Overview: The mathematics of ancient Egypt was fundamentally different from our math today. Contrary to what people might think, it wasn't a primitive forerunner of modern mathematics. In fact, it can’t be understood using our current computational methods. Count Like an Egyptian provides a fun, hands-on introduction to the intuitive and often-surprising art of ancient Egyptian math. David Reimer guides you step-by-step through addition, subtraction, multiplication, and more. He even shows you how fractions and decimals may have been calculated—they technically didn’t exist in the land of the pharaohs. You’ll be counting like an Egyptian in no time, and along the way you’ll learn firsthand how mathematics is an expression of the culture that uses it, and why there’s more to math than rote memorization and bewildering abstraction.
Reimer takes you on a lively and entertaining tour of the ancient Egyptian world, providing rich historical details and amusing anecdotes as he presents a host of mathematical problems drawn from different eras of the Egyptian past. Each of these problems is like a tantalizing puzzle, often with a beautiful and elegant solution. As you solve them, you’ll be immersed in many facets of Egyptian life, from hieroglyphs and pyramid building to agriculture, religion, and even bread baking and beer brewing. Fully illustrated in color throughout, Count Like an Egyptian also teaches you some Babylonian computation—the precursor to our modern system—and compares ancient Egyptian mathematics to today’s math, letting you decide for yourself which is better.
Count Like an Egyptian A Hands-on Introduction to Ancient Mathematics by David Reimer Book |
Count Like an Egyptian A Hands-on Introduction to Ancient Mathematics by David Reimer Book Read Online Chapter One
THE WORDS OF THE GODS
Hieroglyphic Numbers
In the primal waters at the dawn of time, the Egyptian god Ptah brought himself into being. This bearded god had skin blue as the night sky, and he carried a scepter whose form combined the Egyptian symbols of stability, dominion, and life. In his heart, Ptah conceived of the world, and his tongue turned his thoughts into words. At the sound of his voice, the universe changed. The amorphous eight gods of the Ogdoad, including the primeval waters, darkness, chaos, and the invisible power, came together. There they formed the primeval mound, the first piece of the earth. The act drained the power of the Ogdoad and the mound became their tomb, but their sacrifice created the birthplace of the sun, the father of the Egyptian pantheon.
This mound was the center of the earth, which the Egyptians believed resided right in the middle of their nation. The Egyptians called the central part of the world the Mansion of the Life Force of Ptah, which the ancient Greeks translated as Aigyptos, the origin of our word Ã¢€Å“Egypt.Ã¢€
The magic of PtahÃ¢€™s words created the world, and words in ancient Egypt had real power. This was especially true for hieroglyphics, which the Egyptians called the words of the gods. These artistic writings, along with other magical diagrams, cover the walls of their tombs and temples. But hieroglyphs are more than mere writing. When Egyptians wanted just to write, they used the hieratic script, a simplified form of hieroglyphs. They used the hieroglyphs only when their words needed a small portion of the same power that Ptah had used to create the world. They used the magic of words to protect themselves from the evil that was in the spirit world.
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Ptah, the god of creation.
Such spells usually took the form of either monologues or stories. In monologues, Egyptians spoke directly to the gods, and they would plead with a god for his or her assistance. However, the words contained so much power that these spells contained threats directed at the gods. The magic in the monologuesÃ¢€™ words was apparently strong enough to prevent divine retribution for their harsh words. Similarly, words infused stories, the second form of a magical spell, with divine power. Hence telling a tale of a god healing another god had the power to heal.
The diagrams that accompanied the hieroglyphs were also magic. Spells granted them the ability to come to life to serve the dead or protect the living. One such spell, the opening of the mouth, allowed the spirits both of the dead and the divine to enter or leave a mummy, statue, or drawing. PtahÃ¢€™s name literally translates into the words Ã¢€Å“the opener,Ã¢€ interpreted precisely in this sense. Ptah, in fact, was the patron god of the craftsmen who built and decorated the tombs and temples of ancient Egypt.
These craftsmen had to create the images according to precise specifications because of their mystical nature. Important objects needed to have more magic and hence needed to be drawn bigger. They also had to be drawn with attention to mathematical proportion so they wouldnÃ¢€™t come to life misshapen and malformed. These Ã¢€Å“magical blueprintsÃ¢€ required that all the parts were carefully detailed. Hence the figures took on odd poses to clearly depict each essential body part. Many of the poses also possessed symbolic value and in turn conveyed different occult powers. Egyptians were quite capable of accurately drawing figures in natural postures, but these images were not art but, rather, detailed specifications for their afterlife.
