Mesopotamia - people.clas.ufl.edu · Algebra developed in Mesopotamia far more than in Egypt Had working understanding of ‘combining like terms’, multiplying both sides of an

Mesopotamian WritingMesopotamian Mathematics

Conclusion

Mesopotamia

Douglas Pfeffer

Douglas Pfeffer Mesopotamia


Conclusion

Table of contents

1 Mesopotamian Writing

2 Mesopotamian Mathematics

3 Conclusion



Conclusion

The Era and the SourcesCuneiform Writing

Outline



3 Conclusion



Conclusion


Location and Timeline



Conclusion


The Era

4000 BCE to 3000 BCE was an incredible millenium ofcultural development

Metal (beginning of Bronze Age) and widespread use of thewheelEgypts first dynasty began around 3100 BCESumerians united the mesopotamian region and built incrediblecanal systems for irrigation to control the erratic flooding ofthe Tigris and Euphrates rivers

Not predictable like that of the Nile in Egypt

Began a widespread use of writing called cuneiform



Conclusion


Fertile Crescent Valley and Babylon

The Fertile Crescent Valley, unlike Egypt, has no naturalbarriers from invasion

Semitic Akkadians took over the valley in 2276 BCE underSargon the Great and began unifying the regions disparate,Sumerian culture

Of importance was the adoption of native cuneiform writingReign ended in 2221 BCE

Other invaders dominated the region until about 1900 BCE

The era from 1894 BCE to 539 BCE is referred to asBabylonian

In 539 BCE, Babylon fell to Cryus of Persia. The cityremained, but the empire endedIts mathematics persisted through to the start of the commonera



Conclusion


Sources

Soft clay tablets baked in the hot sun or in an oven

Thousands of tablets contained information like laws, taxaccounts, school lessons, etc.

Tablets, unlike Egyptian papyri, stand the test of time quitewell

There exists a much larger body of mathematical knowledge

From one site in Nippur, there are approx. 50,000 tablets

Interestingly, Egyptian hieroglyphics were still translated first

Took until early 1800s when a German philologist F.W.Grotefend started translatingLate 1800s finally saw a full translation of Mesopotamianmathematics



Conclusion


Cuneiform Writing

Earliest vestiges occur in tablets from Uruk circa 3000 BCE

Based in picture-writing

In 3000 BCE, approx. 2000 symbols were in use. By 2300BCE (Sargon invasion), only 700 were in use.

Instead of pictures, it now used combinations of wedgesOriginally, cuneiform was written vertically, left-to-right. By2300 BCE, it switched to horizontal right-to-leftThe early stylus was a truncated cone with thin strokes forsmall values and think strokes for larger values

Around 2000 BCE, the numbering system simplified greatly toa single width stylus



Conclusion


Babylonian Numerals

By 1800s, the numeral system simplified greatly:

Remained this way for millenia



Conclusion


Babylonian Numerals

The Mesopotamian number system operated on a Base 60Theories as to why vary, but most agree it was due to‘metrology’

60 has 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30 as divisors

While no culture uses Base 60 anymore, quantities like timeand angle measures still use it



Conclusion


Positional Notation

The Babylonians utilized positional notation.

Small whole numbers < 60

— = 1, | = 10. Used repetition to reach desired numberSimilar to Egyptian hieroglyphics

Whole numbers ≥ 60Differed greatly from hieroglyphics The same symbolrepresented a different value depending on where it wasrelative to the other symbols!

This positional notation is much like todays: 222 uses thecipher ‘2’ three times – each with a different meaningEx: || || || = 2(602) + 2(60) + 2 = 7322

Invented approx. 2000 BCE



Conclusion


The Empty Position

Our knowledge of Mesopotamian mathematics comes fromtwo main time frames:

Old Babylonian Age (c. 1800 BCE)Seleucid Period (c. 300 BCE)

The Old Babylonian Age had a wealth of mathematicalcontribution, but the notion of an empty position didn’tappear in sources until the Seleucid Period.

The empty position is essentially what we use 0 for in todaysnumeration

Before 300 BCE, it was impossible to tell whether || || meant2(60) + 2 or 2(602) + 2.Unfortunately, this ambiguity was settled by appealing tocontext.



