AMA1D01C { Egypt and Mesopotamia · Babylon Mesopotamian civilization I Emerged in the Tigris and Euphrates river valleys around 3500 BC I Many kingdoms (e.g. Sumer, Akkad, Assyria)

AMA1D01C – Egypt and Mesopotamia

Dr Joseph Lee, Dr Louis Leung

Hong Kong Polytechnic University

Dr Joseph Lee, Dr Louis Leung AMA1D01C – Egypt and Mesopotamia

Outline

Cultures we will cover:

I Ancient Egypt

I Ancient Mesopotamia (Babylon)

I Ancient Greece

I Ancient India

I Medieval Islamic World

I Europe since Renaissance


References

These notes follow the following book:

I Calinger, R. A Contextual History of Mathematics.Prentice-Hall, 1999.

I Katz, V. A History of Mathematics: an Introduction.Addison-Wesley, 1998.


Introduction

“Origin” of mathematics:

I Earliest motivation: counting, tax collection, measurement,building, trade, calendar making, ritual practices


Babylon

Mesopotamian civilization

I Emerged in the Tigris and Euphrates river valleys around 3500BC

I Many kingdoms (e.g. Sumer, Akkad, Assyria) rose over thenext 3000 years

I The one based in the city of Babylon (present-day Hillah)conquered the entire area around 1700 BC

I In these notes we use the terms “Babylon” and“Mesopotamia” interchangeably

I Mesopotamia is also known as the “Fertile Crescent”.I Writing was done by styli (singular: stylus, a pointed device

used to scratch out letters) on clay tabletsI Thousands of such tablets have been excavated and well

documented by archeologistsI Many of these tablets contain mathematical problems,

solutions and tables


Egypt

I First dynasty to rule over both Upper Egypt (“upper” in thegeographic sense, as in “upriver”, the river being the Nile)and Lower Egypt (“lower” geographically, as in “downriver”)dated from around 3100 BC

I Much of what we know about ancient Egyptian mathematicscomes from the two following sources


Egypt and Mesopotamia

Figure: Map of Egypt and Mesopotamia over contemporary politicalboundaries. The modern city of Hillah (the site of ancient Babylon) ismarked. Source: Google Map.


Egypt and Mesopotamia

Figure: Map of Mesopotamia. Source: Encyclopedia Britannica.


Egypt

Rhind Mathematical Papyrus

I Named for Scotsman Alexander Henry Rhind (1833-1863)

I Bought by Rhind at a market in Luxor, Egypt in 1858

I Has remained in the British Museum since 1864

Moscow Mathematical Papyrus

I Also known as the Golenishchev Papyrus

I Purchased by Vladimir Semyonovich Golenishchev in 1893 andlater sold to the Moscow Museum of Fine Arts (Alexander IIIMuseum of Fine Arts, now known as the Pushkin StateMuseum of Fine Arts)


Egypt

Papyrus: https://www.wikihow.com/Make-Papyrus


https://www.wikihow.com/Make-Papyrus

Arithmetic

ARITHMETIC


Counting - Hieroglyph

Egyptian numerals (hieroglyphic system, used for writing on templewalls or carving on columns):

I Each power of ten has a symbol

I Numbers are represented by corresponding repetitions of suchsymbols

I For example, the number 367 is represented by 3 copies of thesymbols for 100, 6 copies of the symbol for 10, and 7 copiesof the symbol for 1


Hieroglyph

Figure: Hieroglyphic numerals. Source:http://www-history.mcs.st-and.ac.uk/HistTopics/Egyptian_numerals.html


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


Hieroglyph

Figure: Hieroglyphic numerals-examples. Source:http://www-history.mcs.st-and.ac.uk/HistTopics/Egyptian_numerals.html




Counting - Hieratic

Egyptian numerals (hieratic system, used for writing on papyrus(something like paper made from the pith of the papyrus plant)):

I Each number from 1 to 9, each multiple of 10 from 10 to 90,each multiple of 100 from 100 to 900, and each multiple of1000 from 1000 to 9000, has its own symbol

I Example: 37 is represented by the symbol for 7 next to thesymbol for 30

I Example: 243 is represented by the symbol for 3, 40 and 200.