Words had so much power that they were often dangerous, even to their users. The bad parts of a magical story could harm someone as easily as the good parts could help. So when a tale included an evil event such as a murder, it often skipped these parts or made a vague reference to such events. Even the symbols used to make up words presented some danger. Imagine the frustration youÃ¢€™d feel if your soul woke up shortly after your funeral only to be chased around your tomb by the spirit of a venomous snake. This would have happened because some craftsman didnÃ¢€™t take the proper precautions when writing a word containing the j sound, whose symbol takes the form of a cobra. A better-trained craftsman would have drawn the snake sliced up or impaled for the safety of the deceased.
There is no mathematics written in hieroglyphs, but numbers are used for the occasional date or quantity. They use a straight vertical line, A, to represent the number one. This is no surprise since virtually every culture uses a similar symbol to represent 1 just as we do. This practice is tens of thousands of years old and far predates writing, which is a mere five thousand years old. It seems to have been started by hunter-gatherers who used notched bones or sticks to record quantities. While itÃ¢€™s easy to cut a straight line with a knife across a piece of wood, a curved shape, like our 2, would be needlessly difficult. So, when a denizen of the ice age needed to remember the number 5, he or she would make five straight cuts into a stick. The Egyptians carried on this practice in their writing. Hence, the Egyptian 3 appears as AAA, just like three notches on a bone.
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The hieroglyphic symbol for the j sound can become a venomous snake in the afterlife.
Unlike their contemporaries, such as the Mesopotamians, the Egyptians didnÃ¢€™t group their 1s in specific patterns. For example, 4 could be written in one line as AAAA, or in two rows of two as AA. This is consistent with their other hieroglyphs, since the layout was concerned more with the aesthetic look of the word than with a systematic layout. For example, the word Ã¢€Å“dayÃ¢€ could be written Hru. These three symbols represented a hut, a mouth, and a quail chick and made the sounds of h, r, and w respectively. Because the first two symbols were short compared to the picture of the quail, it was often written as below, filling up the space on the temple wall more uniformly.
The numbers 1 and 3 had special use in Egyptian hieroglyphs. As weÃ¢€™ve seen, the symbol u can represent the sound made by the letter w, but it could also represent an actual quail. In order to help the reader distinguish between the two, the Egyptians wrote a symbol identical to the number 1 below the drawing when they wished to identify the object and not the sound. Similarly they could pluralize the object by writing the number 3 below it. For example, the following depicts both the singular and plural of fish.
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The Egyptian word Ã¢€Å“day.Ã¢€
The system of writing numbers as a bunch of 1s has a serious flaw. Look at the number AAAAAAAAAAAAAAAAAAAAA. ItÃ¢€™s far from obvious that this is the number 21. Too many lines blur together making them difficult to count. The Egyptians, like most ancient cultures, used symbols to represent groups larger than 1. For example, they used S, a picture of a cattle hobble, to represent the number 10. Using the S and the A symbols, they represented numbers up to 99. For example, the number 21 could be written SSA.
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A fish and many fish.
For larger numbers they used the symbols D, F, G, and H to represent 100; 1,000; 10,000; and 100,000 respectively. The pictures represent a coil of rope, a lotus flower, a finger, and a tadpole. So we can write the number 251,342 as follows:
H H G G G G G F D D D S S S S A A
The pictures used for the numbers may give clues to how the words were pronounced. Words in ancient Egypt were usually spelled without vowels. If we used a similar system, we could use the symbol, Image, to represent the word Ã¢€Å“bell.Ã¢€ However, it could also be used to write Ã¢€Å“ballÃ¢€ and Ã¢€Å“bull.Ã¢€ The symbol for cattle hobble, S, is composed of three consonants: m, j and w. So the Egyptian word for 10 could sound something like Ã¢€Å“mojaw,Ã¢€ Ã¢€Å“mijow,Ã¢€ and so on. We need to remember that Egyptian is a dead language and no one is really sure how any of the symbols sounded. Egyptologists have made intelligent guesses based on their knowledge of the ancient language Coptic, which evolved from Egyptian. But this problem is compounded when we realize that even if we knew all the written sounds, we still donÃ¢€™t know what vowels go in between. So when we see the coiled rope symbol, D, we believe the consonants are s, h, and t and can only guess the vowel. While most of us can think of a few interesting words using these letters, each adjacent pair has an unknown vowel sound between them suggesting 100 is pronounced something like Ã¢€Å“sehetÃ¢€ or Ã¢€Å“sahot.Ã¢€
The number for a million is depicted by a man holding his hands in the air, J. Egyptians used this word to represent extremely large numbers in exactly the same way we do when we say weÃ¢€™ve done something Ã¢€Å“a million times.Ã¢€ ItÃ¢€™s difficult to say what the pose means. Some have suggested itÃ¢€™s the arms of man outstretched, overwhelmed when confronted with the concept of infinity. ItÃ¢€™s also reminiscent of the pose the air god Shu takes while holding up the sky, restraining her from embracing her lover, the earth, and crushing all things between heaven and earth. The symbol, J, also stands for each of the Heh gods. These are the spirits of the Ogdoad, who died to form the earth and coincidentally help Shu hold up the sky.
