VII.
Numerals of the land of pyramids
[Contd. A Journey to the Wonderland of Math.by Ajay Kumar Chaudhuri]
Another counting system and
numerals of an ancient Civilization about 5000 years from now flourished in a
region of the Dark Continent, as Europeans called so, which is at present the
world’s second largest and second most populous continent Africa.
The first hominids,
whose close relatives are gorilla, chimpanzee, great apes and like evolved in
the jungles of Africa 10 to 5 million years ago. It is established that we, the
humans, are the descendant of the hominids. So that Dark Continent is our
original birth place.
Surprisingly, a
torch of civilization was kindled in a region of the land of mysteries and
little known of its habitants of that very Dark Continent, known as Egypt, the
famous land of pyramids
Geographically
Egypt relates to two regions: North Africa and South East Asia. It has coast
lines on both the Mediterranean Sea and the Red Sea. Egypt borders Libya to the
west, the Gaza Strip and Israel to the east and Sudan to the South. The ancient
Egyptian Civilisation developed in North eastern Africa, concentrated along the
lower reaches of the Nile River. It is one of the six civilisations globally to
arise.
Ancient Egypt is a
land of mysteries on various counts No other civilisation has so attracted the
minds of scholars and common people alike. Mystery surrounds its origin, its
religion, its monumental architectures, colossal temples, pyramids and enormous
Sphinx, an Egyptian mythological figure having the body of a lion and the head
of a man, ram or hawk. The Egyptian pyramids, one of the Seven Wonders of the world,
are the most famous of all the ancient monuments.
Just as life arose
from the primordial soup, a solution rich in organic compounds in the primitive
oceans of the Earth from which life is thought to be originated, so also the
seeds of civilisation were first shown along the banks of the river Nile. This
mighty river, which flows from the heart of Africa to the Mediterranean Sea,
nourished the growth of the pharaonic kingdom. It is worthy to mention that
pharaoh was the most important and powerful person in the kingdom. He was the
head and high priest of every temple. The people of Egypt revered him to be a
half -man, half-god, yet they did not consider him as their king. Pharaoh was
the common title of the Monarchs of ancient Egypt from the First Dynasty 3150
BC until the conquest of Alexander the Great in 305 BC.
The long narrow
flood plain of the Nile was highly attractive for life, attracting people,
animals and plants to its banks. In pre-dynastic periods nomadic hunters
settled in the valley around 6000 BC and began to grow crops for livelihood.
Seen as a gift from the gods, the annual flooding of the river deposited
nutrient rich silt over the land, creating ideal conditions for growing wheat,
flax and other crops. For this reason sometimes Egypt is called ‘The gift of
the Nile.’ The first important public project of this fledgling Society was the
building of irrigation canals for agriculture purposes. The Egyptian believed
in rebirth. The Sun was a principal deity whose passage across the sky
represented the eternal cycle of birth, death and rebirth. The Pharaohs were
seen as divine representatives on Earth who, through rituals, ensured the
continuation of life. After death they became immortal, joining the gods in the
after world.
The Egyptians
believed that both the body and the soul were important for human existence, in
life as well as in death. So their funerary practices, such as mummification
and burial in tombs were designed to assist the deceased find their way in the
after world. They filled the tombs with food, tools, domestic wares, treasures,
in a word, all necessities of life to ensure the soul’s return to the body so
that deceased would live happily even after life.
The most impressive
Egyptian tombs are the famous pyramids, shaped like the sacred mounds where the
gods first appeared, as runs the story of creation. These were incredibly the
largest structure ever built. Egypt’s Great Pyramids at Giza are one of the
world’s most amazing achievements. Built around 2530 BC, the largest pyramid
towers are about 500 ft high and covers 13 acres. They are among the world’s
top tourist attractions and subject of both serious study and wild speculation.
The gigantic
pyramids were the targets for tomb robbers. Their plundering jeopardized the
hope for eternal life. Subsequent generations of kings hid their tomb in the
Valley of the Kings in an attempt to elude the robbers. Despite efforts to hide
the entrances, thieves managed to find the tombs, pillaging and emptying them
of their treasures.
One tomb however
was spared and that was Tutankhamun’s. Tutankhamun was an Egyptian pharaoh of
the 18th dynasty who ruled during 1332 to 1323 BC and died at a
quite young age of 19. Although his resting place was disturbed twice by
robbers, the entrance was resealed and remained hidden for over 3000 years. The
British archeologist Howard Carter and George Herbart discovered almost intact
tomb of Tutankhamun in 1922. It is considered the greatest archeological find
in history. The most important of the artifacts discovered from this tomb are
the pharaoh’s gold coffins and mask. Tutankhamun’s mummy remains in his tomb,
the only pharaoh to be left in the valley of kings.
