22
MATHEMATICS IN EGYPT AND MESOPOTAMIA SERAP DEMİR YASEMİN DUMAN PELİN CANBAZ ALİ VEDAT ÖZKAN

Egypt Mesopotamia

Embed Size (px)

Citation preview

Page 1: Egypt Mesopotamia

MATHEMATICS IN

EGYPT AND

MESOPOTAMIA

SERAP DEMİR

YASEMİN DUMAN

PELİN CANBAZ

ALİ VEDAT ÖZKAN

Page 2: Egypt Mesopotamia

MATHEMATICS IN EGYPT AND MESOPOTAMIA

HIEROGLYPHIC NOTATION

Why did we divide the past of humanity into eras and periods with particular reference to cultural

levels and characteristics? Because it helps us to remember what was in the past easily. Now let‘s look

at mathematics in its periods and eras so in this lecture we will go back almost 4000 years to the

earliest recorded mathematics which can be found in ancient Mesopotamia and Egypt and I will talk

about hieroglyphic notation, Rhind papyrus, unit fractions and algebraic notation in Egypt.

Our major knowledge of Egyptian culture come extensive sources of Egyptian hieroglyphics (sacred

signs). Hieroglyphics remained indecipherable until 1799 when in Alexandria the trilingual Rosetta

stone was discovered. The discovery was due to Napoleon Bonaparte's attempted seizure of Egypt in

1798. While the invasion was a military disaster, it was a great scientific achievement. Napoleon had

taken 167 scholars with him to make a complete investigation of ancient and modern Egypt. One of

their major finds was the black basalt Rosetta Stone. It contains three panels that each with a different

form of writing (Greek, demotic, hieroglyphic) of the same text. Jean Francois Champollion (1790-

1832) was able to decipher the hieroglyphics based on the Greek text.

The records of mathematics we have from ancient civilizations are the textbooks used to train such

bureaucrats. These are primarily sets of problems with solutions. From Egypt we have only two books:

One of them the Rhind papyrus that, I will talk about, was probably copied around the 17th century BC

by a scribe named Ahmes That scribe developed the text from a manuscripts believed to have been

written 200 years earlier. The papyrus roll is about 13 inches wide and 18 feet long. It is called Rhind

papyrus because it had been bought in 1858 in a Nile resort town by a Scottish antiquary Henry Rhind

And other important source of Egyptian mathematics from this period is the Moscow mathematical

papyrus

So how do we know what the Egyptian language of numbers is? It has been found on the writings on

the stones of monument walls of ancient time. Numbers have also been found on pottery, limestone

plaques, and on the fragile fibers of the papyrus. The language is composed of hieroglyphs, pictorial

signs that represent people, animals, plants, and numbers. Egyptian hieroglyphic numeration was

easily disclosed. But in fact these two problem books show us how the ancient Egyptians represented

number.

Our modern representation of number is quite sophisticated. We use 10 digits, the digits 0 through 9; if

we want to represent a number that is greater than 9, we reuse these digits; but we put them in places

that extra value.

Page 3: Egypt Mesopotamia

For example:

The two digits 1 and 3 can be combined to write 13 or 31; but 3 may represent three 1s or three 10s,

depending on where it is placed in the written representation

In our modern system the original 10 digits can be put in different positions to indicate the number of

10s, 100s, 1000s and so on

The system used in ancient Egypt was much simpler. This system represents numbers with strokes: 1

is a single stroke, 2 is 2 strokes and so on

We will see echoes of this in our modern representations numbers. The written numeral 2 for example

is based on 2 horizontal strokes connected by a curve

The stroke system becomes cumbersome in representing large numbers. Thus The Egyptians devised a

system that

A single vertical stroke represented a unit 1

An inverted wicket or heel bone was used for 10

A snare somewhat resembling a capital letter C stood for 100

A lotus flower for 1000

A bent finger for 10000

A burbot fish resembling a pollywog for 100000

And a kneeling figure (perhaps god of unending) for 1000000

Sometimes the smaller digits were placed on the left and sometimes the digits were arranged

vertically. The symbols themselves occasionally were reversed in orientation so that the snare C might

be convex toward either the right or left.

