Ancient EgyptDr. Patrick Perry

Associate Professor of Mathematics

Mathematics Across the Ages

Fall 2010

Mathematics Across the Ages

Reference: http://aleph0.clarku.edu/%7Edjoyce/mathhist/earth.html

This map shows the major areas of mathematical development prior to the twentieth century.

We will discuss discus the mathematical developments of most of the regions highlighted.

Ancient Egypt

Reference: http://en.wikipedia.org/wiki/Ancient_Egypt

The civilization began around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh, and it developed over the next three millennia.

The rule of the pharaohs officially ended in 31 BC when the early Roman Empire conquered Egypt and made it a province.

Ancient Egypt

Reference: http://en.wikipedia.org/wiki/Ancient_Egypt

Ancient Egypt was an ancient civilization in eastern North Africa, concentrated along the lower reaches of the Nile River in what is now the modern nation of Egypt.

Ancient Egypt

Reference: http://en.wikipedia.org/wiki/Egyptian_hieroglyphs

Hieroglyphic writing dates to c. 3200 BC, and is composed of some 500 symbols.

The Rosetta Stone is an Ancient Egyptian artifact which was instrumental in advancing modern understanding of hieroglyphic writing.

A section of the Papyrus of Ani showing hieroglyphs.

Papyrus

Reference: http://en.wikipedia.org/wiki/Papyrus

Papyrus is a thick paper-like material produced from the papyrus plant.

A section of the Egyptian Book of the Dead written on papyrus

In a dry climate like that of Egypt, papyrus is stable, highly rot-resistant.

In European conditions, papyrus seems only to have lasted a matter of decades.

Ancient Egypt

Reference: http://www-groups.dcs.stand.ac.uk/~history/HistTopics/Egyptian_mathematics.html

It is from the Rhind papyrus and the Moscow papyrus that most of our knowledge of Egyptian mathematics comes….

Rhind papyrus

Moscow papyrus

Rhind Papyrus

Reference: http://www-groups.dcs.stand.ac.uk/~history/HistTopics/Egyptian_mathematics.html and http://www.math.tamu.edu/%7Edallen/masters/index.html

It is 18 feet long and 13 inches wide.

It is also called the Ahmes Papyrus after the scribed that last copied it. It is a collection of 84 exercises designed primarily for students of mathematics. Included are exercises in fractions, notation, arithmetic, algebra, geometry, and mensuration.

The Rhind Mathematical Papyrus named for A.H.Rhind who purchased it at Luxor in 1858. Origin: written around 1650 BC by the scribe Ahmes.

Moscow Papyrus

The Moscow Mathematical papyrus was purchased by V.S.Golenishchev and sold to the Moscow Musuem of Fine Arts. Origin 1700 BC. It is 15 feet long and 3 inches wide.

The papyrus contains only about 25 practical examples. The author is unknown.

Reference: http://www-groups.dcs.stand.ac.uk/~history/HistTopics/Egyptian_mathematics.html and http://www.math.tamu.edu/%7Edallen/masters/index.html

Egyptian Numbers The Egyptians had a decimal system using seven different symbols.

1 a single stroke. 10 a hobble for cattle. 100 a coil of rope. 1,000 a lotus plant. 10,000 a finger. 100,000 a tadpole or frog1,000,000 a god with arms raised above his head.Reference: http://www.discoveringegypt.com

Egyptian MultiplicationThe Egyptian algorithm for multiplication was based on continual doubling process.

For example, to multiply 12 by 13, the scribe would set down the following lines:

‘1 12

‘2 24

‘4 48

‘8 96

Since 1+4+8 = 13,

we get 12 * 13 = 12 + 48 + 96 = 156

Reference: A History of Mathematics by V. Katz, 3rd ed.

This multiplication algorithm shows that the scribes were somehow aware that every positive integer could be uniquely expressed as the sum of powers of two.

Egyptian Fractions

Reference: http://www.ics.uci.edu/%7Eeppstein/numth/egypt/ and http://en.wikipedia.org/wiki/Egyptian_mathematics

The ancient Egyptians used a number system based on unit fractions: fractions with one in the numerator.

Numbers such as 2/7 were represented as sums of unit fractions (e.g. 2/7 = 1/4 +1/28). Further, the same fraction could not be used twice (so 2/7 = 1/7 + 1/7 is not allowed).

