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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. It can be shown that every positive rational number can be written in the form of an Egyption fraction.

Harmonic series can be used to prove every positive rational number can be written in the form of an Egyption fraction.

Computing Egyptian fractions

An algorithm for calculating Egyptian fractions represented for a given rational number between 0 and 1, was the 800AD Islamic method. This followed the greedy algorithm and has been attributed to James Joseph Sylvester, 1891. Another reference to the algoritm found in Chapter 7 of the Liber Abaci of Leonardo of Pisa (1202)).

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

1. 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.)
2. Subtract the found unit fraction from r.
3. 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 the first unit fraction is 1/2.
• 19/20 − 1/2 = 9/20.
• 20/9 = 2 with some remainder, so the second unit fraction is 1/3.
• 9/20 − 1/3 = 7/60
• 60/7 = 8 with some remainder, so the third unit fraction is 1/9.
• 7/60 − 1/9 = 1/180 which is itself a unit fraction.

One result is

${\displaystyle {\frac {19}{20}}={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{9}}+{\frac {1}{180}}}$

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. Ahmes' method optimally found short series with small last term denominators, such as converting 19/20 into:

${\displaystyle {\frac {19}{20}}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{5}}}$

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

19/20 = 1/2 + 18/40,

and solving 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

19/20 = 1/2 + (10 + 8)/40

= 1/2 + 1/4 + 1/5

Ahmes' solution appears.

Had 1/2 failed, as it often did (prime number denominators provided relatively difficult problems to find optimal solutions, as Ahmes' 2/nth table confirms) the next alternative, 2/3 would have been tested, and so forth. Red Auxiliary numbers, a least common multiple technique, denoted a scribal sorting routine that assisted Ahmes' work. Ahmes was often required to chose between more than two Egyptian fraction series, again as the 2/nth table clearly shows. The selection of 1/42 as the first partition for 2/43, is an interesting example, in that it 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.

This ancient method can often be done in one's head, given a little practice.

Fractions in Egypt

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 x 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.

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

1. rhetorical algebra, wherein the problem was stated in words of the language of the ancient mathematician;
2. syncopated algebra, wherein some words of the problem were abbreviated, for easier comprehension;
3. 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, used the word "aha" meaning "heap" or "set" for the unknown.

This was shown in the Rhind Mathematical Papyrus (Second Intermediate Period) in the British Museum in London in a translation of one of its "aha" (heap or set) problems:

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

"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."

In modern symbolic form, x + x/7 = 8x/7 = 19, or x = 133/8. Proof: 133/8 + 133/(7 · 8) = 133/8 + 19/8 = 152/8 = 19.

A clearer way to state the above shows that Ahmes' algebra was mentally worked within a remainder arithmetic context, or

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

as Ahmes saw the problem in terms of unit fractions.

Note the fractions in this problem. Ancient Egyptians calculated by 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. These sums of unit fractions have, therefore, become known as "Egyptian fractions".

That is, given any remainder arithmetic problem, noting that the RMP includes 60 plus examples of this 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 often factored from the Egyptian fraction series, thereby decreasing the difficulty of the vulgar fraction conversion process in a major way.

Notation

A hieroglyph indicating the Eye of Horus denoted the fractional solidus, with a number hieroglyph written below this "open mouth" icon to denote the denominator of the fraction.

${\displaystyle ={\frac {1}{3}}}$

${\displaystyle ={\frac {1}{10}}}$

So, ${\displaystyle {\frac {5}{7}}={\frac {1}{2}}+{\frac {1}{6}}+{\frac {1}{21}}}$

is one representation of an Old Kingdom Horus-Eye series.

However, Ahmes (1650 BCE) would have 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 writen 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:

(represented in Latin script as R2R6R21).

Note the special case for ${\displaystyle {\frac {1}{2}}}$. The three special cases are:

${\displaystyle ={\frac {1}{2}}}$

${\displaystyle ={\frac {2}{3}}}$

${\displaystyle ={\frac {3}{4}}}$

See also

• Egyptian numerals
• Eye of Horus
• Vulgar fraction