Page 4: Egypt Mesopotamia

EXAMPLES:

1

=

10

=

100

=

1000

=

2

=

20

=

200

=

2000

=

3

=

30

=

300

=

3000

=

4

=

40

=

400

=

4000

=

5

=

50

=

500

=

5000

=

AHMES PAPYRUS AND UNITFRACTIONS AND ARITHMETIC OPERATIONS

If we had to depend on ceremonial and astronomical material there is a limit to extent of mathematical

information tombstones and calendar and pictures of Egyptian contributions. Mathematics is far more

than counting and measuring the aspects generally featured in hieroglyphic inscriptions. Fortunately

we have other sources of information these are Egyptian papyri one of the famous papyri is Rhind

papyrus. What is the Rhind papyrus?

As I said at the beginning; Rhind papyrus was probably copied around the 17th century BC by a scribe

named Ahmes. That scribe developed the text from a manuscripts believed to have been written 200

years earlier. The papyrus roll is about 13 inches wide and 18 feet long. It contains 87 math problems,

including equations, volumes of cylinders and prisms, and areas of triangles, rectangles, circles and

trapezoids, and fractions.

The Egyptians used unit fractions, which are fractions with one in the numerator, in the Rhind

Papyrus. In order to simplify things, the Egyptians included an important 2/n table in the papyrus, so

they could look up the answers to arithmetic problems. This table showed the number 2 divided by all

the odd numbers from 3 to 101. Where today we think of 3/5 as a single irreducible fraction, Egyptian

scribes thought of it as reducible to sum of the three unit fractions 1/3, 1/5 and 1/15. We get

Page 5: Egypt Mesopotamia

mathematical formula by looking the problem 61 in the papyrus so it means there is no formulation in

that time

Problem 61 in the Rhind mathematical papyrus gives one formula: (taking third)

and 2∕n₌2∕(n+1)+2∕n(n+1) (halving procedure). This can be stated equivalently

as (n divisible by 3 in the latter equation) Other possible formulas are

(n divisible by 5)

(where k is the average of m and n)

This formula yields the decomposition for n = 101 in the table.

Ahmes was suggested to have converted 2/p (where p was a prime number) by two methods, and

three methods to convert 2/pq composite denominators. Others have suggested only one method was

used by Ahmes which used multiplicative factors similar to least common multiply

At last; ONE PROBLEM in the Rhind Papyrus specifically by using these formulas why did not write

2/15 as a sum of unit fractions

Egyptians had some appreciation of general rules and methods above and this represents an important

step in the development of mathematics. The fundamental arithmetic operation in Egypt was addition

and our operations of multiplications and division were performed in Ahmes‘s day through successive

doubling or ‗duplation‘ our own word multiplication in fact suggestive of the Egyptian process.

A multiplication say 69 by 19 would be performed by adding 69 to itself to obtain 138 then adding

this to itself to reach 276 applying duplation again to get 552 and once more to obtain 1104 which is

18 times 69. The result of multiplying 69 by 19 is 1104+138+69 that is 1311

So Egyptians had developed a high degree of artistry in applying the duplation and the unit fractions

concept is apparent from the calculations in the problems of Ahmes. . For division duplation process is

Page 6: Egypt Mesopotamia

reversed and the divisor is successively doubled instead of multiplicand. So Egyptians had developed

a high degree of artistry in applying the duplation and the unit fractions concept is apparent from the

calculations in the problems of Ahmes. Problem 70 calls for the quotient when 100 is divided by

7+1∕2+1∕4+1∕8 the result is obtaining as follows:

Doubling the divisor successively we first obtain 15+1/2+1/4 then 31+1/2 and finally 63which is 8

times the divisor. Moreover two thirds of the divisor is known to be 5+1/4. Hence it follows that the

divisor when multiplied by 2/63 will produce 1/4. From the 2/n table one knows that 2/63 is

1/42+1/126 hence the desired quotient is 12+2/3+1/42+1/126.let‘s look this problem;

As you can see in 2/n table all these calculations show us that Egyptian are very clever people they

found the solution to problems that even we cannot solve today.

To conclude Egyptians used more clever processes to do calculations and they write such numbers of

the form 2/n in a simpler way where n is between 3 and 101 by using unit fractions.

Page 7: Egypt Mesopotamia

EXTRA KNOWLEDGE ABOUT UNIT FRACTION

Ancient Egypt lived near river Nile and their lives depended on the water level of the river Nile.

Arithmetic and geometry improved much in Egypt because Egyptians had to measure and check the

water level of the river Nile so that they could calculate and measure the land.