We call a formula representing a sum of distinct unit fractions an Egyptian fraction.

Example from the Rhind Papyrus

5 + 1⁄2 + 1⁄7 + 1⁄14 (= 5 5⁄7) 4 1 1

11 3 33

3 33

Example An “eye” symbol was written above a number to denote it is a fraction.

Reading Egyptian Hieroglyphics

The hieroglyphics are read from the right to the left. Thus we get 1/6 + 1/ 18.

This example shows 7 + 1/9.

Special symbols were used for 1/2 and 2/3. These are shown here.

1/2 2/3

Reference: http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egypt_geometry.html#ahmes10

Computing Egyptian Fractions

4/13 = 1/4+1/18+1/468

The page http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html has a calculator for converting modern fractions to Egyptian fractions.

Convert 4/13 to an Egyptian fraction. Searching to find the first unit fraction smaller than 4/13, we find 1/4 < 4/13 < 1/3. Thus we take 1/4 as the first part.

Now 4/13 – 1/4 = 3/52. Searching to find the first unit fraction smaller than 3/52, we find 1/18 < 3/52 < 1/17. Thus 1/18 is the second part.

Now 3/52 – 1/18 = 1/468. Since 1/468 is a unit fractions, we are done!

This is a modern solution. It is not known how the Egyptians solved such questions.

Reference: http://www.math.buffalo.edu/mad/Ancient-Africa/best-egyptian-fraction.html

The first section of the Rhind papyrus is a table of the division of 2 by every odd integer from 3 to 101.

2/3 = 1/2 + 1/6 2/5 = 1/3 + 1/15 2/7 = 1/4 + 1/282/9 = 1/6 + 1/18 2/11 = 1/6 + 1/66 2/13 = 1/8 + 1/52 + 1/1042/15 = 1/10 + 1/30 2/17 = 1/12 + 1/51 + 1/68 2/19 = 1/12 + 1/76 + 1/1142/21= 1/14 + 1/42 2/23 = 1/12 + 1/276 2/25 = 1/15 + 1/752/27 = 1/18 + 1/54 2/29 = 1/24 + 1/58 + 1/174 + 1/232

2/31 = 1/20 + 1/124 + 1/155

2/33 = 1/22 + 1/66 2/35 = 1/25 + 1/30 + 1/42

2/37 = 1/24 + 1/111 + 1/296

2/39 = 1/26 + 1/78 2/41 = 1/24 + 1/246 + 1/328

2/43 = 1/42 + 1/86 + 1/129 + 1/301 2/45 = 1/30 + 1/902/47 = 1/30 + 1/141 + 1/470

2/49 = 1/28 + 1/196 2/51 = 1/34 + 1/102

2/53 = 1/30 + 1/318 + 1/795

2/55 = 1/30 + 1/330 2/57 = 1/38 + 1/114

2/59 = 1/36 + 1/236 + 1/531

2/61 = 1/40 + 1/244 + 1/488 + 1/610

2/63 = 1/42 + 1/126 2/65 = 1/39 + 1/195 2/67 = 1/40 + 1/335 + 1/536

2/69 = 1/46 + 1/1382/71 = 1/40 + 1/568 + 1/710

2/73 = 1/60 + 1/219 + 1/292 + 1/365

2/75 = 1/50 + 1/150 2/77 = 1/44 + 1/308 2/79 = 1/60 + 1/237 + 1/316 + 1/790

2/81 = 1/54 + 1/162 2/83 = 1/60 + 1/332 + 1/415 + 1/498

2/85 = 1/39 + 1/195 2/87 = 1/58 + 1/174 2/89 = 1/60 + 1/356 + 1/534 + 1/890

2/91 = 1/70 + 1/130 2/93 = 1/62 + 1/186 2/95 = 1/60 + 1/380 + 1/5702/97 = 1/56 + 1/679 + 1/776

2/99 = 1/66 + 1/198 2/101 = 1/101 + 1/202 + 1/303 + 1/606

There is still debate ongoing as to what algorithm Ahmes used to derive these results.

Rhind Problem #3

Reference: A History of Mathematics by V. Katz, 3rd ed.

Problem 3 of the Rhind Papyrus asks how to divide 6 loaves among 10 men.