Hieratic

Figure: Hieratic numerals. Source:http://www-history.mcs.st-and.ac.uk/HistTopics/Egyptian_numerals.html




Hieratic

Figure: Hieratic numerals - Example. Source:http://www-history.mcs.st-and.ac.uk/HistTopics/Egyptian_numerals.html




Arithmetic

Egyptian algorithms for addition and multiplication in thehieroglyphic system

I Addition: Combine the symbols, then convert

I Subtraction: Convert (i.e., borrowing, if necessary), thensubtract

I Multiplication: continuous doubling

I There is no evidence, however, to show how the doubling wasdone

I Depends on the fact that every number can be written as asum of powers of 2. We are not sure, however, if theEgyptians knew this fact.

I Maybe they just observed by experimentation

I Division is the inverse of multiplication, therefore the numbera/b would be phrased as“muliply by b so that we get a”


Arithmetic

Example: 12× 13x 12× x1 122 244 488 96

I Keep doubling.

I Doubling the last row will give x = 16 and 16 > 13

I Find numbers in the first column so that the sum is 13:13 = 1 + 4 + 8, therefore 12× 13 = 12 + 48 + 96 = 156


Arithmetic

Algorithms for the hieratic system

I no evidence to show how addition was done

I addition tables probably existed


Arithmetic

Egyptian fractions

I The Egyptians used only unit fractions, with the singleexception of 2/3

I To write a reciporcal, in hieroglyphics they put a flat circleabove the number and in hieratics they put a dot over thenumber

I Problem 3 of the Rhind Papyrus: How to divide 6 loavesamong 10 men?

I Each man gets 12 + 1

10 . To use a notation more similar to thehieroglyphic system, we write 2 + 10

I Observation: Giving each man half a loaf plus one-tenth of aloaf requires less cutting than giving each man sixtenths-of-a-loaf.


Arithmetic

Figure: Reciprocals in hieroglyphics. Source:http://www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_numerals.html


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


Arithmetic

Check:1 2 102 1 54 2 3 15

8 4 3 10 30

I How was the doubling done? The first section of the RhindPapyrus is a table which contains numbers of the form 2 timesn where n is an odd integer between 3 and 101

I To check the correctness of the answer, need to know adding

4 3 10 30 and 1 5 gives 6

I It is conjectured (guessed with a certain level of confidence)that addition tables existed


Modern proof

Modern proof that every fraction can be written as a sum of unitfractions (Greedy method)

I Given ab , let c be the smallest integer such that 1

c ≤ab <

1c−1

I Consider ab −

1c = ac−b

bc

I From ab <

1c−1 , we get ac − a < b, so ac − b < a

I Any time we subtract the biggest possible unit fraction fromab , the numerator becomes smaller

I A decreasing sequence of non-negative integers must reach 0in finitely many steps


Modern proof

Example: Consider 1721

I Note 12 <

1721 < 1. Do 17

21 −12 = 13

42 .

I Note 14 <

1342 <

13 . Do 13

42 −14 = 5

84 .

I Note 117 <

584 <

116 . Do 5

84 −117 = 1

1428 .

I Therefore 1721 = 1

2 + 14 + 1

17 + 11428

I Note, however, the expression is not unique, since 1721 is also

equal to 12 + 1

6 + 17 , which is easier to work with.

I Practical concern: cutting a pizza into 7 equal slices is easierthan cutting it into 21 (or 1428) equal slices


Counting

Babylonian numerals (Base 60):

I Numbers smaller than 60 are written in base 10

I More precisely a mixed-base system

I In these notes (following Katz, using Otto Neugebauer’sconvention), we write, for example, a, b, c ; d , e, f for thenumber a · 602 + b · 60 + c + d · 1

60 ,+e · 1602

+ f · 1603

, wherea, b, c , . . . are numbers ≥ 0 and ≤ 59.

I A place-value system (i.e., the value of a symbol depends onwhere it is placed).


Babylonian numerals

Figure: Babylonian numerals. Source:http://www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_numerals.html




Babylonian numerals

Figure: Babylonian numerals. Source:http://www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_numerals.html




Babylonian numerals

Figure: The Babylonian mixed-base system is still in use today.Source:http://www.hertzelectronics.com.au/digital-clocks-hours-minutes-seconds.html


http://www.hertzelectronics.com.au/digital-clocks-hours-minutes-seconds.html

http://www.hertzelectronics.com.au/digital-clocks-hours-minutes-seconds.html

Babylonian numerals

How did we get the number on the previous page?

I 1× 603 + 57× 602 + 46× 60 + 40 = 424000.