The style of showing fractions in Egypt is more restricted than our modern representation. Egyptians

had a fraction notation as 1/2,1/3,....1/n. But this notation does not cover 2/5, 3/4 completely. Since

they had only the notation of unit fraction, they expressed non unit fractions as sum of the unit

fractions. The distinctive aspect of this statement is that unit fractions which were used were selected

differently from each other. So 3/5 is not equal to 1/5+1/5+1/5 because 1/5‘s are same. For example

3/4=1/2+1/4

6/7=1/2+1/3+1/42

Page 8: Egypt Mesopotamia

The 2/n table from the Rhind Mathematical Papyrus

2/3 = 1/2 + 1/6 2/5 = 1/3 + 1/15 2/7 = 1/4 + 1/28

2/9 = 1/6 + 1/18 2/11 = 1/6 + 1/66 2/13 = 1/8 + 1/52 + 1/104

2/15 = 1/10 + 1/30 2/17 = 1/12 + 1/51 + 1/68 2/19 = 1/12 + 1/76 + 1/114

2/21= 1/14 + 1/42 2/23 = 1/12 + 1/276 2/25 = 1/15 + 1/75

2/27 = 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 = 1/30 + 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 + 1/90 2/47 = 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 + 1/138 2/71 = 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 = 1/51 + 1/255

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 + 1/570 2/97 = 1/56 + 1/679 + 1/776

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

ALGEBRAIC PROBLEMS

The Egyptian problems which are the best classified as arithmetic can also be put into class of

algebraic. The form of linear equations are or where a, b and c are

known and x is unknown. The unknown is called as ‗aha‘ or ‗heap‘. The solution given by Ahmes is

not that of modern textbooks, but is characteristic of a procedure known as ‗method of false position‘

or ‗the rule of false‘. The method of false position is by assuming a convenient but incorrect answer

and then adjusting it appropriately. For example: we can look at Problem 15 given as ‗A quantity

(any) plus one-fourth of it becomes 15. What is the quantity?‘ in other words: where x

is unknown.

The solution that is given by Ahmes is like that: Assume the answer is 4. He notes that 4 + 1/4 · 4 = 5.

To find the correct answer, he must multiply 4 by the quotient of 15 by 5, namely 3.

Page 9: Egypt Mesopotamia

The Rhind Papyrus has several similar problems, all solved using false position. The step-by-step

procedure that the scribe followed can be considered as an algorithm for the solution of a linear

equation of this type. Even though there is no discussion of how the algorithm was discovered or why

it works, it is evident that the Egyptian scribes understood the basic idea of a linear relationship

between two quantities-that a multiplicative change in the first quantity implies the same

multiplicative change in the second.

Many of the ‗aha‘ calculations in the Rhind Papyrus are practice exercises. For instance: Problem 79 is

about 7 houses, 49 cats, 343 mice, 2401 heads of wheat, 16807 hekats. The problem evidently called

not for the practical answer. The scribe was introducing the symbolic terminology houses, cats and so

on, for the first power, second power and so on. And this could be useful for finding the number of

measures of grain that were saved.

GEOMETRIC PROBLEMS

The generally accepted account of the origin of geometry is that it came into being in ancient Egypt,

where the yearly inundations of the Nile demanded that the size of landed property be resurveyed for

tax purposes. Indeed, the name ―geometry,‖ a compound of two Greek words meaning ―earth‖ and

―measure,‖ seems to indicate that the subject arose from the necessity of land surveying.

It often said that the ancient Egyptians were familiar with Pythagorean Theorem. But there is no clue

to prove this thought. Nevertheless there are some geometric problems in the Ahmes Papyrus. For

example: Problem 51. It is asked that the area of an isosceles triangle. Ahmes solved it by taking half

of what we call the base and multiplying this by the altitude. He justified his method by suggesting

that isosceles triangle can be thought as two right triangles that one of which can be shifted in position,

so that together the two triangles form a rectangle.

Same thought used in the Problem 52 that asked the area of the isosceles trapezoid. The Egyptian rule

for finding the area of a circle has long been regarded as one of the noticeable achievements of the

time. In Problem 50 of the Rhind Papyrus reads, "Example of a round field of diameter 9. What is the

area? Take away 1/9 of the diameter; the remainder is 8. Multiply 8 times 8; it makes 64. Therefore,

the area is 64. We find the Egyptian rule to be equivalent to giving a value of about .

A hint is given by problem 48 of the same papyrus, in which is shown the figure of an octagon

inscribed in a square of side 9.