Today Egyptian
archeologists are still making important discoveries and the scientific study
of royal mummies which is shedding new light on the genealogy of pharaohs. The
ongoing deciphering of hieroglyphic writings and research on the life of
peasants are also answering many questions related to the evolution of Egyptian
culture. The pharaonic religion gives the impression that Egyptians were
preoccupied with death; however there are ample indications that they were a
happy lot who knew how to enjoy life.
The early Egyptians
settled along the fertile Nile Valley as early as about 6000 BC, which has
already been mentioned. They began to record the pattern of lunar phases and
the seasons both for agricultural and religious purposes. They used
measurements using body parts such as fingers, palm, length from elbow to
fingertips etc. to measure land, buildings, as found in early Egyptian history.
They developed a decimal numeric system based on ten fingers.
We learnt the
Egyptian language of numbers from the writings on stones of ancient monument
walls. Numbers have also been found on pottery, lime stone plaques and fragile
fibers of the papyrus. The language was composed of hieroglyphs, pictorial
signs that represent people, animals, plants and numbers.
They used a single
vertical line to mean one, two such lines to represent the number two and
continued up to nine lines to refer nine. By now, there were a lot of lines. So
they introduced a new symbol for ten. Then they carried on adding lines for
units and ten symbols for ten until they reached hundred, which needs a new
symbol and so on.
In fact, the Egyptians
used written numeration that was changed into hieroglyphic writing, which enable them to
express whole numbers up to 1000,000. It had a decimal base and allowed for the
additive principle. In this notation there was a special sign for every power
of ten. It is already mentioned how they wrote numbers from one to nine by
using vertical lines. Reaching ten, they introduced a sign with the shape of an
upside down U. For 100 a spiral rope, for 1000, a lotus blossom; for 10,000, a
raised finger, slightly bent; for 100,000, a tadpole; and for 1000,000, a
kneeling genie with raised arms. Thus the symbols are as in Table No.- 3.
Table No.3
This hieroglyphic
numeration was a written version of a concrete counting system using material
objects. To represent a number, the sign for each decimal order was represented
as many times as necessary. To make it convenient to read the repeated signs,
they were placed in groups of two, three or four and arranged vertically.
The Egyptians had no idea of nothingness and so they had no
symbol for zero. But they had idea of a very very large number which is bigger
than any number that’s ever been conceived. In our modern concept we call such
number as “infinity”. The Egyptians used a circle like symbol ( ♎ )
for such a number, which might imply that you round forever without finding an
end.
Egyptians loved all
big things, such as big buildings, big statues and big armies. To build
pyramids they would have needed a good number system. They would have needed to
work out how much stones were required and when; otherwise, the workmen
employed to build the pyramid would have been sitting idle for sometimes. Also
to fed these workers, it was necessary to calculate the amount of food to be
stored. All these required big numbers. They had mathematical skill to perform
addition, subtraction and even multiplication with these numbers. They had also
idea of fractions and symbol for it.
Now let us see how
the Egyptians wrote bigger numbers and perform mathematical operations by
applying the additive principle as is shown in Table No.4.
Table No.4
The technique of writing numbers
was the largest decimal order would be written first. The numbers were written
from right to left. For example, the number 45306 in our decimal system would
be written in hieroglyphic notation
as
which means as per our decimal representation 40000 + 5000 +
300 + 6.
So the largest decimal order
(40000) was written first, then the other descending decimal orders from right
to left.
Then if we like to add our numbers 546 and 465 in the
Egyptian way, it will as follows:
So obeying the rule of carrying,
the sum will be as simple as : I⋂△⊸
that is, the sum will be 1011 or 1000 + 10 + 1 as shown in
Egyptian form.
The operation
of subtractions may be done, as we do, except that when one has to borrow, it
is done with writing ten symbols instead of a single one.
If we subtract 437 from 645 in Egyptian system, it will be :
that is, 645 – 437
= 208.
The multiplication
of Egyptian numbers was not so easy as the operation of addition, yet they used
a cunning method for multiplication. All they had to do was to divide one of
the numbers repeatedly by 2 and simultaneously to multiply the other by 2 in
each step. Let us take a concrete example of multiplication of 46 by 75. It
will be convenient to take the smaller one, here 46, as the first number and
the bigger 75, as the second. So 46 is divided by 2 several times until to
reach one. Of course, sometimes it cannot be done, if the number is odd. In
that case 1 is subtracted before halving it. The other number, 75 is multiplied
by 2 the same number times.