Rhind papyrus

The solution is given that each man get 1/2 + 1/10 loaves.

We divide five loaves in half and the sixth one we divide into tenths. It is then clear to all that every man has the same portion of bread. The modern answer would be

3/5, from which it is not clear how exactly we should divide the loaves.

The first six problems of the Rhind papyrus ask how to divide n loaves between 10 men where n =1 for Problem 1, n = 2 for Problem 2, n = 6 for Problem 3, n = 7 for Problem 4, n = 8 for Problem 5, and n = 9 for Problem 6

Ancient Egypt: Examples

Problem 24 of the Rhind Papyrus: a quantity added to a quarter of that quantity become 15. What is the quantity?

Modern solution would be:

X + 1/4 x = 15

5/4 x = 15

X = 15 * 4/5

X = 12

Since the Egyptians did not have Modern algebra, they had to provide detailed solutions for the such problems. Problems 24-29 are similar to this problem.

Reference: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Egyptian_papyri.html

Rhind Problem 24 cont’

Reference: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Egyptian_papyri.html

Problem 24 of the Rhind Papyrus: a quantity added to a quarter of that quantity become 15. What is the quantity?

Ahmes guesses the answer x = 4. This is to remove the fraction in the x/4 term. Now with x = 4 the expression x + x/4 becomes 5. This is not the correct answer, for the expression is required to equal 15. However, 15 is 3 times 5 so taking 3 times his guess of x = 4, namely x = 12, gives Ahmes the correct result.

Ahmes uses the "method of false position”.

In modern notation the problem is to solve x + x/4 = 15.

Area of Circle

29

63.617252

Reference: A History of Mathematics by V. Katz, 3rd ed.

Problem 50 of the Rhind Papyrus asks what is the area of a round field of diameter 9.

The solution is take away 1/9 of the diameter; the result is 8. Multiply 8 times 8; it makes 64. Therefore, the area is 64. (from Rhind papyrus)

The modern answer would be

2 22561( )9 81A d d r The solution is algebraically

given by the modern expression:

Problems 48-55 concern finding areas of regions.

Rhind Prob. 50 cont’

3.141592654

2 22561( )9 81A d d r

Reference: A History of Mathematics by V. Katz, 3rd ed. And http://www.africanexecutive.com/modules/magazine/articles.php?article=4075

Equating these two “solutions” shows that the Egyptians were using 256/81 as the “value” for Pi.

The Egytian solution is algebraically given by the modern expression:

2A rThe modern expression for the area is:

2563.160493827

81

Area of Triangles

Problem 17 MMP. Given a triangle with area 20 (setjat) and a length that is 1/3+1/15 of its breadth; Find the length and breadth. (The answer is length 10 and breadth 4)

Problem 51 RMP Example of producing [the area] of a triangle (spdt) of land. What is the area of a triangle of height 10 khet and a base of 4 khet? The answer is computed to be 20 setjat. (1/2*4*10).

It is clear that they knew that the area of a triangle is 1/2*base*height. Problem 51 in RMP and problem 17 in MMP use this formula. Reference:

http://euler.slu.edu/escher/index.php/The_Mathematical_Papyri

Rectangles

Problem 6 MMP: A rectangle is 12 setjat [in area] has a breadth 1/2 1/4 [i.e. 3/4] of its length.[Calculate it]

Calculate 1/2 1/4 to get 1. The result is 1 1/3 Take this 12 setjat 1 1/3 times. The result is 16.Calculate its square root. The result is 4 for its length [and] 1/2 1/4 of it is 3 for the breadth

Reference: http://euler.slu.edu/escher/index.php/The_Mathematical_Papyri

In modern terms, we would solve 3/4 x * x = 12 which gives x 2 = 16 and so x =4.

Ahmes solution

Rhind Problem 56Problem 56 RMP. Example of reckoning a pyramid 360 in its ukha-thebet (length of the base) and 250 in its peremus. Cause thou that I know the seked of it.

The seked is the slope of an object. (RISE/RUN)

Reference: http://www.math.washington.edu/~greenber/Rhind.html

The royal cubit was equal to 7 palms. One palm was equal to 4 fingers.

You are to take half of 360; It becomes 180. You are to reckon with 250 to find 180. Result:          1/2  + 1/5 + 1/50. A cubit being 7 palms, you are to multiply by 7.

