Egyptian mathematics

Egyptian mathematics refers to the style and methods of mathematics performed by scribes in Ancient Egypt, as understood from the rare discoveries of ancient papyri: in particular, the Rhind, or Ahmes, Mathematical Papyrus (RMP), dating from the Second Intermediate Period (though the author identifies it as a copy of a now lost Middle Kingdom papyrus), and the Moscow Mathematical Papyrus (MMP). The Rhind Mathematical Papyrus was donated together with the Egyptian Mathematical Leather Roll (EMLR), containing 26 Egyptian fraction examples, to the British Museum by Henry Rhind's estate in 1863; in addition, the Reisner Papyrus, housed in the Boston Museum of Fine Arts, the Akhmim (Cairo) Wooden Tablet (AWT), housed in Cairo's main museum, and several other texts, including the Berlin, Kahun, Michigan and about 2,000 medical prescriptions written in additional texts inform our understanding of Egyptian mathematics.

Egyptian addition and multiplication methods employed the method of doubling and halving a known number to approach certain solutions, and the method of false position may have been used for division and simple algebra problems as reported by scholars that propose additive intellectual connections to the Old Kingdom. Allied with Old Kingdom decimal number systems, Middle Kingdom unit fractions, and tables of common results, scribes solved a range of complex mathematical problems, though few scholars agree on the exact methods used during the Old Kingdom, Middle Kingdom and later times.

The Old Kingdom 'additive' scholars report that Egyptians often confined themselves to applications of practical arithmetic with many problems addressing how a number of loaves can be divided equally between a number of men. Most of the modern 'additive' scholars believe that the Egyptians could not and did not think of numbers as abstract quantities, but always thought of specific collections objects like 8 objects when 8 was mentioned. The problems written in the Moscow and Rhind Mathematical Papyri can be seen as expressed in a practical instructional context, though three abstract definitions of number, and other higher forms of arithmetic, have been reported by scholars working solely with the hieratic texts. The three abstract definitions can be found in the Akhmim Wooden Tablet, the EMLR and the Rhind Mathematical Papyrus. The higher forms of arithmetic include the use of Egyptian fraction series as non-additive subtraction and division remainders.

Overview

Circa 3100 BC Egyptians introduced the earliest known decimal system, allowing the use of large numbers and also fractions in the form of unit fractions and Eye of Horus fractions.^*. By 2700 BC, Egyptian construction techniques included precision surveying, marking north by the sun's location at noon. Clear records began to appear by 2,000 BC citing approximations for pi and square roots. Exact statements of number, written arithmetic tables, algebra problems, and practical applications with weights and measures also began to appear around 2,000 BC.

Two of the oldest mathematics texts discovered so far are the Moscow Mathematical Papyrus (MMP), and the Akhmim Wooden Tablet (AWT) which are Egyptian Middle Kingdom papyri and tablets dated circa 2050 BC - 1800 BC. Like many ancient mathematical texts, the MMP can be seen as "word problems" or "story problems", some of which may have been intended as entertainment. One problem considered a method for finding the volume of a frustrum (truncated pyramid) with sides of 2 and 4 units and a height of 6: "Add together this 16 with this 8 and this 4. You get 28. Compute a third of 6. You get 2. Multiply 28 by 2. You get 56. Behold: it is 56. You have found right." Van der Waerden, 1961, Plate 5

The AWT lists five example divisions of a volume unit, named a hekat, beginning with a hekat unity valued as 1 = 64/64. The AWT did not contain entertaining information. The divisions by 3, 7, 10, 11 and 13 of the hekat unity were all exact and were calculated and proven by the scribe using serious methods. The scribal notes wihtin the tablet(s) reported five two-part answers, the first half being a quotient. Example, divide 1/3 by first introducing a hekat unity: (64/64)/3 finds quotient 21, since 64/3 = 21 with a remainder of 1. Or, writing as a binary quotitent 21 = 16 + 4 + 1 meant that (16 + 4 + 1)/64 = 1/4 + 1/16 + 1/64, as recorded by the scribe. The second half of the two-part answer processed the remainder, 1, in terms the division by 3 by first factoring out a constant common divisor named ro = 1/320, and then converting the modern remainder 1/(3*64) to the scribal (5/3)*ro (since 1/64 = 5/320). The final scribal step converted the vulgar fraction 5/3 to an Egyptian fraction series, writing the remainder as (1 + 2/3)ro. Generally the five AWT division problems show that remainders (5R/n)*1/320 converted (5R/n) to an Egyptian fraction followed by ro. The scribe proved his two-part result by multiplying his answer by the initial divisor. In the n =3 case, the proof was written right to left without using the (+) sign, or, + 1/16 + 1/64) + (1 + 2/3) *1/320* 3 = + 1/64 = 64/64.

Hana Vymazalova published in 2002 a fresh copy of the AWT that showed that all five AWT divisions had been exact after all, thereby updating Daressy's 1906 discussion of the subject that had only found 1/3, 1/7 and 1/10 had been exact (Daressy had inadvertently introduced typos that limited his seeing that 1/11 and 1/13 had been exact).