Page 10: Egypt Mesopotamia

There is no statement of the problem, however, only a bare computation of and

. If the scribe had inscribed a circle in the same square, he would have seen that its area

was approximately that of the octagon. Since the octagon has area 7/9 that of the square, the scribe

might have simply put A = (7/9) = (63/81)

No theorem or formal proof is known in Egyptian mathematics, but some of the geometric

comparisons made in the Nile Valley, such as those on the perimeters and areas of circles and squares,

are among the first exact statements in history concerning curvilinear figures.

A TRIGONOMETRIC RATIO

For the construction of the pyramids it had been necessary to obtain a uniform slope for the faces. In

spite of the usage of tangent of an angle which means ratio of the ‗rise‘ and ‗run‘ in the modern world,

in Egypt cotangent of an angle was used. Problems 56–60 of the Rhind papyrus clearly inform us on

how the slope of a pyramid might be calculated using the seqt, that is, the horizontal displacement of a

sloping surface for a vertical height of 1 cubit, being the distance from the elbow to the extremity of

the middle finger. In other words, ancient Egyptian surveyors would measure or calculate how much

the sloping surface had ‗moved‘ from the vertical line at the height of 1 cubit. The seqt of the face of a

pyramid was the ratio of run to rise. They basically constructed a right-angled triangle in which the

hypotenuse corresponded to the sloping surface, the height to 1 cubit, and the horizontal top to the

seqt. First it measured in hands. Then it measured in cubits which is a measure of length, being.

In problem 56 asked to find the seqt of a pyramid that is 250 ells or cubits high and has a square base

360 ells on a side. The scribe solved the question as follows:

He gave the seqt as hands per ell.

Page 11: Egypt Mesopotamia

In other pyramids problems, the seqt turns out to be . Problem of the Great Pyramid, 440 ells wide

and 280 high, the seqt was .

There are many stories about presumed geometric relationships among dimensions in the Great

Pyramid, some of which are patently false. For instance, the story that the perimeter of the base was

intended to be equal to the circumference of a circle of which the radius is the height of the pyramid is

not in agreement with the work of Ahmes. The ratio of perimeter to height is indeed very close to ,

which is just twice the value of often used today for π; but we must recall that the Ahmes value for

π is about ,, not . That Ahmes' value was used also by others. For instance; the volume of a

cylinder is found by multiplying the height by the area of the base, the base being determined

according to Ahmes' rule.

MOSCOW PAPYRUS

Much of our information about Egyptian mathematics has been derived from the Rhind or Ahmes

Papyrus which is the most extensive mathematical document from ancient Egypt; but there are other

sources as well. There is an important papyrus known as the Moscow Papyrus. The Moscow papyrus

contains only about 25, mostly practical, examples. The author is unknown.

In this papyrus there are two problems which have special significance. Problems 10 and 14 compute a

surface area and the volume of a frustum respectively. The remaining problems are more common in

nature.

Problem 14: It asks the volume of frustum.

The scribe directs one to square the numbers two and four and to add to the sum of these squares the

product of two and four. The result is twenty eight. Multiply this by one third of six. And then the

scribe concludes with the words, ‗See, it is 56; you have found it correctly.‘ The modern formula is

Page 12: Egypt Mesopotamia

. Nowhere in Egypt is this formula written out. But it evidently was known to the

Egyptians. If b=0 then the formula becomes the formula of a pyramid. But how these results were

arrived at by the Egyptians is not known. An empirical origin for the rule on volume of a pyramid

seems to be a possibility, but not for the volume of the frustum. It has been suggested that the

Egyptians may have proceeded here as they did in the cases of the isosceles triangle and the isosceles

trapezoid—they may thought to break the frustum into parallelepipeds, prisms, and pyramids and

replace the pyramids and prisms by equal rectangular blocks.

Problem 10: It asks the surface area of what looks like a basket with a diameter of . He used the

formula is where x is . The answer is 32. is the Egyptian

approximation of . The answer would correspond to the surface of a hemisphere of diameter .

Later analysis indicates that the "basket" may have been a roof. The calculation in this case calls for

nothing beyond knowledge of the length of a semicircle; and the uncertainty of the text makes it

admissible to offer still more primitive interpretations, including the possibility that the calculation is

only a rough estimate of the area of a roof. In any case, we seem to have here an early estimation of a

curvilinear surface area.