Conclusion


The Empty Position

By around 300 BCE, when Alexander the Great died and theSeleucid period began, a cipher was finally introduced for theempty position: //

|| // || = 2(602) + 0(60) + 2|| || = 2(60) + 2Unfortunately, there was still ambiguity. The Babylonianpositional system was relative, not absolute.Thus it was impossible to distinguish between

|| ||2(60) + 22(602) + 2(60)2(603) + 2(602)

Nevertheless, an incredible improvement over Egyptianhieroglyphics!



Conclusion


Sexagesimal Fractions

Incredibly, the Babylonians seamlessly extended the positionalsystem to fractions

Ex: || ||, which we saw could have been equal to 2(60) + 2,could also have been equal to:

2 + 2(60−1)2(60−1) + 2(60−2)

While still suffering from ambiguity, it was a definitiveimprovement over the Egyptian unit fractions

This treatment mimics the modern decimal system

An advantage of the modern decimal system and ourpositional notation is that 23.45× 9.876 is no more difficult tocompute than 2, 345× 9, 876The Babylonian system behaved similarly and they utilized thisfreedom often.



Conclusion

Approximations and AlgorithmsTables and ArithmeticAlgebraPythagorean Triples and Geometry

Outline



3 Conclusion



Conclusion


Approximations

Modern discussions of sexagesimal numbers writes:

2, 17; 8, 50 = 2(60) + 17 + 8(60−1) + 50(60−2).

A tablet currently in the Yale collection contains analgorithmic approximation to

√2.

It computes the value to 2 sexagesimal places:

1; 24, 51, 10 = 1 + 24(60−1) + 51(60−2) + 10(60−3) = 1.414222

Actual value:√

2 = 1.41421...



Conclusion


The Square Root Algorithm

Let x =√a be the desired root.

Let a1 be a first guess at x . Let b1 = aa1

be another guess at x .Either a1 < x < b1 or b1 < x < a1.Therefore a2 = a1+b1

2 is a better approximation. Considerb2 = a

a2. Again, either a2 < x < b2 or b2 < x < a2.

Choose a3 = a2+b2

2 , continue.

On the Yale tablet, a1 = 1; 30. Their final approximation wasa3 = 1; 24, 51, 10.

Sometimes this algorithm is attributed to other peoples andcultures

Greek scholar Archytas (c. 400 BCE)Heron of Alexandria (c. 100 BCE)‘Newtons Algorithm’

An early instance of an infinite procedure.



Conclusion


Tables

A large portion of the Babylonian cuneiform mathematicaltablets are ‘table texts’.

Multiplication tables, reciprocal tables, squares, cubes, squareroots, cube roots, etc.

Ex:

2 303 20 Observe: The product of each line is 60.4 155 12 This is a table of reciprocals6 10 (unit fractions in Egyptian)8 7,309 6,40 1

8 = 7(60−1) + 30(60−2)10 612 5



Conclusion


Reciprocal Tables

Observe that 17 and 1

11 are missing.

7 and 11 were considered ‘irregular’ since their sexagesimalexpansions are non-terminatingSimilar to the numbers 1

3 ,17 , and 1

9 in our decimal system

Again, an opportunity to investigate the infinite, but theBabylonians seeming skirted around it

At one point, however, a scribe noted that

0; 8, 4, 16, 59 ≤ 1

7≤ 0; 8, 34, 18.



Conclusion


Arithmetic

Multiplication was carried out easily be the Mesopotamiancultures

Division was far more sophisticated than Egyptian duplationBabylonians divided two numbers by multiplying the dividendby the reciprocal of the divisor (using the tables)

Ex: Much like today, we can compute:

34

5= 34·1

5= 34·0.2 = 34·2, then move one decimal place over.

They computed:

34

5= 34 · 1

5= 34 · 12, then move one sexagesimal place over

= 6; 48 = 6 4860 .



Conclusion


Algebraic Equations

Algebra developed in Mesopotamia far more than in Egypt

Had working understanding of ‘combining like terms’,multiplying both sides of an equation to clear fractions, andinstances of adding quantities like 4ab to (a− b)2 to yield(a + b)2

Factoring posted no great challenge

Note: They did not use letters to represent unknowns sincethe alphabet didn’t yet exist

Used words like ‘length’, ‘breadth’, ‘area’, and ‘volume’Interestingly, they routinely added ‘area’ and ‘volume’,suggesting that these words were abstract quantities and notconcrete representations



Conclusion


Linear Equations

Egyptian algebra capped out at linear equations, Babyloniantexts show that these were handled easily

A tablet from the Old Babylonian Age (c. 1800 BCE)encounters a system of linear equations:{

14 width + length = 7 hands

width + length = 10 hands

The scribe provided two solutions. The first was essentially asolution by ‘inspection’. The second, however, first multipliedthe first equation by 4, then subtracted the two equations toyield 3 width = 18, or width = 6 and therefore length = 4

An incredible instance of what we call the ‘method ofelimination’.



