Ancient Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt 3000 to c. , from the

Old Kingdom of Egypt In ancient Egyptian history, the Old Kingdom is the period spanning c. 2700–2200 BC. It is also known as the "Age of the Pyramids" or the "Age of the Pyramid Builders", as it encompasses the reigns of the great pyramid-builders of the Fourt ...

until roughly the beginning of Hellenistic Egypt. The ancient Egyptians utilized a numeral system for counting and solving written mathematical problems, often involving

multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...

and

fractions A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...

. Evidence for Egyptian mathematics is limited to a scarce amount of surviving sources written on

papyrus Papyrus ( ) is a material similar to thick paper that was used in ancient times as a writing surface. It was made from the pith of the papyrus plant, ''Cyperus papyrus'', a wetland sedge. ''Papyrus'' (plural: ''papyri'') can also refer to a d ...

. From these texts it is known that ancient Egyptians understood concepts of

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, such as determining the

surface area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of ar ...

and

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The def ...

of three-dimensional shapes useful for

architectural engineering Architectural engineers apply and theoretical knowledge to the engineering design of buildings and building systems. The goal is to engineer high performance buildings that are sustainable, economically viable and ensure the safety health. Archi ...

, and

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

, such as the

false position method In mathematics, the ''regula falsi'', method of false position, or false position method is a very old method for solving an equation with one unknown; this method, in modified form, is still in use. In simple terms, the method is the trial and er ...

and

quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not quadr ...

Overview

Written evidence of the use of mathematics dates back to at least 3200 BC with the ivory labels found in Tomb U-j at Abydos. These labels appear to have been used as tags for grave goods and some are inscribed with numbers. Further evidence of the use of the base 10 number system can be found on the

Narmer Macehead The Narmer macehead is an ancient Egyptian decorative stone mace head. It was found in the “main deposit” in the temple area of the ancient Egyptian city of Nekhen (Hierakonpolis) by James Quibell in 1898. It is dated to the Early Dynastic ...

which depicts offerings of 400,000 oxen, 1,422,000 goats and 120,000 prisoners. Archaeological evidence has suggested that the Ancient Egyptian counting system had origins in Sub-Saharan Africa. Also, fractal geometry designs which are widespread among Sub-Saharan African cultures are also found in Egyptian architecture and cosmological signs. The evidence of the use of mathematics in the

Old Kingdom In ancient Egyptian history, the Old Kingdom is the period spanning c. 2700–2200 BC. It is also known as the "Age of the Pyramids" or the "Age of the Pyramid Builders", as it encompasses the reigns of the great pyramid-builders of the Fourt ...

(c. 2690–2180 BC) is scarce, but can be deduced from inscriptions on a wall near a

mastaba A mastaba (, or ), also mastabah, mastabat or pr- djt (meaning "house of stability", " house of eternity" or "eternal house" in Ancient Egyptian), is a type of ancient Egyptian tomb in the form of a flat-roofed, rectangular structure with inwar ...

Meidum Meidum, Maydum or Maidum ( ar, ميدوم, , ) is an archaeological site in Lower Egypt. It contains a large pyramid and several mudbrick mastabas. The pyramid was Egypt's first straight-sided one, but it partially collapsed in ancient times. Th ...

which gives guidelines for the slope of the mastaba. The lines in the diagram are spaced at a distance of one cubit and show the use of that

unit of measurement A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multip ...

. The earliest true mathematical documents date to the

12th Dynasty The Twelfth Dynasty of ancient Egypt (Dynasty XII) is considered to be the apex of the Middle Kingdom by Egyptologists. It often is combined with the Eleventh, Thirteenth, and Fourteenth dynasties under the group title, Middle Kingdom. Some s ...

(c. 1990–1800 BC). The

Moscow Mathematical Papyrus The Moscow Mathematical Papyrus, also named the Golenishchev Mathematical Papyrus after its first non-Egyptian owner, Egyptologist Vladimir Golenishchev, is an ancient Egyptian mathematical papyrus containing several problems in arithmetic, geom ...

, the

Egyptian Mathematical Leather Roll The Egyptian Mathematical Leather Roll (EMLR) is a 10 × 17 in (25 × 43 cm) leather roll purchased by Alexander Henry Rhind in 1858. It was sent to the British Museum in 1864, along with the Rhind Mathemat ...

