46 Lessons in Early Geometry, part 9/10 / provisional version in freestyle English / a corrected version will follow in March, April or May (hopefully) / Franz Gnaedinger / February 2003 / www.seshat.ch
Gregory Peacock and Augustus de Morgan, the founding fathers of symbolic algebra, freed the algebraic operations from the restrictions of conventional numbers. A similar and no less radical step of emancipation occurred several millennia ago, when the numbers were freed from objects, and when counting passed on to calculating and computing (from Latin computare, con-putare, think together, add up in the mind). As long as the numbers belong to objects you can mainly count:
one fish two fish three fish many fish
1 loaf of bread, 2 loaves, 3 loaves, 4 loaves, 5 loaves
But when you free the numbers from the objects and see them as ideal objects in their own right you can perform all kinds of new tasks. Consider for example this problem. 700 loaves of bread shall be divided among four men called A B C D, in such a way that A obtains the double share of C, and B obtains the double share of D, while A and C together obtain the double share of B.
Let me try to solve this problem by playing with beans of different color, here represented by the letters a b c d:
a a a a a a a a c c c c b b b b b b d d d
8 beans 4 beans 6 beans 3 beans
I found a total of 8+4+6+3 = 21 beans, which represent 700 loaves of bread.
A's share will be '21 of 8 times 700 loaves, makes 266 "3 loaves.
B's share will be '21 of 6 times 700 loaves, makes 200 loaves.
C's share will be '21 of 4 times 700 loaves, makes 133 '3 loaves.
D's share will be '21 of 3 times 700 loaves, makes 100 loaves.
Essentially this calculation is carried out in problem no. 63 of the Rhind Mathematical Papyrus, here given in Eric Peet's translation: "Example of dividing 700 loaves among 4 men, 2/3 to one, 1/2 to another, 1/3 to another, 1/4 to another. Let me know the share of each of them." It would hardly be possible to solve such a task properly with 700 loaves placed on several boards in front of you and four men waiting for their correct share, while it is fairly easily solved in a symbolical way, by using beans and a further abstraction, bare numbers, which can represent any object you like.
We are so used to the benefit of numbers that we can hardly estimate the radically new concept of free numbers, which must have appeared a strange if not crazy idea to Stone Age man: I can well see fish; I can even count fish: one fish, two fish, three fish, many fish; but I can't see any numbers, let alone can I catch numbers, bring them home, have them cooked and nourish my family ...
Peacock and De Morgan "advocated algebraic freedom" (Victor J. Katz, A History of Mathematics, Addison-Wesley 1998). In a similar way one may say that Ahmes in the Rhind Mathematical Papyrus advocates numerical freedom: Dare consider numbers as beings in their own right, and you shall be able to solve a variety of demanding problems. Trust me, and learn from me. Did you solve the problem of the 700 loaves? Did you like it? Really? So I tell you a secret. I folded many further problems into my numbers, to be solved by advanced learners. Play with my numbers, and you will find out what I mean. You need a clue? Well then. Each man got his share of loaves. Now each one has to pay '8 of his share as tribute, say, to the fishermen of the village. How do you proceed? You may simply divide the numbers of loaves by 8. Or you may start from the numerical shares and transform them into pairs of unit fractions, in such a way that one fraction is the part a man can keep, and the other one the part he has to pay for the fish, and only then proceed to loaves. Choosing that way you will find an elegant solution:
'21 of 8 makes '3 '21 '21 of 700 makes 33 '3 loaves
'21 of 6 makes '4 '28 '28 of 700 makes 25 loaves
'21 of 4 makes '6 '42 '42 of 700 makes 16 "3 loaves
'21 of 3 makes '8 '56 '56 of 700 makes 10 '2 loaves
Let me inform you about a visit to my astrologer. She kindly allowed me to have a look into her Fine Magic Bablefish Crystal Ball (trademark reg.), and was able to establish a connection between me and Ahmes (- with a crystal ball, thousand miles are less than an inch, and thousand years less than a second, you know). I was pleased to see Ahmes, and asked him the following question:
Good morning, professor Ahmes, nice to meet you. May I ask you a question? I run across a funny equation of yours:
1 hekat times 3 '7 times '22 of 7 makes 1 hekat
What can this possibly mean? First of all, what is a hekat? A measure of grain, we know that much, but how is it defined?
Ahmes: Good morning, whoever you are, and wherever your voice comes from. A hekat is a measure of grain, you are correct. 30 hekat equal 1 cubic cubit, and one hekat is defined as a right parallelepiped of these numbers:
'2 royal cubit x '3 royal cubit x '5 royal cubit
Me: So simple?
Ahmes: Well, if you understand the basic idea, we can always go a step further. Consider our subdivision of the royal cubit into 7 palms or 28 fingers or 56 Re marks or 84 Shu marks or 112 Tefnut marks or 140 Geb marks or 168 Nut marks or 196 Osiris marks or 224 Isis marks or 252 Seth marks or 280 Nephtys marks or 308 Horus marks or 336 Imsety marks or 364 Hapy marks or 392 Duamutf marks or 420 Qhebsenuf marks or 468 Thoth marks.
Now you can define a square hekat as follows:
28 Re marks x 28 Shu marks x 28 Geb marks
Or like this:
210 by 140 by 84 Qhebsenuf marks
cubic diagonal 266 Qhebsenuf marks
Me: Amazing. First it looks so simple, and suddenly we are in the middle of a complex problem. Now tell me what you mean with your funny number 3 '7 in combination with a hekat.
Ahmes: You are free to play with all my numbers and inventing problems of your own.
Me: Can it possibly be the number of the circle? and might there be another hekat in a round shape, perhaps a cylinder?