Conclusion


Quadratic Equations

Babylonians not only handled linear equations easily, but seemto have tackled quadratic equations as well. Something Egyptdidn’t do.

A tablet from the Old Babylonian Age asks: What is the sideof a square if the area less the side is 14, 30?

The scribe writes the solution: “Take half of 1, which is 0; 30,and multiply 0; 30 by 0; 30, which is 0; 15; add this to 14, 30 toget 14, 30; 15. This is the square of 29; 30. Now add 0; 30 to29; 30, and the result is 30, the side of the square.”



Conclusion


Quadratic Equations

In modern notation, the problem asked for the solution tox2 − x = 870. Applying the quadratic formula to x2 − bx = c ,(the) solution should be

x =

√(b

2

)2

+ c +b

2.

where b = 1 and c = 14, 30 = 870.

The scribe wrote: “Take half of 1, which is 0; 30, and multiply0; 30 by 0; 30, which is 0; 15; add this to 14, 30 to get14, 30; 15. This is the square of 29; 30. Now add 0; 30 to29; 30, and the result is 30, the side of the square.”This is exactly the correct procedure!



Conclusion


Quadratic Equations

Historically, since complex numbers didn’t exist until the late1500s, quadratic equations of the form x2 + bx + c = 0 withb, c > 0 were never considered.

Quadratic equations came in three types:

x2 + px = q, x2 = px + q, and x2 + q = px .

In Mesopotamian tablets dating to 2000 BCE, examples ofeach type appear and are solved.

Of interest are instances of classic questions regarding pairs ofnumbers whose sum and product are fixed. The solutionsinvolve solving systems of equations that reduce to aquadratic equation.

We still see these types of problems in texts today.



Conclusion


Cubic Equations

Unlike the Egyptians, Babylonians considered cubic equationsCubics of the form x3 = a were solved via a table of cubes

For non-integer solutions, they used linear interpolation toapproximate

Similarly, cubics of the form x3 + x2 = a were solved byreferencing an n3 + n2 table.

For general ax3 + bx2 = c , they reduced to the previous case

by multiplying through by a2

b3 to convert it into(axb

)3

+(axb

)=

ca2

b3.

This substitution is an incredible observation when oneremembers that they didn’t have variables

Unfortunately, there are no sources indicating a solution tothe general cubic ax3 + bx2 + cx = d .



Conclusion


Pythagorean Triples

A tablet in the Plimpton collection at Columbia (c. 1800 BCE)



Conclusion


Plimpton Tablet

The tablet translates into the following table:1,59,0,15 1,59 2,49 11,56,56,58,14,50,6,15 56,7 1,20,25 21,55,7,41,15,33,45 1,16,41 1,50,49 31,53,10,29,32,52,16 3,31,49 5,9,1 41,48,54,1,40 1,5 1,37 51,47,6,41,40 5,19 8,1 61,43,11,56,28,26,40 38,11 59,1 71,41,33,59,3,45 13,19 20,49 81,38,33,36,36 8,1 12,49 91,35,10,2,28,27,24,26,40 1,22,41 2,16,1 101,33,45 45,0 1,15,0 111,29,21,54,2,15 27,59 48,49 121,27,0,3,45 2,41 4,49 131,25,48,51,35,6,40 29,31 53,49 141,23,13,46,40 56 1,46 15

θ

ca

b

Analysis suggests the table is a proto-trigonometry table.

Column 2 is a, Column 3 is c , and Column 1 is cb = sec2(θ).



Conclusion


Plimpton Tablet

While neither the Egyptians nor the Babylonians had anymodern sense of an ‘angle’, they wrestled with geometricrelations

Of interest, the first line is decreasing

First row = 1, 59, 0, 15 ≈ sec2(45◦)Last row = 1, 23, 13, 14, 40 ≈ sec2(31◦)

The choice of lengths is theorized to have been chosen w.r.tthe following:

Given 2 regular sexagesimal integers p and q with p > q, form

p2 − q2, 2pq, and p2 + q2.

