’s sequence. It appears as sequence A000058 in the On-Line Encyclopedia of Integer Sequences.
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Nice post! For what it's worth the greedy algorithm is also applied to the knapsack problem. A nice introduction to that can be found in John Guttag's Introduction to Computation and Programming using Python.
Excellent history too of ancient math! I never knew how such civilizations as ancient Egypt dealt with mathematics. Further posts on ancient understanding would be really interesting.