MATHEMATICAL WEAKNESSES

For many years it had been assumed that the Greeks had learned the fundamentals of geometry from

the Egyptians, and Aristotle argued that geometry had arisen in the Nile Valley. That the Greeks did

borrow some elementary mathematics from Egypt is probable, for the use of unit fractions emphasized

in Greece and Rome well into the medieval period. The knowledge indicated in Egyptian papyri is

mostly of a practical nature. The Ahmes and Moscow papyri may have been only manuals intended

for students, but they nevertheless indicate the direction and tendencies in Egyptian mathematical

instruction; further evidence provided by inscriptions on monuments, fragments of other mathematical

papyri, and documents from related scientific fields serves to confirm the general impression. It is true

that our two chief mathematical papyri are from a relatively early period, a thousand years before the

Page 13: Egypt Mesopotamia

rise of Greek mathematics, but Egyptian mathematics seems to have remained remarkably uniform

throughout its long history. The fertile Nile Valley has been described as the world's largest oasis in

the world's largest desert. Geometry may have been a gift of the Nile, but the Egyptians did little with

the gift. For more progressive mathematical achievements one must look to the more turbulent river

valley known as Mesopotamia.

Mesopotamia

The mesopotamian civilization is older than the Egyptian,having developed in the Tigris and

Euphrates valleys beginning sometime in the fifth millennium BC. Many different governments ruled

this region over the centuries. Initially, there were many small city-states, but then their education

system and so bureaucratic system developed. Therefore one of the city states had expanded his rule to

much of Mesopotamia and instituted a legal system to help regulatehis empire. Sumerian and

Babylonian were most important of these dynasties. These were an advanced civilisation building

cities and supporting the people with irrigation systems, a legal system, administration, and even a

postal service. The Sumerians had developed an abstract form of writing based on cuneiform

symbols. Their symbols were written on wet clay tablets which were baked in the hot sun and many

thousands of these tablets have survived to this day. The later Babylonians adopted the same style of

cuneiform writing on clay tablets.

Cuneiform numbers

For small numbers, the Babylonians represented the numbers in a similar way that Egyptians had. A

vertical wedge symbol was used to represent numbers up to 10 and a horizontal wedge represented 10.

s

Page 14: Egypt Mesopotamia

However, the Babylonians stopped at 60. They use the symbol of 1 as the symbol of 60. The

babylonians used sexagesimal system, base-60 system. For writing numbers greater than 60, they just

repeated the symbols in different columns, just as we do, except that where for us a '1' in the 'tens'

column means 10, for the Babylonians a in the 'sixties' column meant 60.

Example:

: 1x(60)+15=75

Example:

: 1x60+40=100

Example:

:

Example:

Write the number 10000 in cuneiform numbers.

One disadvantages of Babylonian‘s number system is that they had no special way to mark an empty

column. This meant that their forms for number 122 and 7202 were same. For might mean

either 2*60+2 or . A gap was often used to indicate that a whole sexagesimal place was

missing, but this rule was not strictly applied and confusion could result. Someone recopying the

tabletmight not notice the empty space, and would put the figures closer together, thereby altering the

value of the number. (Only in a positional system must the existence of an empty space be specified,

so the Egyptians did not encounter this problem.) Later on, oblique wedge symbol used as a

placeholder where a numeral was missing. But since the sign seems to have been used fot intermediate

empty positions only, it did not end all unclearness. This means that the Babylonians never achieved

an absolute positional system. The symbol could represent 2x60+2 or or

or any of indefinitely many other numbers in which two successive positions are

involved.

Page 15: Egypt Mesopotamia

Sexagesimal fractions

The cuneiform number was used not only for 2x60+2, but also or for

or for other fractional forms involving two successive positions.

Example:

Write the number 0.0862 in Babylonian numbers.

For the Babylonians, addition and multiplication of two fractions was no more difficult than was the

addition and multiplication of two whole numbers; and the Mesopotamians were quick to exploit this

important discovery. An old Babylonian tablet from the Yale collection(No.7289) includes the

calculation of the square root of two to three sexagesimal places, the answer being written

. İn modern characters, it is 1;24;51,10.

This Babylonian value for square root of two is differing by about 0.000008 from the true value.

Accuracy in approximations was relatively easy for the Babylonians to achieve with their fractional

notation, the best that any civilization afforded until the time of the Renaissance.