Why base 60?

Theon of Alexandria (Greek scholar, AD 335-405) tried to explainwhy a base-60 system was chosen.

I 60 = 2 · 2 · 3 · 5I 1

2=0;30, 13 = 0; 20, 1

4 = 0; 15, 15 = 0; 12, 1

6 = 0; 10, 110 = 0; 6,

112 = 0; 5, 1

15 =; 4, 120 = 0; 3, 1

30 = 0; 2, can all be written asfinite sexagesimals with one sexagesimal place

I Note all of the five smallest non-unit integers appear asdenominators in the list above.


Arithmetic

Babylonian arithmetic

I Extensive use of multiplication tables proved by tabletspreserved to this day

I However no addition tables have been found

I Since the Babylonian place-value system is similar to ours, wemay assume their adding algorithm is similar to ours.

I Example: add 23,35 to 40,33


Arithmetic

23,35+40,33

I 35+33=1,08

I 23,00+40,00=1,03,00

I Therefore 25,35+40,33=1,04,08


Linear Equations and Linear Systems

LINEAR EQUATIONS



There is evidence of the Egyptians and the Babylonians solvinginteresting linear problems.

I Problem 64, Rhind Papyrus: Arithmetic progression

I Babylonian text VAT8389: Linear system



Problem 64 of the Rhind Papyrus: “If it is said to thee, divide 10hekats of barley among 10 men so that the difference of each manand his neighbour in hekats of barley is 1

8 , what is each man’sshare?” (Gillings, Mathematics in the Times of the Pharaohs)

I Arithmetic progression

I Average is 1 hekat per man

I Add half the common difference 12 ·

18 = 1

16 nine times to getthe largest share 1 + 9

16 (or 1 2 16)

I Subtract 18 from the largest share nine times to get the size of

each share



Figure: Problem 64, Rhind Papyrus



Problem from VAT8389: “One of two fields yields 23 sila per sar,

the second yields 12 sila per sar. The yield of the first field was 500

sila more than that of the second; the areas of the two fields weretogether 1800 sar. How large is each field?” (Katz)

I We have the system{23x −

12y = 500

x + y = 1800

I Assume x and y are both 900.I 2

3 · 900− 12 · 900 = 150

I 500− 150 = 350, and each unit increase in x (with acorresponding unit decrease in y) increases 2

3x −12y by

23 + 1

2 = 76

I Number of such increments needed is 350 divided by 76 , which

is 300I Add 300 to 900 to get x = 1200 and subtract 300 from 900

to get y = 600Dr Joseph Lee, Dr Louis Leung AMA1D01C – Egypt and Mesopotamia

Elementary Geometry

ELEMENTARY GEOMETRY


Elementary Geometry

Problem 50 of the Rhind Papyrus: “Example of a round field ofdiameter 9. What is the area? Take away 1/9 of the diameter; theremainder is 8. Multiply 8 times 8; it makes 64. Therefore, thearea is 64.”

I Area is given by (8d9 )2 = 64d2

81 = 256r2

81

I 25681 = 3.16049 . . .

I How did they come up with this number?


Elementary Geometry

Hint: Problem 48

I Using the octagon to approximate the area of the circle, weget 7d2/9 = 63d2/81

I The Egyptians may be interested in “squaring the circle”, i.e.,finding a number x such that the area of the circle is x2.

I 64/81 is close to 63/81, and√

64/81 = 8/9.


Rhind 48

Figure: Rhind Papyrus Problem 48. Source:http://www.math.tamu.edu/~don.allen/history/egypt_old/egypt.html


http://www.math.tamu.edu/~don.allen/history/egypt_old/egypt.html

http://www.math.tamu.edu/~don.allen/history/egypt_old/egypt.html

Elementary Geometry

The Babylonians also used the formula (C/2)(d/2)

I C is the circumference while d is the diameter

I The formula is correct, since C = 2πr and A = πr2.