Beyond the fact that (64/64)/n = Q/64 + (5R/n)*ro fairly states the 2,000 BC scribal form of hekat division two additional facts show scribal thinking. One is that whenever a divisor n was less than or equal to 64 (with Q being a quotient and R being a remainder), a limit had been reached. Second, to go beyond the limit, hin, ro and other sub-units of the hekat were developed in a one-part format, 10/n hin, and 320/n ro, and so forth, with n being a divisor. The medical texts have been particular difficult to read since the one-part scaling factors related to the oipe, dja and other sub-units have been 'guessed at' by scholars by not showing proven links to the hekat, its parent, or any of its known hin and ro children.

Scribes like Ahmes were also able to go beyond the 64 divisor limit within the two-part structure. The advanced two-part method, a primary level of scribal arithmetic, was described in problem 35 as 100 hekat divided by 70. Ahmes wrote 100*(64/64)/70 = (6400/64)/70 = 91/64 + 30/(70*64. The first part was written (64 + 16 + 8 + 2 + 1)/64 = (1 + 1/8 + 1/32+ 1/64). Ahmes then wrote out the second part as (150/70)*1/320 = (2 + 1/7)ro, as partially reported by Robins-Shute), by following rules set down in the 350 year older Akhmim Wooden Tablet.

Scribes like Ahmes were able to go beyond the 64 divisor limit by working in additive one-part arithmetic. The two-part statements were coverted to hin data, as clearly stated by Ahmes in RMP 80. Ro data was also written in additive one-part statements in other RMP problems. The non-additive two-part statements had been converted to additive one-part statements by showing a secondary level of scribal arithmetic, by writing 10/n hin, 320/n and other hekat sub-units, using n as the hekat divisor.

The Rhind papyrus (circa 1650 BC) is another major Egyptian mathematical text, an instruction manual in arithmetic and geometry. In addition to giving area formulas and methods for multiplication, division and working with unit fractions, it also contains evidence of other mathematical knowledge (see including composite and prime numbers; arithmetic, geometric and harmonic means; and simplistic understandings of both the Sieve of Eratosthenes and perfect number theory linear equations ^*" target="_blank" >as well as summing arithmetic and geometric series [http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egypt_algebra.html#areithmetic%20series.

Finally, the Berlin papyrus, written around 1300 BC, shows that ancient Egyptians had solved two second-order Diophantine equations, though the Berlin method for solving $x^2 + y^2 = 100$ has not been confirmed in a second hieratic text. ^*.

Numerals

Main article: Egyptian numerals

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 number unit. For example, two symbols had been used to write three, thirty, three hundred, and so on, in a system that was superced by the hieratic method (in all situations beyond the most spiritual of texts). 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 as Plato and many other Greeks have fairly reported.

Multiplication

Egyptian multiplication was done by repeated doubling of the number to be multiplied (the multiplicand), and choosing which of the doublings to add together (essentially a form of binary arithmetic), a method that links to the Old Kingdom. The multiplicand was written next to the figure 1; the multiplicand was then added to itself, and the result written next to the number 2. The process was continued until the doublings gave a number greater than half of the multiplier. Then the doubled numbers (1, 2, etc.) would be repeatedly subtracted from the multiplier to select which of the results of the existing calculations should be added together to create the answer.

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
Result	Multiplier		Result	Multiplier
V20V20V20V20:V20V20V20V20	Z1		80	1
V1V1V1V1:V1V1V1V1	V20	/	800	10
V20V20V20:V20V20V20-V1	Z1*Z1		160	2
V20:V20-V1*V1:V1	Z1Z1Z1*Z1	/	320	4
V20:V20-V1-M12	hiero		1120	14

The / denotes the intermediate results that are added together to produce the final answer.

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

or,

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.

example,

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

Note that scribal subtraction and division were complicated matters. Selecting A was the key to solving 2/p conversions to Egyptian fraction series using subtraction or division. Scribal division, therefore, will be discussed in detail within a broader context, elsewhere on Wikipedia, at the proper time.

Fractions

Main article: Egyptian fraction

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 frequently) and 3/4 (used less frequently).

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).

Assume 2

1 2 / ½ 1 / Total 1½ 3

As many times as 3 must be to give 16, so many times must 2 be multiplied to give the answer.

1 3 / 2 6 4 12 / 2/3 2 1/3 1 / Total 5 1/3 16

So: 1 5 1/3 (1 + 4 + 1/3) 2 10 2/3

The answer is 10 2/3.

Check - 1 10 2/3 ½ 5 1/3 Total 1½ 16

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).

Geometry

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 (so 1/9 is subtracted from the diameter, and the resulting figure is multiplied by itself, using the doubling method). 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).

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.

Overview

Numerals

Multiplication

Fractions

Geometry

See also

Notes

External links

Further reading

Gourt Home::Gourt :: Egyptian mathematics

Overview

Numerals

Multiplication

Fractions

Geometry

See also

Notes

External links

Further reading