The number 1 million was used repeatedly in the Egyptian mythos. Perhaps the most important example is the barque of millions. A barque was a boat a god used to sail across the sky, which according to the Egyptians, was made of water. The barque of millions was the sun god Ra, which was navigated by the god Thoth and his wife MaÃ¢€™at across the sky each day. The Ã¢€Å“millionsÃ¢€ refers to all the souls who had achieved salvation and manned the barque as it descended into the nether world each night. Together with Seth, the god of strength and violence, they defended the boat on its journey to the dawn, when the sun god would be reborn anew in order to shine another day.
THOTH, SCRIBES, AND BUREAUCRACY
Hieratic Numbers and Addition
On the day an ancient Egyptian was born, Thoth, the god of scribes and wisdom, would change into his ibis form. He needed this form so he could fly from his barque, the moon, down to the earth, where he would carry out the will of the gods. Unseen by human eyes, Thoth would find one of the bricks of the house in which the baby was born and write down the day the child was fated to die. When that day would finally arrive, the soul of the mortal would once again encounter Thoth in the Hall of Osiris. Here, in the land of the setting sun, which formed the barrier between heaven and hell, the soul would be given final judgment. Regardless of the outcome, Thoth would dutifully record the result.
In order to fully appreciate the importance of Thoth and the scribal class in Egypt, we need to understand the central role of bureaucracy in Egypt. Contemporary movies about the ancient world seem to invariably include scenes of a vast marketplace where the characters are offered a wide array of treats and forbidden goods. This imagery is based on a modern misconception. The economies of ancient civilizations were, by and large, controlled by a central government. The state provided everything its ruling class thought the citizens needed, and the former took what they considered to be their share. When not working for the government, individuals would occasionally exchange goods and services with each other for a few items the government wouldnÃ¢€™t provide. Relatively speaking, it was a small part of the economy, which otherwise was dominated by EgyptÃ¢€™s ruling class.
The pharaoh, governors, and high priests who controlled the government needed an army of bureaucrats to manage the economy. The scribes of Egypt performed this function, documenting every aspect of ancient life. Just like Thoth, they were there from a personÃ¢€™s birth to their death, recording all. Scribes were on the farms, in the storehouses, and on the factory floors. They were even on the battlefields, recording the details of the fight and tabulating the casualties by counting thousands of hands severed from the dead.
The constant need to keep records on every aspect of Egyptian life was a huge drain on the time of the scribes. Hieroglyphics consist of detailed pictures that take a long time to properly write. Apparently, the ancient scribes didnÃ¢€™t have the time or patience to make their records with hieroglyphics, so they invented hieratic. This is essentially a cursive form of hieroglyphics, but it is different enough to be considered its own script.
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The god of wisdom and scribes, Thoth.
Some of the hieratic symbols are recognizable from their hieroglyphic roots. For example, the hieratic number 2 evolved from the two straight lines of the hieroglyphics. When an ancient scribe painted the first line, he wouldnÃ¢€™t lift his brush all the way before painting the second line, making the motion a little faster. This would have the effect of connecting the two vertical lines near the bottom.
Our number 2 was created in much the same way long ago in India. The only significant difference is that they started with two horizontal lines. The curve of the 2 is nothing more than the backstroke to begin the second line. The number 3 evolved in much the same way.
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The transition of the hieroglyph 2 to the hieratic 2.
