Egyptian fraction

An Egyptian fraction is the sum of distinct unit fractions (that is, fractions whose numerators are equal to 1) whose denominators are positive integers, and all of whose denominators differ from each other, for example $\begin{matrix}\frac{1}{2}+\frac{1}{3}\end{matrix}$ . It can be shown that every positive rational number, a/b, can be written in the form of an Egyptian fraction following ancient and modern methods. The ancient methods will be described below.

But first, it should be noted that modern Harmonic series can be used to prove every positive rational number can be written in the form of an Egyption fraction. However, modern methodologies produce awkward and relatively longer unit fraction series compared to the shorter and more concise ancient methods.

The greedy algorithm for computing modern Egyptian fractions

An algorithm for calculating Egyptian fractions represented for a given rational number between 0 and 1, was developed after 800 AD Islamic mathematicians. This modern looking method has used the name greedy algorithm. Its form was later attributed to James Joseph Sylvester, 1891. Another reference to the early algorithm is found in Chapter 7 of the Liber Abaci of Leonardo of Pisa (1202 AD) that was only a secondary recreational math consideration. Fibonacci used another method for his primary conversions of vulgar fractions to Egyptian fractions series. The primary method used by Fibonacci to calculate his Egyptian fraction series looked and acted more finding a parameter t, as used with diophantine indeterminate equations, and therefore Fibonacci himself had not used an algorithm to find his Egyptian fractions.

The recreational math algorithm runs as follow: let r be a rational number, r = a/b

Find the largest unit fraction just less than r. The denominator can be found by dividing b by a, discarding the remainder, and adding one. (If there is no remainder, we are done because r is itself a unit fraction.)
Subtract the found unit fraction from r.
Continue with step 1, using this new smaller value for r.

Example: convert 19/20 into an Egyptian fraction.

20/19 = 1 with some remainder, so our first unit fraction is 1/2.
19/20 − 1/2 = 9/20.
20/9 = 2 with some remainder, so our second unit fraction is 1/3.
9/20 − 1/3 = 7/60
60/7 = 8 with some remainder, so our third unit fraction is 1/9.
7/60 − 1/9 = 1/180 which is itself a unit fraction.

One result is

\begin{matrix}\frac{19}{20} = \frac{1}{2}+\frac{1}{3}+\frac{1}{9}+\frac{1}{180}\end{matrix}

The Liber Abaci also includes two Diophantine-type indeterminate equation versions of the main two vulgar fraction conversion methods that Ahmes used. One method converted n/p vulgar fractions to concise and exact unit fraction series by extending the range of first partitions from p to 2p, and then solving by a parametric method selecting t. The second method converted n/pq vulgar fractions within a modified methodology that can be recognized as also connected to the one used by Ahmes in the Middle Kingdom, 3,000 years earlier.

Note that the modern representation of a given rational number as an Egyptian fraction appears not to be unique, and that the above algorithm almost never yields the shortest (optimal) series. The algorithm, in fact, produces very long and non-Egyptian like looking series.

The ancient method shows that Ahmes, 1650 BCE, optimally found short series with small last term denominators, such as converting 19/20 into:

\begin{matrix}\frac{19}{20} = \frac{1}{2}+\frac{1}{4}+\frac{1}{5}\end{matrix}

Ahmes converted 19/20 by simply testing the first partition 1/2 within a well known ancient method, known as Hultsch-Bruins, or

\begin{matrix}\frac{19}{20} = \frac{1}{2}+\frac{18}{40}\end{matrix}

and solved 18/40 by looking for the divisors of 40, or 20, 10, 8, 5, 4, 2, 1 that add up to 18. Note that by replacing 18 by (10 + 8), or

\begin{matrix}\frac{19}{20} = \frac{1}{2}+\frac{10+8}{40}\end{matrix}

\begin{matrix}= \frac{1}{2}+\frac{1}{4}+\frac{1}{5}\end{matrix}

Had 1/2 failed, as it often did, prime number denominators provided Ahmes relatively difficult problems to find optimal solutions, as the 2/nth table confirms. The next alternative, 2/3 would have been tested by Ahmes, and so forth until one or more answers appeared. Red Auxiliary numbers, a least common multiple technique, denoted a scribal sorting routine that assisted Ahmes' work, by selecting the 'best answer'. Ahmes was often required to chose between more than two Egyptian fraction answers, again as the 2/nth table implies. The selection of 1/42, for example, as the first partition for 2/43, shows that Ahmes had tested eleven even number first partitions between 1/22, 1/24, 1/26, 1/28, 1/30, 1/32, 1/34, 1/36, 1/38, 1/40 and 1/42 before ending his computational task, selecting 1/42 as the 'best answer'.

Ahmes' ancient method was often done in his head, as implied by his brief shorthand notes associated with many of his problems.