Now the next step to be
followed is, every line where 1 has been spared, that is subtracted, to note
what the doubled number has come to, and these numbers to be added together,
ignoring the others. This will be the required result of the multiplication. To
believe or not, is the same as the two numbers multiplied together, without
using multiplication tables at all. The actual operation of 46 x 75 is shown
below.
Halve first
number
|
Odd number
|
Double
second number
|
Doubled on
odd halved numbers
|
46
|
75
|
||
23
|
Subtract 1
|
150
|
150
|
11
|
Subtract 1
|
300
|
300
|
5
|
Subtract 1
|
600
|
600
|
2
|
1200
|
||
1
|
Subtract 1
|
2400
|
2400
|
Total of relevant doubled
numbers
|
3450
|
||
Surprisingly, this method effectively made use of the concept of binary
numbers, over 3000 years before Leibnitz, (1646–1716 AD) a German
mathematician, philosopher who developed the most important branch of
mathematics, Calculus, independently of Isaac Newton (1642 – 1726 AD) in
England, introduced into the west and many more years before the development of
the computer was to fully explore its potential.
The process used to
divide numbers by the Scribes of the Ancient Egyptian culture was very similar
to their method of multiplication. For division by ancient Egyptian method just
follow the following steps :
Step
1: construct a column of side-by-side
two numbers. The left column will contain powers of 2, eventually give the
quotient. The right column will contain multiplies of the divisor. Additional
columns of checks or marks and indentified “doubles’ may be placed as desired.
Step
2: Begin at the top of the left column
and fill the column in order with powers of two until you reach the dividend.
Step
3: On the top of the right column place
the divisor.
Step
4: Create the remaining entries in the
right column by doubling the number in the row above it. Stop when the “double”
is as large as the dividend.
Step
5: Find and check or mark “doubles” in the right column so that the sum is
dividend.
Evidently it is a tough job and sometimes
causes to spin our head.
Step 6: The
sum of the indicated left-column powers of 2 is the quotient.
So, if we like to divide 9960 by 415, let us proceed as follows
1
|
415
|
|
2
|
830
|
|
4
|
1660
|
|
8
|
3320
|
3320
|
16
|
6640
|
6640
|
32
|
12560
|
|
8+16
|
9960
|
From the above table it is clear
that the divisor is multiplied successively by 2 until we a reach a number
greater than the dividend, 9960.
Now look at the
rows of the right hand column and mark those numbers in it whose sum will be
equal to the dividend. If it is so, the number will be exactly divisible. In
that case, the sum of the corresponding number in powers of 2 on the left
column will be the quotient.
In our example the
sum of the products of 8 and 16 with 415 equals the dividend 9960. So, the
number is exactly divisible by 415 and the required quotient will be 8 + 16 or
24.
But if the number
is not exactly divisible, then with much effort we shall fruitlessly try to
find numbers on the right hand column whose sum will be equal to the dividend.
For example, if the dividend be
10072 and the divisor is 415, then proceeding as before we find that 10072 lies
between 9960 and the number 12560 in the last row of the right hand column. In
this case we are unable to find numbers on the right hand column whose sum will
be exactly 10072. So the inference will be: the number in question is not
exactly divisible by 415. Knowing the rule of division that
divisor x quotient + remainder
= dividend.
We can find the quotient and the
remainder.
In our case, the divisor is 415,
quotient 8+16 or 24.
So, the remainder = 10072
– 415x24
= 10072 – 9960
= 112
The ancient Egyptian method of
division may appear cumbrous, yet their method had advantages over our present
method; for we use the base 10 and they used the base 2, which has many
advantages over ours.
Pic.No3. Eye of Horus.
Necessity is heading the human race
through the zigzag and uneven void of inventions right from the age of hunting
and dwelling in caves. In course of evolution, we tried to overcome hurdles and
solve problem far back from that primitive age. The idea of counting, vis-à-vis
notion of mathematics was a great step forward in this direction.
The idea of fractions also grew in
the minds of the ancient Egyptians from practical needs. Suppose Sabina had
employed 8 workers in her field to tilt and clear an irrigation channel and she
had 5 barley loaves to be distributed amongst the workers. The problem was
grave for her, for she had to divide 5 by 8. This problem would pose no
‘hurdle, if it was milk, bear or a sack of grain. Sabina took a cunning method
to distribute the loaves amongst the workers. She thought: What if there were 4
loaves not 5 to be split amongst 8 people? Sabina saw that they all get at
least half a loaf and remaining loaf to be split in 8 parts to distribute
equally.