As the figures grew more complicated, the Egyptian scribes reduced parts of the hieroglyphic symbols to simple strokes. Consider the hieroglyphic 7. Normally it could be written in four vertical strokes on top of three more. The impatient scribe would paint all four as one horizontal brush stroke and zigzag back for the next row. He could not make a stroke for the second row because it would be unclear how many ones it contained. So he would jiggle his brush representing two and follow it with a slash down representing the third one. While the figure hardly looks like the original seven lines, all that really matters is that the scribes recognized this symbol and were able to easily distinguish it from similar figures.
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The evolution of the Hindu 2 into the modern 2.
The ancient scribes of Egypt, like any other accountants, eventually needed to find the total of the values of the objects they inventoried. This is perhaps the most difficult subject for me to write about. I can easily explain how they calculated the volume of an unfinished pyramid, multiplied mixed numbers, and created fractional identities, but I canÃ¢€™t be sure how they added 15 and 12. Both of the ancient math scrolls that have been found regard addition as being too simple to detail. Hence the work of the solutions is not shown. Only the answers are given. With no written record, we can do little but guess.
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The transition of the hieroglyph 7 to the hieratic 7.
On the surface, the problem doesnÃ¢€™t seem that bad. Consider the following addition of 82 and 54. The Egyptians would have written the numbers in hieratic, but IÃ¢€™m using hieroglyphics just because theyÃ¢€™re more recognizable.
The above sum is easily added in oneÃ¢€™s head. We can first combine the ones, A, adding 2 and 4 to get 6. Then we can compute the tens, S, adding 8 and 5 to get 13. We can interpret this as ten 10s and three 10s. Since ten 10s is 100, or D, the answer is D SSS AAAAAA.
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We canÃ¢€™t automatically assume that the ancient Egyptians used this method. Although there are hints in ancient texts, there is no direct evidence as to how they added. We also donÃ¢€™t see their work when they added lists of fractions often having different denominators. I canÃ¢€™t imagine doing such problems in my head. If we know they didnÃ¢€™t do all additions in their head, how can we be sure they did any in their head?
WeÃ¢€™ve fallen into the trap of mathematical familiarity. Subconsciously we think to ourselves, Ã¢€Å“They must do their math the way we do, because our way is the right way.Ã¢€ Of course itÃ¢€™s foolish to think that anything we do is the so-called right way. ItÃ¢€™s often simply one choice of many.
In Mesopotamia at that time, there is some evidence that the people used tokens to perform calculations. In order to explain this in modern terms imagine that you have 82 cents in one hand and 54 in the other. In the first hand you have 8 dimes and 2 pennies. Note the relationship between the 8 dimes and the eight S in the scribeÃ¢€™s number. Similarly in the second hand you have 5 dimes and 4 pennies. All you need to do to add these numbers is to pour the coins from one hand into the other. So you now have 13 dimes and 6 pennies in one hand. You now decide you have too many coins, so you replace 10 dimes with a one-dollar bill. The addition is now complete. You now write D representing the dollar, SSS representing the three dimes, and AAAAAA the six pennies. Note that we never actually added any digits. We simply pushed the piles together and made a currency exchange.
IÃ¢€™m not advocating that the Egyptians used tokens. You could rightly argue that there is no physical evidence of tokens being used in ancient Egypt. I could counter by saying that there is no need for physical tokens. They might have done their mathematics on a dust board. This is roughly the equivalent of doing math on a dirty car window. They could have Ã¢€Å“placed tokensÃ¢€ by making marks with their fingers and Ã¢€Å“picked up tokensÃ¢€ by smearing dirt over the marks they wished to erase. IÃ¢€™m simply pointing out that there are other ways, beside the modern methods, to solve problems. In the absence of evidence, speculation is fine, but we have to understand it for what it is. Ignorance has never stopped me in the past, so letÃ¢€™s add some Egyptian numbers.
EXAMPLE: Add FSSAAAAAA and FFSSSSSSSAAAAA.
LetÃ¢€™s proceed as we would when adding modern numbers. First we add the 1s. This is more difficult because of the way I wrote the numbers. Note that the lines that denote the 1s blur together and are difficult to count. Ancient people recognized this and generally never wrote more than four identical number symbols in a single row. My modern word processor just doesnÃ¢€™t seem to appreciate this difficulty, refusing to do anything but go left to right. If we count them carefully, we will get six 1s in the first number and five in the other. When added these make 10 and 1, or equivalently, SA. We treat the S as we would a carry.