Fractions in Egypt

Rhind Mathematical Papyri

The first known use of Egyptian fractions comes from the Rhind Mathematical Papyrus. The RMP was written by Ahmes and other Egyptian scribes and dates from the Second Intermediate Period wrote out over 100 Egyptian fraction based math problems and solutions.

Mathematical historians sometimes describe algebra as having developed in three primary stages:

rhetorical algebra, wherein the problem was stated in words of the language of the ancient mathematician;
syncopated algebra, wherein some words of the problem were abbreviated, for easier comprehension;
symbolic algebra, where in symbols for operators and operands made comprehension still easier.

Typical of symbolism is denoting "the unknown" by "x". We know from ancient Egyptian texts that Egyptian priests and scribes, in their rhetorical algebra, used the word "aha" meaning "heap" or "set" for the unknown. This was shown in the in a translation of one of its "aha" problems in the papyrus:

Problem 24: A quantity and its ¹/₇ added together become 19. What is the quantity?

One view of the solution given in the papyri is

Assume 7. 7 and ¹/₇ of 7 is 8. As many times as 8 must be multiplied to give 19, so many times 7 must be multiplied to give the required number.

The 'assume' aspect refers to the concept of 'false position' or regula falsi. This is a modern concept that had little to do with the algebraic methods employed by Ahmes, as will detailed below per the RMP problems # 24, 31 and 33.

The 'false position' view of the modern form of symbolic logic says that: x + x/7 = 8x/7 = 19, or x = 133/8. It proof says that 133/8 + 133/(7 · 8) = 133/8 + 19/8 = 152/8 = 19.

However, Ahmes never used the 19/8 term. A historically correct way to solve the above problem, without using 'false position' says that Ahmes' algebra was mentally worked within a remainder arithmetic context, a method that did not use 19/8 type terms. Restating the problem closer to Ahmes' thinking says:

x + 7x = 19

(8/7)x = 19

with the problem becoming, solve the vulgar fraction

(19·7)/8 = 133/8

x = 133/8 = 16 + 5/8 = 16 + (4 + 1)/8

= 16 + 1/2 + 1/8

the answer listed in the RMP.

Note the vulgar fractions and unit fractions in this problem. Ancient Egyptians calculated its final step, its remainders, with unit fractions, such as ¹/₂, ¹/₃, ¹/₄, ¹/₁₀, ....

Any fraction we write with a non-unit numerator was written by ancient Egyptians as a sum of unit fractions no two of whose denominators are the same, for its final step. These sums of unit fractions have, therefore, become known as "Egyptian fractions", a notation that had confused many historians. Ahmes and Egyptian scribes often computed within remainder arithmetic, a fact that was not confirmed in the Akhmim Wooden Tablet, the RMP and other mathematical texts until 2005.

That is, given any remainder arithmetic problem, noting that the RMP includes 60 plus examples of the remainder arithmetic notation, the quotient was either written as a whole number, or a binary series (depending upon the problem). Most importantly the remainder portion was always stated first as a vulgar fraction and then as an Egyptian fraction series. The special case of hekat division shows that a common divisor of 1/320 was factored from the Egyptian fraction series, with its divisor was less than 64, thereby decreasing the difficulty of the vulgar fraction conversion process in a major way.

Two other algebraic problems, noted by RMP #31, and RMP # 33, define two closely related problems, stated in modern terms by:

31: x + (2/3 + 1/2 + 1/7)x = 33

or, x = (42/97) × 33 = 1386/97 = 14 + 28/97 = 14 + 2/97 + 26/97

and 2/97 = 1/56 + 1/679 + 1/776 solved by the 2/nth table, and 26/97 = 1/4 + 1/87 + 1/194 + 1/338 solved by an application of the Hultsch-Bruins method, or selecting 1/4 first term such that:

26/97 − 1/4 = (104 - 97)/(4×97)

= (4 + 2 + 1)/(47×97)

= 1/97 + 1/194 + 1/388

or,

26/97 = 1/4 + 1/97 + 1/194 + 1/388

33: x + (2/3 + 1/2 + 1/7) x = 37

x = (42/97)7times;37 = 1554/97 = 16 + 2/97,

with 2/97 = 1/56 + 1/679 + 1/776, taken from the 2/nth table, as easily calculated by the Hultsch-Bruins method, noted by

2/97 − 1/56 = (112 − 97)/(56×97)

= (8 + 7)/(56×97)

= 1/679 + 1/776

However, Ahmes (1650 BCE) would have selected another series using his standard hieratic shorthand thinking, or

5/7 − 1/2 = (10 − 7)/(2·7)

= (2 + 1)/14

Thus,

5/7 = 1/2 + 1/7 + 1/14

would have been Ahmes' answer. Clearly, Ahmes' calculation would have first been written in hieratic script as all of his 1650 BC (Rhind Mathematical) papyrus problems were written. By the Middle Kingdom Egyptian math was almost exclusively written in hieratic script.