We do not know why the Babylonians decided to have one large unit represent 60 small units and then

adapted this method for their numeration system. This question considered many years ago and has

received different answers over time. One plausible conjecture is that 60 is evenly divisible by many

small integers. Therefore, fractional values of the large unit could easily be expressed as integral

values of the small. Another theory was that the early Babylonians reckoned the year at 360 days, and

a higher base of 360 was chosen first, then lowered to 60. Also the Babylonians divided the day into

24 hours, each hour into 60 minutes, each minute into 60 seconds. Maybe this was the reason, it is

easy wiritng hours, minutes and second in sexagesimal system. But, they are all theories, there is no

certain answer that we know.

Page 16: Egypt Mesopotamia

Fundamental operations

The effectiveness of Babylonian computation did not result from the system of numeration alone.

Mesopotamian mathematicians were skillful in developing procedures. They developed a procedure

for finding square roots that is equivalent to Newton‘s Methods that we know.

Let be the root desired and let be the first approximation to this root; let a second

approximation be found from the equation . If is too small, then is too large, or vice

versa. The next approximation is found by . Since is always too large, the next

approximation will be too small. To obtain better result, one takes the arithmetic mean

Hence, the value of above was found with this method.

This algorithm is equivalent to a two-term approximation to the binomial series, a case with which the

Babylonians were familiar. For finding , the approximation leads to and

, which is in agreement with the first two terms in the expansion of (a2 + b)m and

provides an approximation found in Old Babylonian texts.

Besides approximation of square roots, the Babylonian did many other basic operations such as

addition, multiplication, reciprocals, and exponentiation. Addition was easy, but for multiplication and

reciprocals they needed tables. There are many tables found on cuneiform tablets. Since the place-

value system was based on 60, the multiplication tables were extensive. Any given one listed the

Page 17: Egypt Mesopotamia

multiples of a particular number, say 9, from 1 X 9 to 20 X 9 and then gave 30 X 9, 40 X 9, and 50 X

9 To obtain the product 34 X 9, the scribe simply added the two results 30 X 9 = 4,30 (= 270) and 4 X

9 = 36 to get 5,06 (= 306). For multiplication of two- or three-digit sexagesimal numbers, several such

tables were needed. The exact algorithm the Babylonians used for such multiplications-where the

partial products are written and how the final result is obtained--is not known, but it may well have

been similar to our own. Besides multiplication tables, the Babylonians also used extensive tables of

reciprocals. A table of reciprocals is a list of pairs of numbers whose product is 1 (where the 1 can

represent any power of 60).

2 30

3 20

10 6

16 3,45

25 2,24

40 1,30

48 1, 15

1,04 56,15

1,21 44,26,40

For example, the reciprocal of 48 is the sexagesimal fraction 0;1,15, which represents The

reciprocal tables were used in conjunction with the multiplication tables to do division. Thus the

multiplication table for 1,30 (= 90) served not only to give multiples of that number, but also, since 40

is the reciprocal of 1,30, to do divisions by 40. In other words, the Babylonians considered the

problem to be equivalent to , or, in sexagesimal notation, to 50 X 0; 1 ,30. The

multiplication table for 1,30, part of which appears here, then gives 1,15 (or 1,15,00) as the product.

The appropriate placement of the sexagesimal point gives 1; 15( = 1 1/4) as the correct answer to the

division problem.

1 1,30

2 3

3 4,30

10 15

11 16,30

12 18

30 45

Page 18: Egypt Mesopotamia

40 1

50 1 , 15

One finds among the Old Babylonian tablets some table texts containing successive powers of a given

number, analogous to our modern tables of logarithms. Exponential (or logarithmic) tables have been

found in which the first ten powers are listed for the bases 9 and 16 and 1,40 and 3,45 (all perfect

squares). The chief differences between the ancient tables and our own, apart from matters of language

and notation, are that no single number was systematically used as a base in varied connections and

that the gaps between entries in the ancient tables are far larger than in our tables. Then, too, their

"logarithm tables" were not used for general purposes of calculation, but rather to solve certain very

specific questions.

Algebraic problems

In their cuneiform tablets, also linear equations were found. For example, a problem from tablet YBC

4652 reads: "I found a stone, but did not weigh it; after I added one-seventh and then one-eleventh [of

the total], it weighed 1 mina [= 60 gin]. What was the original weight of the stone?‖ We can translate

this into the modem equation . The scribe just presents the answer, here

. Perhaps solution procedures for such problems are on tablets yet to be discovered.