We now add the two 10s to the seven 10s along with the carry to get ten 10s. This presents us with two minor problems. First of all, the entire 10, S, gets Ã¢€Å“carried.Ã¢€ So we donÃ¢€™t write down any S. The second problem is a little more subtle. WeÃ¢€™re used to seeing the hundreds digit next to the tens digit. When we see 324, we assume the 3 represents the hundreds because itÃ¢€™s before the 2, which is the tens digit. However in the Egyptian number FSSAAAAAA, the F is not a hundreds digit but actually represents 1000. Today we would avoid this confusion by writing 1026, putting a 0 between the tens and thousand. However, when we write zero D, we simply donÃ¢€™t write anything. So if we werenÃ¢€™t paying attention, we might be tempted to add an extra F as our carry because itÃ¢€™s the next digit in the number. However ten 10s is D, not F. Finishing our addition, we add the one and two lotus flowers to get FFF, giving us a final answer of FFFDA. HereÃ¢€™s the work of the above problem without all the discussion. Note that IÃ¢€™ve left room for the missing hundreds and wrote the carries above in the modern way.
SOLUTION:
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Now try this one on your own.
PRACTICE: Add GGG DDDDDDA + GGGGGGGFFDDDDDDDAA.
ANSWER: HFFFDDDAAA
Although we canÃ¢€™t be sure that the Egyptians viewed addition in this way, it seems fairly simple. ItÃ¢€™s not hard to understand why they didnÃ¢€™t devote any precious book space to this elementary subject. Multiplication, on the other hand, is different.
BREAD, BEER, AND PESU
Whole-Number Multiplication
Ask an ancient Egyptian, Ã¢€Å“WhatÃ¢€™s for dinner?Ã¢€ and theyÃ¢€™d probably reply, Ã¢€Å“Bread and beer.Ã¢€ If they were feeling like going wild, they would add some honey or butter. They did eat meat, particularly beef, but this was generally reserved for special occasions, at least for the labor class. The average Egyptian could scrounge up some other food items. They might have a duck to lay the occasional egg or a goat for some milk. They might do some moonlighting on the weekend to earn a few specialty items like fish or figs, but since they were paid in grain, bread and beer were served daily.
You might think that the Egyptians were raging alcoholics, drinking beer with every meal, but their beer was watered down. The purpose of the alcohol was to kill the bacteria in the river water, not to get drunk. The grain to brew your own beer came from the same grain used to bake your own bread. So if you drank too much, you might go hungry. This was no small concern in the ancient world, where having a little belly fat was considered a sign of wealth.
The Egyptians carefully measured the amount of grain that went into a loaf of bread or jug of beer. They measured grain in hekat, a hekat being about a gallon. They measured bread in pesu, the number of loaves that could be made out of 1 hekat of grain. This is different from most modern systems in that the larger the pesu, the smaller the loaf of bread, since the more loaves you made out of 1 hekat, the smaller they would have to be. Perhaps itÃ¢€™s easiest to think of pesu as a fraction. A 10-pesu loaf is a tenth of a hekat in grain and so on. There are some modern equivalents to this system. For example a quart is a quarter of a gallon. This makes for some unusual math since we usually associate larger numbers with greater quantities. With pesu, the reverse is true. For example, two 10-pesu loaves is not 20 pesu but is equivalent to a 5-pesu loaf.
We might ask, why would the Egyptians use this system? Once again we have to set aside modern methods and ask ourselves whether is there a good reason for the Egyptians to have chosen methods different from our own. While it is more difficult to add loaves of bread in the pesu system, perhaps there are problems that are easier in this system. If I want 2 hekat of 25-pesu bread, I need 50 loaves. This is obvious in the pesu system since each hekat has 25 loaves, so 2 hekat have 2 Ãƒ— 25 = 50 loaves. In our system, this problem would be trickier. We would rate a 25-pesu loaf as having 0.04 hekat of grain. To figure out how many loaves make 2 hekat we would have to calculate 2 Ãƒ· 0.04, to obtain 50. Which is easier, multiplying by a whole number or dividing by a decimal? The answer is obvious. This problem naturally leads us to the Egyptian method of multiplication. Consider the following problem:
EXAMPLE: How many loaves of 10-pesu bread can be made out of 7 hekat?