However, to complete the older hieroglyphic script statement, as may have been common before 2,000 BCE, a step that Ahmes did take from time to time, for reasons that will not be detailed here. The reason why details can not be offered is that there are no texts that support Old Kingdom arithmetic statements. Yet, the following hieroglyphic script info may be of interest:

5/7 = 1/2 + 1/6 + 1/21

symbols would have been:

Aa13

D21:Z1*Z1*Z1*Z1*Z1*Z1

D21:V20*V20*Z1

(represented in Latin script as R2R6R21).

So $\begin{matrix}\frac{5}{7} = \frac{1}{2} + \frac{1}{6} + \frac{1}{21}\end{matrix}$ is one representation of an Old Kingdom fraction.

Hultsch-Bruins method

The 1202 AD greedy algorithm therefore had nothing to do with the 3,000 year older Egyptian fraction series found in the RMP 2/nth table. Only four 2/nth series can be found by the greedy algorithm. The 2/p conversions of the 2/nth table were first reconstructed by the 1895 work of F. Hultsch, a German scholar. Today the method is known as the Hultsch-Bruins method since it was independently confirmed by E. M. Bruins in 1945. Simply stated, Ahmes and the Egyptian scribes wrote:

2/p − 1/A = (2A − p)/Ap

or,

2/p = 1/A + (2A − p)/Ap

with A, a highly composite number chosen in the range of

p/2 < A < p,

Scribes allowed the divisors of A, aliquot parts of A, to be optimally selected to solve (2A − p) such that:

2/p = 1/A + (2A − p)/Ap

For example, Ahmes solved 2/19 by selecting A = 12, from several alernatives in the range of 10 to 18. Knowing the divisors of 12 to be: 12, 6, 4, 3, 2, 1 it is clear that the value

(2·12 − 19) = 5 can be additively solved by (4 + 1) or (3 + 2). Ahmes chose the shortest solution with the largest last term by selecting (3 + 2). Ahmes final steps were:

2/19 = 1/12 + (3 + 2)/(12·19)

= 1/12 + 1/76 + 1/114

For the easier 2/pq series, it is clear that three methods were used by Ahmes.

The first, and most often used, allowed

2/pq = 2/A × A/p

to compute all but two series when A = (p + 1).

For example, 2/21 = (2/(3 + 1)) × (3 + 1)/21

= 1/2 × (1/7 + 1/21)

= 1/14 + 1/42

B. The second improved the 2/35 and 2/91 series calculated by the first method. The improved form used a view that was near the form Howard Eves found in a 400 AD Coptic text. The Akhmim P. stated the general case, or

n/pq = 1/pr + 1/qr

where r = (p + q)/n

Setting n = 2, with a little algebraic manipulation, any researcher can catch a glimpse of Ahmes' closely connected version. One suggestion says that 2/pq = 1/A × 1/H, with A = arithmetic mean and H = harmonic mean. Note that this special case remained in use for over 2,000 years.

C. Thirdly, 2/95 was another special case. This method allowed Ahmes to factor 2/95 = 2/19 times 1/5, and use his 2/19 series. Factoring was a highly prized skill in ancient Egypt.

Notation

The hieroglyph D21 (er,"^* among" or possibly re, mouth) was placed above a number to represent a unit fraction. The following hieroglyph was interpreted as a "mouth":

D21:Z1

So the ancient Egyptians designated a number of receivers or a fraction as,

D21:Z1*Z1*Z1

= \frac{1}{3}

D21:V20

= \frac{1}{10}

There were special symbols for $\begin{matrix}\frac{1}{2}\end{matrix}$ , $\begin{matrix}\frac{2}{3}\end{matrix}$ and $\begin{matrix}\frac{3}{4}\end{matrix}$ (Note the special case for $\begin{matrix}\frac{1}{2}\end{matrix}$ ):

Aa13

= \frac{1}{2}

D22

= \frac{2}{3}

D23

= \frac{3}{4}

The parts of the Eye of Horus were a special set of fractions used to represent binary fractions, $\frac{1}{2^k}$ ( $k=1,2,...,6$ ), of a hekat, the primary volume measure for grain. They were also used for such grain products as bread and beer.

Bibliography

Daressy, Georges, Cairo Museo des Antiquities Egyptiennes, Catalogue General Ostraca hierariques, page 190, see 1901 volume numbers 25001-25385 par M.G. Daressy.

Leuneberg, Heinz, Leonardi Pisani Liber Abbaci: Oder Lesevergneugen eines Mathematikers, Wissenschaftsverlag, 1993 ISBN 3-411-15462-4

External links

Mathematics in Egyptian Papyri
Egyptian fraction by David Eppstein. (modern aspects only)

http://historyofegyptianfractions.blogspot.com

Fractions | Elementary arithmetic | Recreational mathematics

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