Egyptian mathematics

Egyptian mathematics refers to the style and methods of mathematics performed in Ancient Egypt. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Ancient Egypt was an Ancient Civilization in eastern North Africa, concentrated along the lower reaches of the Nile River in what is now

1 Introduction
2 Overview
3 Sources
4 Numerals
5 Multiplication
6 Fractions
7 Geometry
8 Hellenistic mathematics in Egypt
9 Islamic mathematics in Egypt
10 See also
11 Notes
12 External links
13 Further reading

Introduction

Egyptian multiplication and division employed the method of doubling and halving (respectively) a known number to approach the solution. The method of false position may not have been used for division and algebra problems. In Numerical analysis, the false position method or regula falsi method is a Root-finding algorithm that combines features from the Bisection method Scribes may have only used Old Kingdom binary numbers, and Middle Kingdom unit fractions, written within RMP 2/n table answers. The Old Kingdom is the name commonly given to that period in the 3rd millennium BCE when Egypt attained its first continuous peak of civilization in complexity and achievement The Middle Kingdom is the period in the history of Ancient Egypt stretching from the establishment of the Eleventh Dynasty to the end of the Fourteenth Dynasty The Rhind Mathematical Papyrus contains among other mathematical contents a Table of Egyptian fractions created from 2/ n. Scribes like Ahmes solved complex mathematical problems, 84 of which are outlined in the Rhind Mathematical Papyrus (RMP), one of which included arithmetic progressions. Ahmes (c 1680 BC-c 1620 BC (more accurately Ahmose) was an Egyptian scribe who lived during the Second Intermediate Period. The Rhind Mathematical Papyrus (RMP (also designated as papyrus British Museum 10057 and pBM 10058 is named after Alexander Henry Rhind, a Scottish In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members

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. The Akhmim wooden tablet, is an Ancient Egyptian artifact that has been dated to 2000 BC, near to the beginning of the Egyptian Middle Kingdom. The Egyptian Mathematical Leather Roll (also referred to as EMLR) was a 10" x 17" leather roll purchased by Alexander Henry Rhind in 1858. The Rhind Mathematical Papyrus (RMP (also designated as papyrus British Museum 10057 and pBM 10058 is named after Alexander Henry Rhind, a Scottish Abstract arithmetic was used to scale hekat, and other weights and measures units. The hekat or heqat (transcribed HqAt) was an ancient Egyptian volume unit used to measure grain bread and beer The hekat included Eye of Horus quotients and Egyptian fraction remainders, scaled to ro, 1/320 of a hekat, or other sub-units. The Eye of Horus ( Wedjat) (previously Wadjet and the Eye of the Moon; and afterwards as The Eye of Ra) or (" Udjat " An Egyptian fraction is the sum of distinct Unit fractions such as \tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{16} 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. The Akhmim wooden tablet, is an Ancient Egyptian artifact that has been dated to 2000 BC, near to the beginning of the Egyptian Middle Kingdom. The Rhind Mathematical Papyrus (RMP (also designated as papyrus British Museum 10057 and pBM 10058 is named after Alexander Henry Rhind, a Scottish The Ebers Papyrus of about 1550 BC is among the most important Medical papyri of Ancient Egypt.

Overview

Circa 2700 BC Egyptians introduced the earliest fully developed base 10 numeration system. The 27th century BC is a Century which lasted from the year 2700 BC to 2601 BC Though it was not a positional system, it allowed the use of large numbers and also fractions in the form of unit fractions and Eye of Horus fractions, or binary fractions. A unit fraction is a Rational number written as a fraction where the Numerator is one and the Denominator is a positive Integer The Eye of Horus ( Wedjat) (previously Wadjet and the Eye of the Moon; and afterwards as The Eye of Ra) or (" Udjat " ^[1]

By 2700 BC, Egyptian construction techniques included precision surveying, marking north by the sun's location at noon. The 27th century BC is a Century which lasted from the year 2700 BC to 2601 BC Surveying is the technique and science of accurately determining the terrestrial or three-dimensional space Position of points and the distances and angles between 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 Akhmim wooden tablet, is an Ancient Egyptian artifact that has been dated to 2000 BC, near to the beginning of the Egyptian Middle Kingdom. The hekat or heqat (transcribed HqAt) was an ancient Egyptian volume unit used to measure grain bread and beer 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. The Rhind Mathematical Papyrus (RMP (also designated as papyrus British Museum 10057 and pBM 10058 is named after Alexander Henry Rhind, a Scottish

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.

Sources

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. The Rhind Mathematical Papyrus (RMP (also designated as papyrus British Museum 10057 and pBM 10058 is named after Alexander Henry Rhind, a Scottish e. , the Egyptian Mathematical Leather Roll and the Akhmim Wooden Tablet. The Egyptian Mathematical Leather Roll (also referred to as EMLR) was a 10" x 17" leather roll purchased by Alexander Henry Rhind in 1858. The Akhmim wooden tablet, is an Ancient Egyptian artifact that has been dated to 2000 BC, near to the beginning of the Egyptian Middle Kingdom. 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 Second Intermediate Period marks a period when Ancient Egypt once again fell into disarray between the end of the Middle Kingdom, and the start of the New Ahmes (c 1680 BC-c 1620 BC (more accurately Ahmose) was an Egyptian scribe who lived during the Second Intermediate Period. 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. An Egyptian fraction is the sum of distinct Unit fractions such as \tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{16} Abstract algebra has an unrelated term Word problem for groups.