We know the answer is 10 Ãƒ— 7, or 70. The Egyptians had a unique method of solving such a multiplication problem. TheyÃ¢€™d begin by writing the numbers 1 and 7 in the first row of their solution. IÃ¢€™m going to also write 10 above the 1 for reasons youÃ¢€™ll soon understand. They then proceeded to double the row repeatedly forming new rows as follows:
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We stop doubling when the first column reaches 8 because if we double it again, it will reach 16 which is bigger than 10.
Our goal now is to find numbers in the first column that add up to 10. Although itÃ¢€™s easy to find them for such a simple problem, letÃ¢€™s do it in a systematic way that will work for more complex problems. Start with the number 8 on the bottom and place a checkmark at the end of the row. Now go up to the next row, 4. If we add 8 and 4 we get 12, which is too big since we want 10. So we ignore the 4 row and proceed to the next row up. Now we add 8 and 2, which is 10. This is of course not bigger than 10 so we check the 2 row. If we didnÃ¢€™t notice that weÃ¢€™re already at 10, we could check the last row but of course adding 1 to our total which is now 10 will give us too much. Our work now looks like this:
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Finally we add up the second column, only using the checked rows. This gives us 14 + 56, or 70, the solution to 10 Ãƒ— 7. The final work appears below.
SOLUTION:
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LetÃ¢€™s try another one.
EXAMPLE: How many loaves of 25-pesu bread can you make out of 3 hekat?
Since there are 25 loaves in each of 3 hekat, we have to determine 25 Ãƒ— 3. As in the first problem, we write 1 and 3 in the first row and double it until the first column is as large as it can be without being greater than 25.
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Check the 16 row and add as we go up, remembering to skip rows that make the number larger than 25. Note that 16 + 8 is 24, which is fine, but 24 + 4 is too big. So is 24 + 2. Finally we note that 24 + 1 gives us the 25 we need.
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We end by adding up the checked entries of the second column getting 75, which is the solution of 25 Ãƒ— 3.
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Try this one on your own. You should get 156.
PRACTICE: Multiply 13 by 12 as an Egyptian would.
The method is remarkably fast and easy when you get used to it. Many people are surprised that it works, but itÃ¢€™s not hard to see why, provided you look at it in the right context. Consider the following multiplication of 6 and 5:
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Instead of writing the numbers, IÃ¢€™m now going to represent them with tokens, specifically pennies and nickels. The first row is 1 and 5, which I will represent with one penny and one nickel. Note that the cash value represents the number. Instead of doubling the numbers, IÃ¢€™m going to double the number of coins. So the second row is two pennies and two nickels and the third row is four pennies and four nickels.
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Beginning the multiplication of 6 and 5 using Ã¢€Å“coins.Ã¢€
Note that each row has exactly the same number of pennies as nickels. This means that the cash value in the right column is five times the value in the left. Finally, note that the cash values are precisely the values in the preceding numerical example.
We want a 6 in the left column, so we check the rows that begin with two and four pennies. When we add these two rows, we get six pennies in the first column and six nickels in the other. Notice that this sum also has the same number of coins in each column. So getting six pennies in the first column means that there will be six nickels in the second. Clearly this means that there will be a cash value of six times the value of a nickel, or 6 Ãƒ— 5, which is precisely the multiplication we wished to solve.
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Each value in the second column is five times that of the first.
The above thought experiment would work on any multiplication of positive whole numbers. For example, if we wanted to multiply 11 by 25, we would simply mimic the above process with quarters, getting the left column to add up to 11 pennies. The bottom of the right column would then have 11 quarters, which has a value of 11 Ãƒ— 25. Similarly we can multiply by any number, such as 7, although we would have to imagine a 7-cent coin.
SIMPLICITY
Whole-Number Division
My eldest daughter just finished the third grade. Her strongest subject is mathematics, which couldnÃ¢€™t make her math-professor father more proud. She did have one trouble spot: memorizing her multiplication tables. This doesnÃ¢€™t surprise me, because in a very real sense, memorizing is not mathematics.
