The traditional Old Kingdom scholars report that Egyptians confined themselves to applications of practical arithmetic with problems additively addressing how a number of loaves can be divided equally between a number of men. Problems in the Moscow and Rhind Mathematical Papyri expressed instructional views. Three views cover abstract definitions of number, and higher forms of arithmetic. Abstract definitions are found in the Akhmim Wooden Tablet, the Egyptian Mathematical Leather Roll and the Rhind Mathematical Papyrus. Abstract arithmetic was used to scale hekat, and other weights and measures units. The hekat included Eye of Horus quotients and Egyptian fraction remainders, scaled to ro, 1/320 of a hekat, or other sub-units. Five hekat two-part statements are defined in the Akhmim Wooden Tablet, and applied 30 times in the Rhind Mathematical Papyrus, and many additional times in other Middle Kingdom texts, such as the Ebers Papyrus, a medical text.
By 2700 BC, Egyptian construction techniques included precision surveying, marking north by the sun's location at noon. Clear records began to appear by 2000 BC citing approximations for π and square roots. Exact statements of number, written arithmetic tables, algebra problems, and practical applications with weights and measures also began to appear around 2000 BC, with several problems solved by abstract arithmetic methods.
For example, the Akhmim Wooden Tablet (AWT) lists five divisions of a unit of volume called a hekat, beginning with one hekat unity valued as 64/64. The hekat unity was divided by 3, 7, 10, 11 and 13, with all answers being exact. The first half of the answers cite a binary quotient, i.e. one hekat (64/64), divided by 3, found a quotient 21 with a remainder of 1. The scribe wrote 21 as (16 + 4 + 1), such that a binary series was obtained by (16 + 4 + 1)/64 = 1/4 + 1/16 + 1/64. The second half of the answer scaled the remainder one (1) to 1/320th (ro) units or 1/(192) = (5/3)*1/320 = (1 + 2/3)*ro.
The scribe combined the quotient and remainder into one statement. The 1/3rd of a hekat answer was written as: 1/4 1/16 1/64 1 2/3 ro. Scribal addition and multiplication signs are not seen. Note that the scribal series was written from right to left. The scribe proved all of his results by multiplying the answers by its initial divisors, finding the initial hekat unity value of(64/64 all five times. The AWT scribe wrote out this exact partitioning method in more detail, a method that was shorteded by Ahmes and other Middle Kingdom scribes. Ahmes' steps did not include the proof aspect, for example. However, Ahmes' partitioning steps, however, did follow the AWT's two-part structure, using it 29 times in Rhind Mathematical Papyrus #81.
Hana Vymazalova published in 2002 a fresh copy of the AWT that showed that all five AWT divisions had been exact, by first parsing the proof steps, returning all five division answers to 64/64. Vymazalova thereby updated Daressy's 1906 incomplete discussion of the subject that had only found 1/3, 1/7 and 1/10 to be exact.
Beyond the fact that (64/64)/n = Q/64 - (5R/n)*ro, with Q = quotient and R = remainder, fairly states the 2,000 BCE scribal form of hekat division, two additional facts reveal early scribal thinking. One fact reveals that whenever the divisor n was between 1/64 and 64 a limit of 64 had been reached. RMP 80 details this two-part limit. Second, to go beyond the divisor n = 64 limit, hin, ro and other sub-units of the hekat were developed. Gillings summaries the RMP data with 29 examples in an appendix, thereby contrasting the two-part statements to the equivalent one-part hin statements. The medical texts and its 2,000 examples also used the extended one-part formats following: 10/n hin for 1/10th of a hekat, and 320/n ro for 1/320th of a hekat for prescription ingredients.
Ahmes was able to go beyond the 64 divisor limit and its two-part remainder arithmetic in other ways, one being to increase the size of the numerator. The two-part hekat partitioning method was described in problem 35 as 100 hekat divided by n= 70. Ahmes wrote 100*(64/64)/70 = (6400/64)/70 = 91/64 + 30/(70*64). The quotient was written as (64 + 16 + 8 + 2 + 1)/64 =(1 + 1/4 + 1/8 + 1/32+ 1/64). Ahmes then wrote the remainder part as (150/70)*1/320 = (2 + 1/7)ro. Finally, the combined 1 1/4 1/8 1/32 1/64 2 1/7 ro answer was written down following the right to left, using no arithmetic addition or multiplication signs, older notation rules set down in the 350 year older Akhmim Wooden Tablet.