I They also used A = C 2/12, obtained by taking d = C/3

I Possible explanation: they divide the circles into sectors andrearranged


Elementary Geometry

Volume

I There are problems in the Rhind Papyrus where the formulaV = Bh was used

I One would expect the Egyptians knew how to calculate thevolumes of pyramids

I No such formula has been found. However, the MoscowPapyrus contained a problem on the volume of a truncatedpyramid


Elementary Geometry

“If it is said to thee, a truncated pyramid of 6 cubits in height, of 4cubits of the base by 2 of the top; reckon thou with this 4,squaring. Result 16. Double thou this 4. Result 8. Reckon thouwith this 2, squaring. Result 4. Add together this 16 with this 8and with this 4. Result 28. Calculate thou 1/3 of 6. Result 2.Calculate thou with 28 twice. Result 56. Lo! It is 56. Thou hasfound rightly.” (Gillings, Mathematics in the Times of thePharaohs)

I 42 = 16, 4 · 2 = 8, 22 = 4, 16 + 8 + 4 = 28, 6/3 = 2,28 · 2 = 56

I If base width is a, top width is b, truncated height is h, thenthe method follows the correct formulaV = (h/3)(a2 + ab + b2).


Calendar

CALENDAR


Calendar

Egyptian calendar

I 12 months of 30 days with 5 additional days

I The priests were aware that the beginning of the year wouldmove through the seasons in 1460-year cycles

I 1460 = 4 · 365, so a shift of 365 days over 1460 years means ashift of 1/4 day per year

I The Egyptians knew that the length of a year wasapproximately 3651

4 days


Calendar

Babylonian calendar

I Months alternate between 29 and 30 days

I Closer to the actual lunar cycle, which averages at about29.5306 days (https://eclipse.gsfc.nasa.gov/SEhelp/moonorbit.html)

I 12 months give us 354 days ((29 + 30) · 6)

I 7 leap years (with 13 months) occur every 19 years

I Lengths of the months were adjusted once in a while toensure there are 6940 days in each 19-year cycle (whichcontains 12 · 19 + 7 = 235 months


https://eclipse.gsfc.nasa.gov/SEhelp/moonorbit.html

https://eclipse.gsfc.nasa.gov/SEhelp/moonorbit.html

Calendar

Babylonian calendar

I Note 6940/235 ≈ 29.5319 and 6940/19 ≈ 365.26

I For comparison, our current estimate of the length of a year is365.2422 days(https://www.grc.nasa.gov/WWW/k-12/Numbers/Math/Mathematical_Thinking/calendar_calculations.htm)

I Better estimate of the month but worse estimate of the yearthan the Egyptians

I The current Jewish calendar is similar to the Babyloniancalendar, with minor modifications


https://www.grc.nasa.gov/WWW/k-12/Numbers/Math/Mathematical_Thinking/calendar_calculations.htm

https://www.grc.nasa.gov/WWW/k-12/Numbers/Math/Mathematical_Thinking/calendar_calculations.htm

Square Roots

SQUARE ROOTS


Square Roots

I The Babylonians had extensive square, square root, cube, andcube root tables

I Very often problems are set up so that the square root is oneof the numbers in the square root table.

I There are problems, however, where the square root of 2 isneeded

I The square root of 2 is given by 1; 25 = 1 + 2560

I Comparison: 1 + 2560 ≈ 1.41667,

√2 ≈ 1.41421


Square Roots

How was the value found?

I Let N = 2, write N = a2 + b = (a + c)2 = a2 + 2ac + c2

I Pick a so that it is very close to√N, so c is small, which

makes c2 small relative to 2ac

I Therefore b ≈ 2ac , or c ≈ b2a = N−a2

2a .

I N ≈ (a + N−a2

2a )2

I For N = 2, let a = 1; 20 = 4/3, then a2 = 1; 46, 40,b = 0; 13, 20, 1/a = 0; 45

I Therefore√

2 =√

1; 46, 40 + 0; 13, 20 ≈1; 20 + (0; 30)(0; 13, 20)(0; 45) = 1; 20 + 0; 5 = 1; 25

I Note 1; 25 = 17/12 and (17/12)2 = 289/144 = 2 + 1144 .