We live in a world that obeys the laws of mathematics. We know that 2 + 2 = 4, not because we memorized this but because every time we had two things and got two more, we ended up with four. Our common sense has been fine tuned by living in a mathematical world. For example, how do we know multiplication by 0 gives 0? I know that two cartons of eggs contain 24 eggs. I can transfer this knowledge into the mathematical fact that 2 Ãƒ— 12 is 24. Now that I need to know the value of 0 Ãƒ— 12, I can simply ask myself, how many eggs are in zero cartons? The answer is clearly 0 so I know 0 Ãƒ— 12 is 0.
Our brains have evolved to deal with mathematics. They even have special locations devoted to mathematical ideas and knowledge. If I were to surgically remove the part of your brain where your knowledge of letters and written words are stored, you would of course no longer be able to write. Oddly enough you would still be able to write down numbers. This is because your subconscious brain recognizes that number knowledge is different and stores their symbol shapes in a separate part of the brain than the location for the symbols of language. Mathematical understanding is different from our knowledge of language. I canÃ¢€™t explain why the word Ã¢€Å“dogÃ¢€ means dog; it just does. We memorize it, and people who speak other languages learn different, arbitrary, words for dog. Mathematical knowledge is not subjective. We donÃ¢€™t arbitrarily decide the value of 2 + 2.
For some reason, our brains donÃ¢€™t recognize our multiplication tables as mathematics. So when my daughter is confronted by the fact that 7 Ãƒ— 8 = 56, her mind regards this as arbitrary memorization and essentially stores this knowledge in the same place it keeps things like advertising jingles. Of course 7 Ãƒ— 8 = 56 is not quite as catchy as a tune sung by a cute five-year-old on the subject of hot-dogs. So we tend to remember one and forget the other.
Memorization in place of understanding seems to have a detrimental effect on early education. Consider the English number system. With the forties we have the numbers Ã¢€Å“forty-one,Ã¢€ Ã¢€Å“forty-two,Ã¢€ and so on. ThereÃ¢€™s an obvious pattern. The word Ã¢€Å“fourÃ¢€ is followed by the syllable ty, which is short for Ã¢€Å“tenÃ¢€ and then itÃ¢€™s followed by a number. So Ã¢€Å“forty-twoÃ¢€ literally means Ã¢€Å“four tens and two.Ã¢€ ItÃ¢€™s very simple. We have the same pattern with the fifties, sixties, and so on. However, in the tens, we run into a problem. We say Ã¢€Å“elevenÃ¢€ and Ã¢€Å“twelveÃ¢€ when we should be writing Ã¢€Å“onety oneÃ¢€ and Ã¢€Å“onety two.Ã¢€ Likewise, the teens still donÃ¢€™t follow the standard pattern. Notice that fourteen doesnÃ¢€™t follow the same pattern as twenty-four and thirty-four. In contrast, the Chinese word for 11 essentially translates to Ã¢€Å“ten one.Ã¢€ The Chinese system follows a fixed pattern for these numbers while ours does not. As a result the mathematical development of English-speaking students slows down compared to the Chinese just at the time when they hit these numbers. Excessive memorization inhibits mathematical growth.
After my daughter finally commits her multiplication table to memory, sheÃ¢€™ll begin to tackle the methods of long multiplication and long division. These are fairly complicated processes. They may seem easy to some of us now, but thatÃ¢€™s because some dedicated teacher forced us to attempt problems over and over until we memorized the algorithm.
You might be asking what this has to do with Egyptian math. What I want to do now is begin to compare our system to theirs. Consider the following long multiplication and the corresponding long division. As you look at them, think about how many memorized facts are required.
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In this multiplication, students need to have memorized 7 Ãƒ— 6, 7 Ãƒ— 5, 3 Ãƒ— 6, and 3 Ãƒ— 5. They also have to adjust the answer to 7 Ãƒ— 5 by the carry of 7 Ãƒ— 6, and the same is true for 3 Ãƒ— 6 and 3 Ãƒ— 5. As they do each digit multiplication, they need to carefully build the number going from right to left. When they start the 3 multiplication, they must remember to go down a line and shift left by one. Finally, they need to compute a sum.
The long division is trickier. They first have to see that 56 does not go into 2 or 20 and then guess how many times 56 goes into 207. This guess is far from obvious, especially to a child in grade school. They take their guess of 3 and multiply it by 56, using the knowledge of 3 Ãƒ— 5 and 3 Ãƒ— 6 and modifying the first by the carry of the second. They also must know to place the answer 168 so its rightmost digit, 8, lies directly underneath the rightmost digit of 207. They then subtract the two answers and repeat the above steps, guessing how many times 56 goes into 392. Finally, they must stop when they get 0, knowing that the process ends when they get any number less than 56.