On the other hand, more detail is given for the solution of pairs of linear equations in two unknowns.

And one of the methods used, making a convenient guess and then adjusting it, shows that the

Babylonians too understood linearity. Here is an example from the Old Babylonian text VAT 8389:

One of two fields yields 2/3 sila per sar, the second yields 1/2 sila per sar (sila and sar are measures for

capacity and area, respectively). The yield of the first field was 500 sila more than that of the second;

the areas of the two fields were together 1800 sar. How large is each field? It is easy enough to

translate the problem into a system of two equations with x and y representing the unknown areas:

A modem solution might be to solve the second equation for x and substitute the result in the first

equation. But the Babylonian scribe here made the initial assumption that x and y were both equal to

900. He then calculated that . The difference between the desired 500 and

the calculated 150 is 350. To adjust the answers the scribe presumably realized that every unit increase

in the value of x and consequent unit decrease in the value of y gave an increase in the "function"

of . He therefore needed only to solve the equation to get the necessary

Page 19: Egypt Mesopotamia

increase . Adding 300 to 900 gave him 1200 for x and subtracting gave him 600 for y, the

correct answers.

QUADRATIC EQUATIONS

In contrast to the Egyptian mathematics, Babylonian mathematics was concerned with 3 term

quadratic equations. There are four types of possible 3 term quadratic equations which are:

1. x2 + px = q

2. x2 = px + q

3. x2 + q = px

4. x2 + px + q = 0

when we restrict q>0, p>0.

However of all those 4 equations, the fourth one does not have any positive roots, therefore

Babylonians were never interested in this, and they were able to solve all the other three. In fact they

were able to recognize that the third equation was equal to the set of equations:

x+y=p and x.y=q

And since they had those values they did the following:

(p/2)=a;

a2-q=[(x-y)/2]

2 ;

(a2-q)

1/2 =(x-y)/2

So now one has both (x-y)/2 and (x+y)/2 so one can find y and x.

CUBIC EQUATIONS

There was no mention of cubics in Egyptian mathematics. It is claimed that this development in

mathematics may be due to the development of algebraic tools in Mesopotamia. Pure cubics were

solved in direct reference to the tablets:

x3=0;7,30 (=1/8 in modern sense) for example was solved directly from the tablets.

Mixed cubics are of form x3+x

2=a. There were tablets for the values of n

3+n

2 for n=1 to n=30. So

when the answer was an integer it could be directly read from tablets. However for the intermediate

values we have tablets which show Babylonians did linear interpolation.

Page 20: Egypt Mesopotamia

A general 3 term cubic equation was also solved by Babylonians. For any positive integer a, b, c;

Babylonians solved equation ax3+bx

2=c by multiplying by a

2/b

3 and hence obtaining:

(ax/b)3+(ax/b)

2=a

2c/b

3

And then solve this as mentioned above for ax/b. Given our modern notation one may assume this may

be rather trivial to see however seeing this without our modern notation is something worth

appreciating.

However there is no record if they were able to solve ax3+bx

2+cx=d. In fact there is not even such a

question.

PLIMPTON 322

Even without Plimpton 322 we must doubt if Babylonian mathematics was only for practical

purposes? For example; the statement ―Area of a square minus length of its one side is equals to 6‖

does not mean anything practically; therefore the early concept of length may be abstract in this

problem. In addition to this we have the Plimpton 322 tablet which has been tried to be understood for

nearly 50 years.

On Plimpton 322 there are four columns and 15 rows. In the first (right most) row there are numbers

from 1 to 15. In the second there is a number c which seems to have been generated from the formula

p2+q

2=c by two positive integers p and q. In the third row there is a number a which is generated by

same p,q using p2-q

2=a. And finally in fourth (left most) row there is b/c, where b=2pq, ie b=(c

2-a

2)

1/2.

Therefore the last row is cos2A in our modern sense where a is the short side of the right triangle with

c as its hypotenuse. Moreover the increments is almost by 1 degrees so it has been claimed by Buck

that these were indeed cos2A and the Babylonians had to same degree concept as we do. However

there is no other evidence to support such a claim. Therefore in 2002 when Robson claimed this was

only another exercise tablet only for applied algebra she won the MAA prize. In her own words it is

"unlikely that the author of Plimpton 322 was either a professional or amateur mathematician. More

likely he seems to have been a teacher and Plimpton 322 a set of exercises." Robson takes an approach

that in modern terms would be characterized as algebraic, though she describes it in concrete

geometric terms and argues that the Babylonians would also have interpreted this approach

geometrically.