Our understanding of ancient Egyptian mathematics has been impeded by the reported paucity of available sources. The most famous such source is the Rhind Mathematical Papyrus, a text that can be read by comparing many of its elements against other texts, i.e., the Egyptian Mathematical Leather Roll and the Akhmim Wooden Tablet. The Rhind papyrus dates from the Second Intermediate Period (circa 1650 BC), but its author, Ahmes, identifies it as a copy of a now lost Middle Kingdom papyrus. The Rhind papyrus contains a table of 101 Egyptian fraction expansions for numbers of the form 2/n, and 84 word problems, the answers to which were expressed in Egyptian fraction notation.
The RMP also includes formulas and methods for addition, subtraction, multiplication and division of sums of unit fractions. The RMP contains evidence of other mathematical knowledge, including composite and prime numbers; arithmetic, geometric and harmonic means; and simplistic understandings of both the Sieve of Eratosthenes and perfect number theory. It also shows how to solve first order linear equations as well as summing arithmetic and geometric series.
Henry Rhind's estate donated the Rhind papyrus to the British Museum in 1863. Also included in the donation was the Egyptian Mathematical Leather Roll, dating from the Middle Kingdom era. Like the Rhind papyrus, the Egyptian Mathematical Leather Roll contains a table of Egyptian fraction expansions.
The Berlin papyrus, written around 1300 BC, shows that ancient Egyptians had solved two second-order, one unknown, equations that some have called Diophantine equations. The Berlin method for solving has not been confirmed in a second hieratic text, though it has been confirmed by a second Berlin Papyrus problem.
Two number systems were used in ancient Egypt. One, written in hieroglyphs, was a decimal based tally system with separate symbols for 10, 100, 1000, etc, as Roman numerals were later written, and hieratic unit fractions. The second, written in a new ciphered one-number-to-one-symbol system was a digital system that was not similar to hieroglyphic system. The hieroglyphic number system existed from at least the Early Dynastic Period. The hieratic system differed from the hieroglyphic system beyond a use of simplifying ligatures for rapid writing and began around 2150 BC. Hieratic numerals used one symbol for each number replacing the tallies that had been used to denote multiples of a unit. For example, two symbols had been used to write three, thirty, three hundred, and so on, in a system that was superseded by the hieratic method. Later hieroglyphic numeration was modified and adopted by the Romans for official uses, and Egyptian fractions in everyday situations.
The Rhind Mathematical Papyrus was written in hieratic. It contains examples of how the Egyptians did their mathematical calculations. Fractions were denoted by placing a line over the letter n associated with the number being written, as 1/n. This method of writing numbers came to dominate the Ancient Near East, with Greeks 1,500 years later using two of their alphabets, Ionian and Doric, to cipher all of their numerals, alpha = 1, beta = 2 and so forth. Concerning fractions, Greeks wrote 1/n as n', so Greek numeration and problem-solving adopted or modified Egyptian numeration, arithmetic and other aspects of Egyptian math.
Example from the Rhind Papyrus
As a short cut for larger numbers, the multiplicand can also be immediately multiplied by 10, 100, etc.
For example, Problem 69 on the Rhind Papyrus (RMP) provides the following illustration, as if Hieroglyphic symbols were used (rather than the RMP's actual hieratic script).
|To multiply 80 × 14|
|Egyptian calculation||Modern calculation|
Hieratic and Middle Kingdom math followed this form of hieroglyphic multiplication.
Subtraction defined in the Egyptian Mathematical Leather Roll (EMLR), an 1800 BC document, included four additive or identity methods, followed by one non-additive, abstract, method that was used five to fifteen times for the 26 EMLR series listed, that looked like this:
1/pq = (1/A)* (A/pq)
with A = 3, 4, 5, 7, 25, citing A = (p + 1) 10 times.
1/8 was written using A = (2 + 1)= 3, the A = (p + 1) case, as used in the RMP 24 times, seeing p = 2, q = 4 and A = 25, following
A = 3: 1/8 = (1/3)*(3/8) = 1/3*(1/4 + 1/8) = 1/12 + 1/24
A = 25: 1/8 = 1/25*(25/8) = 1/5*(25/40)= 1/5 *(24/40 + 1/40)
= 1/5*(3/5 + 1/40) = 1/5*(1/5 + 2/5 + 1/40)
= 1/5 *(1/5 + 1/3 + 1/15 + 1/40)
= 1/25 + 1/15 + 1/75 + 1/200
with the out-of-order 1/25 + 1/15 sequence marking the scribal method of partition.