, the

Lahun Mathematical Papyri The Lahun Mathematical Papyri (also known as the Kahun Mathematical Papyri) is an ancient Egyptian mathematical text. It forms part of the Kahun Papyri, which was discovered at El-Lahun (also known as Lahun, Kahun or Il-Lahun) by Flinders Petrie ...

which are a part of the much larger collection of

Kahun Papyri The Kahun Papyri (KP; also Petrie Papyri or Lahun Papyri) are a collection of ancient Egyptian texts discussing administrative, mathematical and medical topics. Its many fragments were discovered by Flinders Petrie in 1889 and are kept at the U ...

and the

Berlin Papyrus 6619 The Berlin Papyrus 6619, simply called the Berlin Papyrus when the context makes it clear, is one of the primary sources of ancient Egyptian mathematics. One of the two mathematics problems on the Papyrus may suggest that the ancient Egyptians kn ...

all date to this period. The

Rhind Mathematical Papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian, who purchas ...

which dates to the

Second Intermediate Period The Second Intermediate Period marks a period when ancient Egypt fell into disarray for a second time, between the end of the Middle Kingdom and the start of the New Kingdom. The concept of a "Second Intermediate Period" was coined in 1942 by ...

(c. 1650 BC) is said to be based on an older mathematical text from the 12th dynasty. The Moscow Mathematical Papyrus and Rhind Mathematical Papyrus are so called mathematical problem texts. They consist of a collection of problems with solutions. These texts may have been written by a teacher or a student engaged in solving typical mathematics problems. An interesting feature of ancient Egyptian mathematics is the use of unit fractions. The Egyptians used some special notation for fractions such as , and and in some texts for , but other fractions were all written as unit fractions of the form or sums of such unit fractions. Scribes used tables to help them work with these fractions. The Egyptian Mathematical Leather Roll for instance is a table of unit fractions which are expressed as sums of other unit fractions. The Rhind Mathematical Papyrus and some of the other texts contain tables. These tables allowed the scribes to rewrite any fraction of the form as a sum of unit fractions. During the

New Kingdom New is an adjective referring to something recently made, discovered, or created. New or NEW may refer to: Music * New, singer of K-pop group The Boyz Albums and EPs * ''New'' (album), by Paul McCartney, 2013 * ''New'' (EP), by Regurgitator, ...

(c. 1550–1070 BC) mathematical problems are mentioned in the literary

Papyrus Anastasi I {{More footnotes, date=March 2017 Papyrus Anastasi I (officially designated papyrus British Museum 10247) is an ancient Egyptian papyrus containing a satirical text used for the training of scribes during the Ramesside Period (i.e. Nineteenth and ...

, and the Papyrus Wilbour from the time of

Ramesses III Usermaatre Meryamun Ramesses III (also written Ramses and Rameses) was the second Pharaoh of the Twentieth Dynasty in Ancient Egypt. He is thought to have reigned from 26 March 1186 to 15 April 1155 BC and is considered to be the last great mona ...

records land measurements. In the workers village of

Deir el-Medina Deir el-Medina ( arz, دير المدينة), or Dayr al-Madīnah, is an ancient Egyptian workmen's village which was home to the artisans who worked on the tombs in the Valley of the Kings during the 18th to 20th Dynasties of the New Kingdom ...

several

ostraca An ostracon (Greek: ''ostrakon'', plural ''ostraka'') is a piece of pottery, usually broken off from a vase or other earthenware vessel. In an archaeological or epigraphical context, ''ostraca'' refer to sherds or even small pieces of stone ...

have been found that record volumes of dirt removed while quarrying the tombs.

Sources

Current understanding of ancient Egyptian mathematics is impeded by the paucity of available sources. The sources that do exist include the following texts (which are generally dated to the Middle Kingdom and Second Intermediate Period): * The

Clagett, Marshall Ancient Egyptian Science, A Source Book. Volume Three: Ancient Egyptian Mathematics (Memoirs of the American Philosophical Society) American Philosophical Society. 1999 * The

* The

, written around 1800 BC * The Akhmim Wooden Tablet * The

Reisner Papyrus The Reisner Papyri date to the reign of Senusret I, who was king of ancient Egypt in the 19th century BCE. The documents were discovered by G.A. Reisner during excavations in 1901–04 in Naga ed-Deir in southern Egypt. A total of four papyrus ro ...