Observe that these form Pythagorean triples. Restricting tovalues p < 60 and a < b, there are only 38 possible pairs of pand q, the first 15 of which are on the tablet.



Conclusion


Plimpton Tablet

The tablet appears to have been broken at some point, it istheorized that it would have contained the remaining 23 pairsas well as columns for p, q, 2pq, and tan2(θ).

A question arises: Why did the Babylonians record this data?

Some historians argue that this was an exercise in elementarynumber theory and done for solely intellectual reasons (pure)Others argue that it was conducted to aid in the computationof real problems involving the areas of squares whose sides areshared with a right triangle. (applied)



Conclusion


Geometry

It is clear that the Babylonians outclassed the Egyptians inalgebra, but what about geometry?

Recall that the Egyptians had incredible, accurate algorithmsfor volumes of frustums of pyramidsInterestingly, the Babylonians also had formulae for thesecalculations! They used

V = h

[(a + b

2

)2

+1

3

(a− b

2

)2]

where a and b are the side lengths of the top and bottomsquares and h is the height.

It is not too hard to show that this formula reduces to the onethe Egyptians used.



Conclusion


Pythagorean Theorem

The Pythagorean Theorem cannot be found in any Egyptiansources, but it was definitely known to Mesopotamia

A tablet in the Yale collection (c. 1800 BCE) contains adiagram of a square with a diagonal running through it:

3042;2

5,35

1;24

,51,

10



Conclusion


Pythagorean Theorem

3042;2

5,35

1;24

,51,

10

The tablet says, starting with a square of side length 30, thediagonal is approximately 42; 23, 35.

Moreover, it provides the ratio of an arbitrary squares diagonalw.r.t. its side as 1; 24, 51, 10

This is, of course, an approximation to√

2. This shows clearknowledge of the Pythagorean TheoremThis is an example of a very general observation in Babylonianmathematics



Conclusion


Other Geometric Facts

Babylonians (unlike the Egyptians), as early as 1500 BCE,were aware of the following fact:

Theorem

An angle inscribed in a semicircle is always a right angle.

This theorem is usually referred to as the ‘Theorem ofThales’, despite Thales living in 600 BCE!



ConclusionInfluence and Conclusion

Outline



3 Conclusion




Influence

The center of mathematical development began moving fromthe Mesopotamian Valley to Greece around 800 BCE with thedawn of the Iron Age.

A few questions arise:How much mathematics did Greece know?

Hellenic period of Greece began with the death of Alexanderthe Great in 323 BCE. Unfortunately, there are very littlesources available on pre-Hellenic, Greek mathematics. We areunsure what they knew and what they didn’t.




Influence

How much knowledge transfered between Greece, Egypt, andMesopotamia?

Mesopotamian clay tablets stand the test of time far betterthan Egyptian papyrus and Greek parchment, but did theseother civilizations get their hands on the tablets?It would seem Egypt did not have much communication withthe Mesopotamian River Valley since their mathematicsremained inferior.A plausible explanation is given by geography. While only theMediterranean Sea lies between Greece and Egypt,Mesopotamia was separated from both by large swathes ofdesert.




Conclusions

Babylonian mathematics seems superior to that of EgyptBabylonians didn’t seem to have any concept of the moderninterpretation of ‘proof’, but they did check their work as‘proof’ that a particular division worked.

They do not seem to have anything in the way of logicalprinciples that the Greeks would introduce.

Their algebra was incredibly sophisticated. Even if they usedconcrete words such as ‘length’ and ‘width’, these were clearlyplaceholders for abstract quantities.

Much of the Mesopotamian mathematics was of a practicalnature with only a few exceptions given to some recreationalproblems.




“ The Babylonian and Assyrian civilizations have perished;Hammurabi, Sargon and Nebuchadnezzar are empty names; yetBabylonian mathematics is still interesting, and the Babylonianscale of 60 is still used in Astronomy. ”

- G.H. Hardy, A Mathematicians Apology


Documents

Mesopotamia - people.clas.ufl.edu · Algebra developed in Mesopotamia far more than in Egypt Had working understanding of ‘combining like terms’, multiplying both sides of an