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, ^[2] including composite and prime numbers; arithmetic, geometric and harmonic means; and simplistic understandings of both the Sieve of Eratosthenes and perfect number theory^[2]. A composite number is a positive Integer which has a positive Divisor other than one or itself In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Mathematics and Statistics, the arithmetic Mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided The geometric mean in Mathematics, is a type of Mean or Average, which indicates the central tendency or typical value of a set of numbers In Mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of Average. In Mathematics, the Sieve of Eratosthenes is a simple ancient Algorithm for finding all Prime numbers up to a specified integer In mathematics a perfect number is defined as a positive integer which is the sum of its proper positive Divisors that is the sum of the positive divisors excluding It also shows how to solve first order linear equations ^[3] as well as summing arithmetic and geometric series. A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members In Mathematics, a geometric series is a series with a constant ratio between successive terms. ^[4]

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. The Egyptian Mathematical Leather Roll (also referred to as EMLR) was a 10" x 17" leather roll purchased by Alexander Henry Rhind in 1858. 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 Papyrus 6619 commonly known as the Berlin Papyrus is an Ancient Egyptian papyrus document from the Middle Kingdom. In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only The Berlin method for solving $x 2 + y 2 = 100$ has not been confirmed in a second hieratic text, though it has been confirmed by a second Berlin Papyrus problem. ^[5]

Sources other than the ones mentioned above include the Moscow Mathematical Papyrus, the Reisner Papyrus, and several other texts including medical prescriptions found in the Ebers Papyrus. The Moscow Mathematical Papyrus is also called the Golenischev Mathematical Papyrus, after its first owner Egyptologist Vladimir Goleniščev. The Reisner Papyrus is one of the most basic of the hieratic mathematical texts The Ebers Papyrus of about 1550 BC is among the most important Medical papyri of Ancient Egypt.

Numerals

Main article: Egyptian numerals

Two number systems were used in ancient Egypt. The system of Ancient Egyptian numerals was a Numeral system used in ancient Egypt aka Kemet A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. 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. Egyptian hieroglyphs (ˈhaɪərəʊɡlɪf from Greek grc-Grek ἱερογλύφος " sacred carving " also hieroglyphic = grc-Grek The decimal ( base ten or occasionally denary) Numeral system has ten as its base. A unit fraction is a Rational number written as a fraction where the Numerator is one and the Denominator is a positive Integer 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 Archaic or Early Dynastic Period of Egypt immediately follows the unification of Lower and Upper Egypt c 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^[6]

5 + ¹⁄₂ + ¹⁄₇ + ¹⁄₁₄ (= 5 ⁵⁄₇)

Multiplication

Main article: Ancient Egyptian 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. Ancient Egyptian multiplication is a systematic method for multiplying two numbers that does not require the Multiplication table, only the ability to multiply and divide The binary numeral system, or base-2 number system, is a Numeral system that represents numeric values using two symbols usually 0 and 1. 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

800

160

320

[= hiero]

1120

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

See also: Peasant multiplication

Hieratic and Middle Kingdom math followed this form of hieroglyphic multiplication. Ancient Egyptian multiplication is a systematic method for multiplying two numbers that does not require the Multiplication table, only the ability to multiply and divide

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.

For 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

Fractions

Main article: Egyptian fraction

Rational numbers could also be expressed, but only as sums of unit fractions, i. An Egyptian fraction is the sum of distinct Unit fractions such as \tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{16} In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions A unit fraction is a Rational number written as a fraction where the Numerator is one and the Denominator is a positive Integer e. sums of reciprocals of positive integers, 2/3, and 3/4. In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French 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. In Numerical analysis, the false position method or regula falsi method is a Root-finding algorithm that combines features from the Bisection method e. , in modern algebraic notation, what is x if x+½x=16). Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity.

Assume 2

       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.

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

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,

or,

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.

Geometry

The ancient Egyptians knew that they could approximate the area of a circle as follows:^[7]

Area of Circle ≈ [ (Diameter) x 8/9 ]². ^[7]

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. Ahmes (c 1680 BC-c 1620 BC (more accurately Ahmose) was an Egyptian scribe who lived during the Second Intermediate Period. This assumes that π is 4×(8/9)² (or 3. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems 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. Babylonia was an Amorite state in lower Mesopotamia (modern southern Iraq) with Babylon as its capital 125, within 0. 53 percent), but was not otherwise surpassed until Archimedes' approximation of 211875/67441 = 3. Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer 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. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems 111. . .

The two problems together indicate a range of values for Pi between 3. 11 and 3. 16.

Problem 14 in the Moscow Mathematical Papyrus gives the only ancient example finding the volume of a frustum of a pyramid, describing the correct formula:

$V = \frac{1}{3} h(x_1^2 + x_1 x_2 +x_2^2).$

Hellenistic mathematics in Egypt

Main article: Greek mathematics

Further information: Egyptian mathematicians

Islamic mathematics in Egypt

Main article: Islamic mathematics