It is amusing to note that the Egyptian fractions are not only a very practical solution to everyday problems but are also interesting in their own right. They had practical uses in ancient Egyptian method of multiplying and dividing, and every fraction proper or vulgar, can always be written as an Egyptian fraction. There are also many unsolved problems concerning them, which are still a puzzle to mathematicians today.
Along with the
Babylonians and Indians, the Egyptians are largely responsible for the shape of
mathematics as we know now. Their knowledge and mathematical skills were passed
on the Greek, helping them to develop their great store of mathematical
knowledge. The Egyptian mathematicians were so skilled that great Greek
mathematicians such as Thales and Pythagoras learned techniques in Egypt.
Sadly, what we know
about the Egyptian mathematics is scanty and incomplete. Most of the Egyptian
records were stored on papyruses which were fragile and prone to denigration
over years. Also many ancient valuable Egyptian mathematical texts along with
many other important documents were burnt during the fires at the Library of
Alexandria. I think it will be worthy to tell something about this Library of
Alexandria. The history of Alexandria dates back to the city’s founding by
Alexander the great in 331 BC after conquering the Egyptian Empire. The Library
of Alexandria was one of the largest and most significant libraries of the
ancient world. Famous for having been burnt, thus resulting in the loss of many
valuable scrolls and books, It has become a symbol of destruction of knowledge
and culture. The library may have suffered several times fires or acts of
destruction, of varying degrees, over many years. Perhaps the manuscripts were
burnt in stages over eight centuries by different invaders and conquerors of
the Egyptian land.
So we have only a few
spared manuscripts to reveal the skill of the Egyptian mathematicians,
alongside a few hieroglyphic records and Greek sources.
The primary sources of
information about ancient Egyptian mathematics are some inscriptions on papyri
(plural form of papyrus) of those periods. The English word “paper” is derived
from the Egyptian word “papyrus”. But, what is a papyrus? Historically when the
Egyptians developed written language, they felt the need for a medium other
than store to transcribe upon. They found this in their papyrus plant, a
triangular reed. It was light, strong, thin, durable and easy to carry. They
found nothing better for writing for thousands of years, It is believed that
they have been using this as early as 4000 BC and continued until 11th
century AD.
There are two
primary sources and a number of secondary sources on ancient Egyptian
mathematics. The primary sources are the Rhind (or Ahmes) papyrus and the
Moscow papyrus. They contain 112 problems with solutions. Among the Secondary
sources there are three papyri of about 1800 BC. They are Egyptian Mathematical
Leather Roll which contained a table of 26 decompositions of unit fractions,
the Berlin papyrus having two problems of simultaneous equations – one of the
second degree and the Reisher papyrus containing volume calculation.
The Rhind papyrus
(also called Ahmes papyrus) is named after the British Collector, Rhind who
acquired it in 1858. It was copied by a Scribe Ahmes (or Ahmos) around 1650 BC
from another document, written around 2000 BC, which possibly in turn, was
copied from a document of around 2650 BC. The Rhind papyrus is kept in the
British museum and contains mathematical problems with solutions.
The Moscow papyrus
was copied by an unknown Scribe, around 1850 BC, It was brought to Russia
during the middle of the 19th century. It is preserved in the museum
of Fine Arts in Moscow. It contains mathematical problems of simple equations
and their solutions.
The Egyptian
Mathematical Leather Roll (around 1850 BC) is a table consisting of 26
decompositions into unit fractions.
The Berlin papyrus
(about 1800 BC) contains among other things, two problems in simultaneous
equations, one of which is of second degree.
The Reisner Papyri
(about 1800 BC) consists of four fragments of rolls containing calculations of
volumes of temples.
The Kahun papyrus
(around 1800 BC) contains six mathematical fragments. A considerable portion of
Kahun papyrus is still not translated.
Our first knowledge
of mankind’s use of mathematics beyond mere counting comes from the Egyptians
and Babylonians. Both civilisation developed mathematics that was similar in
some ways but different in others. The mathematics of Egypt, at least what is
known from the papyri, can essentially be called applied arithmetic. It was
practical information communicated via example on how to solve specific
problems.
[To continue]
Reference: Internet.
Image credit: Table of ancient Egyptian numerals:Mark Millmore (https://discoveringegypt.com/ )
Eye of Horus:Benoit Stella ( https://commons.wikimedia.org/wiki/File:Oudjat.SVG )
[To continue]
Reference: Internet.
Image credit: Table of ancient Egyptian numerals:Mark Millmore (https://discoveringegypt.com/ )
Eye of Horus:Benoit Stella ( https://commons.wikimedia.org/wiki/File:Oudjat.SVG )
