VII.Numerals of the land of pyramids

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 18^th 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 11^th 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 19^th 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 )