Confirmation of the EMLR (1/A)* (A/pq), with A = (p + 1) rule is found 24 times in the RMP 2/nth table, using the form
2/pq = (2/A)* (A/pq), with A = (p + 1)
example, 2/27, a = 3, q = 9
2/27 = 2/(3 + 1)*(3 + 1)/9 = 1/4*(1/3 + 1/9) = 1/12 + 1/36
Another subtraction method is seen in the RMP 2/nth table as first suggested by F. Hultsch in 1895, and confirmed by E.M. Bruins in 1944, or
2/p - 1/A = (2A - p)/Ap
2/p = 1/A + (2A -p)/Ap
where the divisors of A, from the first partition, were used to additively find (2A - p), thereby exactly solving (2A -p)/Ap.
2/19 - 1/12 = (24 - 19)/(12*19)
with the divisors of 12 = 6, 4, 3, 2, 1 being inspected to find (24 - 19) = 5 taken only from the divisors of 12. Optimally (3 + 2) was selected, by Ahmes and other scribes, over (4 + 1) such that,
2/19 = 1/12 + (3 + 2)/(12*19) = 1/12 + 1/76 + 1/114
Rational numbers could also be expressed, but only as sums of unit fractions, i.e. sums of reciprocals of positive integers, 2/3, and 3/4. The hieroglyph indicating a fraction looked like a mouth, which meant "part", and fractions were written with this fractional solidus, i.e. the numerator 1, and the positive denominator below. Special symbols were used for 1/2 and for two non-unit fractions, 2/3 (used often) and 3/4 (used less often).
Problem 25 on the Rhind Papyrus may have used the method of false position to solve the problem "a quantity and its half added together become 16; what is the quantity?" (i.e., in modern algebraic notation, what is x if x+½x=16).
1 2 /
½ 1 /
Total 1½ 3
As many times as 3 must be multiplied to give 16, so many times must 2 be multiplied to give the answer.
1 3 /
4 12 /
1/3 1 /Total 5 1/3 16
1 5 1/3 (1 + 4 + 1/3)
2 10 2/3
The answer is 10 2/3.
1 10 2/3
½ 5 1/3Total 1½ 16
A more likely and direct approach to solve this class of problem is given by: x + (1/2)x = 16, using these steps
1. (3/2)x = 16, 2. x = 32/3, 3. x = 10 2/3.
Problem 31 sets the problem "q quantity, its 1/3, its 1/2 and its 1/7, added together, become 33; what is the quantity?" In modern algebraic notation, "what is x if x + 1/3 x + 1/2 x + 1/7 x =33?" The answer is 14 1/4 1/56 1/97 1/194 1/388 1/679 1/776, or 14 and 28/97. To solve the problem as Ahmes wrote his answer 28/97 had to be broken up into 2/97 and 26/97, and solved the two separate vulgar fraction conversion problems using Hultsch-Bruins (without using false position, as other algebra problem may have been solved).
The remainder arithmetic solution, the historical method that is most likely, for x + (1/3)x + (1/2)x + (1/7)x = 33 looks like this:
1. 97/42 x = 33, 2. x = 1386/97, and 3. x = 14 + 28/97.
with, 2/97 - 1/56 = (112 - 97)/(56*97) = (8 + 7)/(56*97) = 1/679 1/776,
and 26/97 - 1/4 = (104-97/(4*97) = (4 + 2 + 1)/(4*97)= 1/97 1/194 1/388,
2/97 = 1/56 1/670 1/776,
26/97 = 1/4 1/97 1/194 1/388
such that, writing out x = 14 + 28/97 in an ordered unit fraction series
4. x = 14 1/4 1/56 1/97 1/194 1/388 1/679 1/776, as written by Ahmes.
The ancient Egyptians knew that they could approximate the area of a circle as follows:
Problem 50 of the Ahmes papyrus uses these methods to calculate the area of a circle, according to a rule that the area is equal to the square of 8/9 of the circle's diameter. This assumes that π is 4×(8/9)² (or 3.160493...), with an error of slightly over 0.63 percent. This value was slightly less accurate than the calculations of the Babylonians (25/8 = 3.125, within 0.53 percent), but was not otherwise surpassed until Archimedes' approximation of 211875/67441 = 3.14163, which had an error of just over 1 in 10,000. Interestingly, Ahmes knew of the modern 22/7 as an approximation for pi, and used it to split a hekat, hekat x 22/x x 7/22 = hekat; however, Ahmes continued to use the traditional 256/81 value for pi for computing his hekat volume found in a cylinder.
Problem 48 involved using a square with side 9 units. This square was cut into a 3x3 grid. The diagonal of the corner squares were used to make an irregular octagon with an area of 63 units. This gave a second value for π of 3.111...
The two problems together indicate a range of values for Pi between 3.11 and 3.16.