Pythagorean Theorem

“PYTHAGOREAN” THEOREM


Pythagorean Theorem

A table with four columns was found on the Babylonian TablePlimpton 322

y(reconstructed) (x/y)2 x d (x/y) + (d/y)(reconstructed)

120 0.9834028 119 169 2.4 13456 0.9491586 3367 4825 2.37 24800 0.9188021 4601 6649 2.34 313500 0.8862479 12709 18541 2.31 472 0.8150077 65 97 2.25 5360 0.7851929 319 481 2.22 62700 0.7199837 2291 3541 2.16 7


Pythagorean Theorem

y(reconstructed) (x/y)2 x d (x/y) + (d/y)(reconstructed)

960 0.6845877 799 1249 2.13 8600 0.6426694 481 769 2.08 96480 0.5861226 4961 8161 2.025 1060 0.5625 45 75 2 112400 0.4894168 1679 2929 1.92 12240 0.4500174 161 289 1.87 132700 0.4302388 1771 3229 1.85 1490 0.3871605 56 106 1.8 15


Pythagorean Theorem

A guess on how to generate Pythagorean Triples

I x2 + y2 = d2

I Dividing both sides by y2, we get (x/y)2 + 1 = (d/y)2

I Let u = x/y , v = d/y

I We have u2 + 1 = v2, or v2 − u2 = 1, or (v + u)(v − u) = 1


Pythagorean Theorem

Example (Katz)

I (v + u) = 2; 15, (v − u) = 0; 26, 40

I Solving for v and u we get v = 1; 20, 50 = 1 + 25/72 andu = 0; 54, 10 = 65/72

I Multiply each value by 1, 12 = 72 gives x = 65 and d = 97

I Note: The value v + u for every line form a decreasingsequence of regular sexagesimal numbers of no more than fourplaces, giving evidence which leads us to the guess above


Quadratic Equations

QUADRATIC EQUATIONS


Quadratic Equations

The Babylonians conisdered the following system{x + y = b

xy = c

I Find length and width given perimeter and area

I There was no general formula. Problems were presented withconcrete numbers

I In tablet YBC4663 they considered x + y = 612 , xy = 71

2

I First half 612 to get 31

4 , then square 314 to get 10 9

16 . From10 9

16 they subtracted 712 to get 3 1

16 , then take the square rootto get 13

4

I Length is 314 + 13

4 = 5 and width is 314 − 13

4 = 112


Quadratic Equations

In modern notation (assume x > y):

I ( x+y2 )2 = xy + ( x−y

2 )2

I Therefore x−y2 =

√( x+y

2 )2 − xy =√

(b2 )2 − c

I x = b2 + x−y

2 and y = b2 −

x−y2

I x = b2 +

√( x+y

2 )2 − xy = b2 +

√(b2 )2 − c and

y = b2 −

√( x+y

2 )2 − xy = b2 −

√(b2 )2 − c


Quadratic Equations


Quadratic Equations

Another problem considered by the Babylonians{x − y = b

x2 + y2 = c

I x2 + y2 = 2( x+y2 )2 + 2( x−y

2 )2

I Therefore x+y2 =

√c2 − (b2 )2

I x = x+y2 + x−y

2 =√

c2 − (b2 )2 + b

2 ,

y = x+y2 −

x−y2 =

√c2 − (b2 )2 − b

2


Quadratic Equations


Quadratic Equations

Another problem considered by the Babylonians (from BM13901)

x2 +4

3x =

11

12

I The problem was asked with concrete numbers

I However, generalizing the method and presenting it in modern

notation, we get x =√

(b2 )2 + c − b2 (b is the coefficient of x

on the left and c is the constant on the right

I Most likely obtained by a geometric method: if x2 + bx = c ,then (x + b

2 )2 = (b2 ) + c


Quadratic Equations


Quadratic Equations

A solution to x2 − bx = c (b > 0) may be found in a similar way

I In modern notation, x =√

(b2 )2 + c + b2

I Geometric method: if x2 − bx = c , then (x − b2 )2 = (b2 )2 + c

I Cannot be thought of as the same type of problems asx2 + bx = c

I The geometric meaning is different, so the scribes gave adifferent procedure for finding a solutions

I We may guess that they did not have “abstract” algebra (i.e.,the formal “pushing around” of symbols).


Quadratic Equations


Quadratic Equations

Problems of the form x2 + c = bx were not considered

I Problems of the same essence were solved, i.e., the systemx + y = b, xy = c

I However equations of the form x2 + c = bx did not appear onthe tablets

I Guess: the scribes were not comfortable with equations havingmore than one solutions, so they set up their problems withtwo variables instead.


References

I Katz, V. A History of Mathematics: an Introduction.Addison-Wesley, 1998.


Documents

AMA1D01C { Egypt and Mesopotamia · Babylon Mesopotamian civilization I Emerged in the Tigris and Euphrates river valleys around 3500 BC I Many kingdoms (e.g. Sumer, Akkad, Assyria)