, dated to the early

Twelfth dynasty of Egypt The Twelfth Dynasty of ancient Egypt (Dynasty XII) is considered to be the apex of the Middle Kingdom by Egyptologists. It often is combined with the Eleventh, Thirteenth, and Fourteenth dynasties under the group title, Middle Kingdom. Some ...

and found in Nag el-Deir, the ancient town of Thinis * The

(RMP), dated from the

(c. 1650 BC), but its author,

Ahmes Ahmes ( egy, jꜥḥ-ms “, a common Egyptian name also transliterated Ahmose) was an ancient Egyptian scribe who lived towards the end of the Fifteenth Dynasty (and of the Second Intermediate Period) and the beginning of the Eighteenth Dyna ...

, identifies it as a copy of a now lost Middle Kingdom papyrus. The RMP is the largest mathematical text. From the New Kingdom there are a handful of mathematical texts and inscriptions related to computations: * The

, a literary text written as a (fictional) letter written by a scribe named Hori and addressed to a scribe named Amenemope. A segment of the letter describes several mathematical problems. * Ostracon Senmut 153, a text written in hieratic * Ostracon Turin 57170, a text written in hieratic * Ostraca from Deir el-Medina contain computations. Ostracon IFAO 1206 for instance shows the calculation of volumes, presumably related to the quarrying of a tomb.

Numerals

Ancient Egyptian texts could be written in either

hieroglyph A hieroglyph (Greek for "sacred carvings") was a character of the ancient Egyptian writing system. Logographic scripts that are pictographic in form in a way reminiscent of ancient Egyptian are also sometimes called "hieroglyphs". In Neoplatoni ...

s or in

hieratic Hieratic (; grc, ἱερατικά, hieratiká, priestly) is the name given to a cursive writing system used for Ancient Egyptian and the principal script used to write that language from its development in the third millennium BC until the ris ...

. In either representation the number system was always given in base 10. The number 1 was depicted by a simple stroke, the number 2 was represented by two strokes, etc. The numbers 10, 100, 1000, 10,000 and 100,000 had their own hieroglyphs. Number 10 is a hobble for cattle, number 100 is represented by a coiled rope, the number 1000 is represented by a lotus flower, the number 10,000 is represented by a finger, the number 100,000 is represented by a frog, and a million was represented by a god with his hands raised in adoration. Princess Nefertiabet before her meal-E 15591-IMG 9645-gradient

Princess Nefertiabet before her meal-E 15591-IMG 9645-gradient

Egyptian numerals date back to the

Predynastic period Prehistoric Egypt and Predynastic Egypt span the period from the earliest human settlement to the beginning of the Early Dynastic Period around 3100 BC, starting with the first Pharaoh, Narmer for some Egyptologists, Hor-Aha for others, with th ...

. Ivory labels from Abydos record the use of this number system. It is also common to see the numerals in offering scenes to indicate the number of items offered. The king's daughter Neferetiabet is shown with an offering of 1000 oxen, bread, beer, etc. The Egyptian number system was additive. Large numbers were represented by collections of the glyphs and the value was obtained by simply adding the individual numbers together. Counts tomb 75 gizeh-lepsius

The Egyptians almost exclusively used fractions of the form . One notable exception is the fraction , which is frequently found in the mathematical texts. Very rarely a special glyph was used to denote . The fraction was represented by a glyph that may have depicted a piece of linen folded in two. The fraction was represented by the glyph for a mouth with 2 (different sized) strokes. The rest of the fractions were always represented by a mouth super-imposed over a number.

Multiplication and division

Egyptian multiplication was done by a 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 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 shortcut for larger numbers, the multiplicand can also be immediately multiplied by 10, 100, 1000, 10000, etc. For example, Problem 69 on the

Rhind Papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian, who purchase ...

(RMP) provides the following illustration, as if Hieroglyphic symbols were used (rather than the RMP's actual hieratic script). The '' Yes check

'' denotes the intermediate results that are added together to produce the final answer. The table above can also be used to divide 1120 by 80. We would solve this problem by finding the quotient (80) as the sum of those multipliers of 80 that add up to 1120. In this example that would yield a quotient of 10 + 4 = 14. A more complicated example of the division algorithm is provided by Problem 66. A total of 3200 ro of fat are to be distributed evenly over 365 days. First the scribe would double 365 repeatedly until the largest possible multiple of 365 is reached, which is smaller than 3200. In this case 8 times 365 is 2920 and further addition of multiples of 365 would clearly give a value greater than 3200. Next it is noted that + + times 365 gives us the value of 280 we need. Hence we find that 3200 divided by 365 must equal 8 + + + .