        1                7         1/2           3 + 1/2         1/5        1 + 1/3 + 1/15         1/50        1/10 + 1/25

Its seked is 5 1/25 palms.

180 1*7 5

250 25

Modern solution

Rhind Problem 57Problem 57 RMP. A pyramid 140 in its ukha-thebt (length of the base), and 5 palms, 1 finger in its seked. What is the peremus thereof?

The peremus is the height of an object.

Reference: http://euler.slu.edu/escher/index.php/The_Mathematical_Papyri

The royal cubit was equal to 7 palms. One palm was equal to 4 fingers.

You are to divide 1 cubit by the seked doubled, which amounts to 10 1/2.

You are to reckon with 10 1/2 to find 7 for this one cubit. Reckon with 10 1/2. Two-thirds of 10 1/2 is 7. You are to reckon with 140, for this is the ukha thebt. Make two-thirds of 140, namely 93 1/3. This is the peremus thereof.

3

193140

2/)145(

7

Truncated PyramidExample 14. (Moscow Papyrus) Example of calculating the volume of a truncated pyramid. The base is a square of side 4 cubits, the top is a square of side 2 cubits and the height of the truncated pyramid is 6 cubits.

Reference: http://www-history.mcs.st-and.ac.uk/HistTopics/Egyptian_papyri.html

The Egyptian knew the formula for the volume: V = (h / 3) (a2 + ab + b2).

Example of calculating a truncated pyramid. If you are told: a truncated pyramid of 6 for the vertical height by 4 on the base by 2 on top: You are to square this 4; result 16. You are to double 4; result 8. You are to square this 2; result 4. You are to add the 16 and the 8 and the 4; result 28. You are to take 1/3 of 6; result 2. You are to take 28 twice; result 56. See it is of 56. You will find (it) right.

Geometric SumProblem 79 RMP There are seven houses; in each house there are seven cats; each cat kills seven mice; each mouse has eaten seven grains of barley; each grain would have produced seven hekat. What is the sum of all the enumerated things.

Houses

7

Cats 49

Mice 343 1 2801

Heads of barley

2401 2 5602

Hekats of barley

16807

4 11204

Total 19607

total 19607

The table shown is given as the solution. Can you determine what the last two columns represent?

Reference: http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html

Erdős–Straus conjecture

Reference: http://mathworld.wolfram.com/Erdos-StrausConjecture.html

The Erdős–Straus conjecture concerns the length of the shortest expansion for a fraction of the form 4/n.

Does an expansion

exist for every n?

It is known to be true for all n < 1014, and for all but a vanishingly small fraction of possible values of n, but the general truth of the conjecture remains unknown.

4 1 1 1

n x y z

Example: 4/17 = 1/5+1/29+1/1233+1/3039345

4/17 = 1/6 + 1/17 + 1/102

Egyptian MathematicsSymbol based number system

Unit fractions were used

Mathematics built around addition

Problems on papyri show examples from which students would generalize to actual problems at hand. (Cook Book Math..)

Sophisticated geometry known for areas and volumes

Reference:http://www.math.tamu.edu/%7Edallen/masters/index.html

Ancient Egypt

More Good Stuff:

1. At the site http://www-groups.dcs.st-and.ac.uk/~history/ find and read the article: An overview of Egyptian mathematics

2. At the site http://www.discoveringegypt.com/index.htm find and read the section on Egyptian Numerals. Try working out the variety of mathematics problems given on this site.

3. At the site http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fractions/egyptian.html read the material on Egyptian fractions. You can also use the “calculator” on this page to find Egyptian fractions for some of the homework problems asked for this course. (This is a long page with way more than you’ll want to read. At least read from top to A Calculator to convert a Fraction to an Egyptian Fraction. )

More on Ancient Egypt

More Good Stuff:

1. At the site http://www.bbc.co.uk/ahistoryoftheworld/objects/y1T3knf-T66RwWyEt_cZBw there is an audio tour

of the British museum where the Rhind papyrus is stored.

2. Ein mathematisches handbuch der alten Aegypter (Papyrus Rhind des British Museum) (1877) can be downloaded at http://www.archive.org/details/einmathematische00eise . This is the first publication on the Rhind papyrus, but it is in German.