When you looked at the above multiplication and division, it may not have seemed complicated, but note that when I describe the system in words just how involved it seems to be. ItÃ¢€™s like riding a bicycle. It seems easy because weÃ¢€™ve blocked out the memory of how many times we fell as we waited for our muscles to learn how to make it Ã¢€Å“easy.Ã¢€
Now look at the same multiplication and division done the Egyptian way.
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The first thing that should catch your eye is that the multiplication and the division look almost exactly the same. ThatÃ¢€™s because they are basically the same. Each element in the second column is 56 times larger than the corresponding number in the first column. Hence when we make 37 in the first column, we get 37 Ãƒ— 56 in the second. Reversing the same logic, every number in the first column is equal to the number in the second divided by 56. So if we make 1792 in the second column, we get 1792 Ãƒ· 56 in the first.
LetÃ¢€™s look at an example.
EXAMPLE: Compute 133 Ãƒ· 7 as an Egyptian would.
We start by putting 1 and 7 in the first row. Above the second column, put 133. Continually double each row until the second column approaches but does not exceed 133. In this example we stop when we get to 112 since 112 Ãƒ— 2 is 224, which is greater than 133.
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Start at the bottom in the right column and check off the 112. Now we add the numbers going upward, making sure the sum doesnÃ¢€™t exceed 133. So we try 112 + 56, which is 168. Since this number is bigger than 133, we go to the next row up. Then we try 112 + 28, which is 140 and still too big. When we try 112 + 14 we get 126, which is fine, so we check the 14 row. Finally we try 126 + 7, which is 133, our number, and we check off the 7 row. Now we add up the elements in the first column in the checked off rows giving 16 + 2 + 1, or 19, the solution to 133 Ãƒ· 7.
SOLUTION:
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Sometimes we get remainders. Consider the following problem:
EXAMPLE: How many hekat are there in 81 loaves of 15-pesu bread?
Recall that to determine the number of loaves you multiply the number of hekat of grain by the pesu. To find the number of hekat, you divide the number of loaves by the pesu. So we want to compute 81 Ãƒ· 15. Once again we make a table, with the first row being 1 and 15. We label the second column with an 81. Now we double the first row until weÃ¢€™re about to pass 81 in the second column and add rows going up.
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Notice that as we add going up we get to 75 and are forced to stop. This is because 15 goes into 75 but not 81. WeÃ¢€™re 6 short. This seems like a problem, but it really isnÃ¢€™t. The 5 we get as our answer is 75 Ãƒ· 15. When we divide 81Ãƒ· 15 we still get 5 but with 6 remaining, which is sometimes expressed as Ã¢€Å“5 remainder 6.Ã¢€ Try these two. The second will have a remainder.
PRACTICE:
a) Divide 187 by 17
b) Divide 100 by 21.
ANSWERS:
a) 11
b) 4 remainder 16
LetÃ¢€™s return to the lesson of this section. In order to learn modern long multiplication, we need to memorize a 10-by-10 multiplication table. Then we need to learn how to multiply a single digit by a longer number, adding carries to later multiplications as we go. For a long multiplication we have to do this repeatedly, lining up numbers in a shifting pattern. Finally these numbers need to be added. Long division is even worse. We must continually guess how many times some ugly number like 37 goes into an even uglier number like 207. Once this guess is made, we need to multiply our answer by 37, which takes a fair amount of time, as we see above. Then we must line this up just right and subtract. This procedure has to be repeated multiple times.
Now letÃ¢€™s compare this process to the Egyptian system. First of all, the only number you need to multiply by is 2. In fact, you donÃ¢€™t really need to know this because itÃ¢€™s the same as adding a number to itself. You do this a few times, add parts of the columns, and youÃ¢€™re done. Division is almost exactly the same as multiplication. ItÃ¢€™s all simple addition.
I teach Egyptian mathematics at my college. I can teach my students how to do Egyptian whole-number multiplication and division in less than five minutes. They do it faster than I can and almost error free. Now compare that process to the two or so years my daughter will spend in class using drills and rote memorization in order to learn her arithmetic. I suspect that somewhere in the heavens, Thoth, the scribal god, is looking down at us with a smirk on his face thinking, Ã¢€Å“WhoÃ¢€™s primitive now?Ã¢€
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