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