Algebra

Egyptian algebra problems appear in both the

Rhind mathematical papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian, who purchas ...

and the

Moscow mathematical papyrus The Moscow Mathematical Papyrus, also named the Golenishchev Mathematical Papyrus after its first non-Egyptian owner, Egyptologist Vladimir Golenishchev, is an ancient Egyptian mathematical papyrus containing several problems in arithmetic, geom ...

as well as several other sources. Aha problems involve finding unknown quantities (referred to as Aha) if the sum of the quantity and part(s) of it are given. The

also contains four of these type of problems. Problems 1, 19, and 25 of the Moscow Papyrus are Aha problems. For instance problem 19 asks one to calculate a quantity taken times and added to 4 to make 10. In other words, in modern mathematical notation we are asked to solve the

linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficie ...

: :

\frac 3 2 \times x + 4 = 10.\

Solving these Aha problems involves a technique called method of false position. The technique is also called the method of false assumption. The scribe would substitute an initial guess of the answer into the problem. The solution using the false assumption would be proportional to the actual answer, and the scribe would find the answer by using this ratio. The mathematical writings show that the scribes used (least) common multiples to turn problems with fractions into problems using integers. In this connection red auxiliary numbers are written next to the fractions. The use of the Horus eye fractions shows some (rudimentary) knowledge of geometrical progression. Knowledge of arithmetic progressions is also evident from the mathematical sources.

Quadratic equations

The ancient Egyptians were the first civilization to develop and solve second-degree ( quadratic) equations. This information is found in the Berlin Papyrus fragment. Additionally, the Egyptians solve first-degree algebraic equations found in

Geometry

There are only a limited number of problems from ancient Egypt that concern geometry. Geometric problems appear in both the

(MMP) and in the

(RMP). The examples demonstrate that the Ancient Egyptians knew how to compute areas of several geometric shapes and the volumes of cylinders and pyramids. * Area: ** ''Triangles:'' The scribes record problems computing the area of a triangle (RMP and MMP). ** ''Rectangles:'' Problems regarding the area of a rectangular plot of land appear in the RMP and the MMP. A similar problem appears in the

in London. R.C. Archibald Mathematics before the Greeks Science, New Series, Vol.73, No. 1831, (Jan. 31, 1930), pp. 109–121 Annette Imhausen Digitalegypt website: Lahun PapyrusIV.3
/ref> ** ''Circles:'' Problem 48 of the RMP compares the area of a circle (approximated by an octagon) and its circumscribing square. This problem's result is used in problem 50, where the scribe finds the area of a round field of diameter 9 khet. ** ''Hemisphere:'' Problem 10 in the MMP finds the area of a hemisphere. * Volumes: ** ''Cylindrical (cylinder)'': Several problems compute the volume of cylindrical granaries (RMP 41–43), while problem 60 RMP seems to concern a pillar or a cone instead of a pyramid. It is rather small and steep, with a seked (reciprocal of slope) of four palms (per cubit). In section IV.3 of the

the volume of a granary with a circular base is found using the same procedure as RMP 43. ** ''Rectangular (Cuboid):'' Several problems in the

(problem 14) and in the

(numbers 44, 45, 46) compute the volume of a rectangular granary. ** ''Truncated pyramid (frustum)

Frustum In geometry, a (from the Latin for "morsel"; plural: ''frusta'' or ''frustums'') is the portion of a solid (normally a pyramid or a cone) that lies between two parallel planes cutting this solid. In the case of a pyramid, the base faces are ...

:'' The volume of a truncated pyramid is computed in MMP 14.

The Seqed

Problem 56 of the RMP indicates an understanding of the idea of geometric similarity. This problem discusses the ratio run/rise, also known as the seqed. Such a formula would be needed for building pyramids. In the next problem (Problem 57), the height of a pyramid is calculated from the base length and the ''seked'' (Egyptian for the reciprocal of the slope), while problem 58 gives the length of the base and the height and uses these measurements to compute the seqed. In Problem 59 part 1 computes the seqed, while the second part may be a computation to check the answer: ''If you construct a pyramid with base side 12 ubitsand with a seqed of 5 palms 1 finger; what is its altitude?''

References

External links

Egyptian ArithmeticIntroduction to Early Mathematics
{{DEFAULTSORT:Ancient Egyptian Mathematics Ancient Egyptian society Egyptian mathematics