She claimed it was an exercise set for the solution of the quadratic equation x-1/x=c using intermediate

steps:

v1 = c/2, v2 = v12, v3 = 1 + v2, and v4 = v3

1/2, from which one can calculate x = v4 + v1 and

1/x = v4 - v1.

Page 21: Egypt Mesopotamia

Robson argues that the columns of Plimpton 322 can be interpreted as the following values, for regular

number values of x and 1/x in numerical order:

v3 in the first column, (left-most)

v1 = (x - 1/x)/2 in the second column, and

v4 = (x + 1/x)/2 in the third column.

Then in the broken section (further to the left) there is 1/x and x.

GEOMETRY

An excavation in Susa shows Babylonians took ratio of the perimeter of the regular hexagon to

circumference of circumscribed circle as 0;57,36 thus we can conclude that Babylonians adopted 3.1/8

as pi, which is indeed as good as the Egyptian approximation. A further strength of Babylonian

geometry is that they seem to understand the concept of similarity. A tablet in Baghdad museum

shows that Babylonian used ratio of side squares of similar triangles is equal to their area ratios.

We know Babylonians did not state ―this is an approximation‖ when they did an approximation. For

example they found area of a 4-sided polygon as (a+c)(b+d)/4 which we know is only a simple

approximation. Another example is the volume of a frustum of bottom side a and top side as b. The

correct volume formula is:

V=h[(a+b)2/4+(a-b)

2/12]

However the Babylonians generally omitted the (a-b)2/12 term.

There is evidence that Babylonians were also familiar with the concepts later called Thales' theorem

and Pythagorean' theorem. The former is stated as any triangle placed on a half circle is a right triangle

and the latter is not explicitly stated but was used to solve problems as follows:

If the top of a ladder, standing vertical to ground, moves 3 units down and if the length of the ladder is

9 units then how much did the bottom end of the ladder was displaced?

MATHEMATICAL WEAKNESSES

Although we have no general concept written explicitly by looking at the overwhelming number of

examples created for school boys we can feel that Babylonians should be aware of some basic

principles of mathematics. One of the main weaknesses of pre-Hellenic mathematics is that they lack

the concept of approximation. Another one is the lack of proof and the question of solvability.

Therefore one may assume pre-Hellenic civilization was not interested in mathematics except for the

Page 22: Egypt Mesopotamia

practical purposes. However as examples such as area – length show there is strong evidence to doubt

if everything was done for the sake of practice?

Finally critics claim that Babylonian mathematics lacked abstraction. However as area – width

example shows they may be using other words instead of our modern x.

REFERENCES:

1) Neugebauer, Otto (1969) [1957], The Exact Sciences in Antiquity (2 ed.), Dover Publications, pp.

36–40, ISBN 978-048622332-2.

2) Neugebauer, O.; Sachs, A. J. (1945), Mathematical Cuneiform Texts, American Oriental Series, 29,

New Haven: American Oriental Society and the American Schools of Oriental Research, pp. 38–

41.

3) Katz, Victor J., ―A History of Mathematics: An Introduction‖,1998

4) Burton, D.,―The History of Mathematics An Introduction 6th edition‖

5) Eves,H. ―An Introduction to the History of Mathematics‖,1990

6) Buck, R. Creighton (1980), "Sherlock Holmes in Babylon", American Mathematical Monthly

(Mathematical Association of America) 87

7) Robson, Eleanor (2001), "Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton

322", Historia Math. 28 (3): 167–206

8) Robson, Eleanor (2002), "Words and pictures: new light on Plimpton 322", American

Mathematical Monthly (Mathematical Association of America) 109 (2): 105–120,

9) Boyer, C.B. & Merzbach U.C. ―A History of Mathematics‖, John Wiley & Sons

10) http://sosmate.blogcu.com/misirli-kesri-ile-matematik-egitimi/887714

11) http://matematikdersanesi.net/yazilar/62/babil-ve-misirda-matematik-ve-geometri/

12) http://www.britishmuseum.org/explore/highlights/highlight_objects/aes/r/rhind_mathematical_pap

yrus.aspx

13) http://www.math.wichita.edu/history/topics/num-sys.html#egypt

14) http://mathworld.wolfram.com/RhindPapyrus.html