Ahmes: Fine guessing. Replace the first hekat in my equation by the Qhebsenuf definition:
210 Qm x 105 Qm x 84 Qm x 3 '7 x '7 of 22 = 1 hekat
Now transform the equation as follows:
'4 x 105 Qm x 105 Qm x 3 '7 x '11 x 3136 Qm = 1 hekat
Can you also guess what the first part means?
Me: Possibly a circle of the diameter 105 Qhebsenuf marks or, wait, 7 fingers? If so, the circle would be the cross-section of the cylinder, while the second part would be the height. '11 x 3136 fingers are - give me some time - 19 '165 fingers.
Ahmes: Let go the small fraction '165 and keep the 19 fingers. Thus you got a hekat in the shape of a cup, whose inner diameter measures 7 fingers, whose inner circumference measures 22 fingers, and whose inner height measures 19 fingers.
Me: Let me check it --- yes, you are right, the resulting volume is a hekat, with only a tiny mistake. Thank you for this fine lesson. By the way ... hello? professor Ahmes? hello?
He was gone. My astrologer failed in getting another connection. Well then, we shall try again tomorrow.
My astrologer polished her crystal ball and connected me again with Ahmes, whom I saw standing by a fisherman in the market place this time, asking for the price of the fish he just bought, whereupon the fisherman goes
Fisherman to Ahmes: Give me as much as you like, but in money, please, not in numbers. You know that I still don't really understand your profession. Mine is a regular one. I catch fish and sell 'em on the market. My neighbor is a farmer, he grows emmer, onions, and dates, and sells them on the same market. We work a lot and just manage to make a living, while you are in the number business and earn a good living by catching, planting and selling things nobody can see and touch, let alone cook and eat, and nourish a family with. I understand well the meaning of one fish, two fish, three fish, but I still don't grasp the meaning of a number by itself. What can it possibly be good for? Explain me, professor, and I shall give you a free extra fish, one of the red ones over here, if you please ...
Ahmes, to the fisherman: Are you making fun of me?
Fisherman, to Ahmes: Come on, professor, you know I like you and your family, and I always put my very best fish aside for you. Only thing is I do not understand what you are doing. I see well that a man can make a living from fish and grain, but how can a man possibly make a living from things nobody can see and touch and smell and cook and eat? I should perhaps ask the gods to send me invisible fish to catch, for they sell at much higher price than the ones I offer here ...
Ahmes, to the fisherman: You old rascal are trying to entangle me in yet another discussion on 'number theory' of yours, until, finally, I see no other way out than paying you some extra money for your fish. Which is very fine fish, that much is true. My wife always sends me to buy it from you, because it is the best. You are working hard for making a living, and I always pay a good price. Here is the money for the fish, and here the usual extra money for making me laugh. Some day I shall take you with me to my seminary and let you have a discussion with my pupils. I bet they will have a hard stand arguing with you on the topic of visible fish, the ones you catch and sell, and invisible fish, as you kindly call my numbers. I am sure they can learn from your direct and colorful way of arguing. But now let me go, please.
Fisherman, to Ahmes: Thank you ever so much, professor, and have a nice day, also in the name of my wife and five children.
Ahmes, to the fisherman: Goodbye, goodbye ...
I listened quietly and hoped to exchange a few words with the professor upon his returning home, but alas, my astrologer's crystal ball got blind again. I would have liked to mention how the idea of pure numbers evolved over time, how 'nothing' became a number, namely zero; how the negative numbers were introduced, then so-called imaginary numbers, complex numbers, quaternions and octonions, and even surrealistic numbers; how all the new numbers puzzled people, and that the concept of number will probably evolve ever further, causing much more puzzlement in the future, and then, within a relatively short time, each new number will turn out to be very useful, in one way or another, whereupon it will be fully accepted within the mathematical community. Allow me one more word. New expansions of numbers may be expected where the free flow of natural numbers on any level and the free unfolding of number patterns, which are the 'footprints' of operations working, are getting stopped by the restriction of a convention. A thorough study of all kinds of number patterns may well enhance the further progress in mathematics.
My astrologer lightly touched her crystal ball, murmuring some secret words. I got in contact with Ahmes again and asked him about the meaning of problem no. 23 in the Rhind Mathematical Papyrus.
Me: Good morning, professor Ahmes. May I ask another question? Sometimes you give us a clue as to what you have in mind, for example when you mention a hekat. On other occasions you tell us only the numbers, for example in the case of this problem:
'4 '8 '10 '30 '45 plus ??? makes "3
'4 '8 '10 '30 '45 plus '9 '40 makes "3
Ahmes: Well, you already solved my problem.
Me: Nothing more about it?
Ahmes: I told you before that you are free to invent any other problems to my numbers. But if you ask me for a clue I would suggest that you multiply the above numbers by a factor of 360, thus you will obtain
90+45+36+12+8 plus 40+9 makes 240
191 plus 49 makes 240
Now consider the three numbers as diameters of three circles and calculate their circumferences, using the value 3 '7 we have used before, and, very important, rounding all results.
Thus you get the following numbers:
191 x 3 '7 makes 573 + 27 (rounded) makes 600
49 x 3 '7 makes 147 + 7 makes 154
240 x 3 '7 makes 720 + 34 (rounded) makes 754
Proceed like this and you obtain three values for the secret number living in the circle:
'191 of 600 '600 of 191 makes '4 '30 '40 '100
If you know the circumference of a column or a circular pond and wish to calculate its diameter, multiply the circumference by the above unit fraction series.
'49 of 154 makes '7 of 22 or 3 '7
We know that value already.
'240 of 754 makes '120 of 377 or 3 '10 '24 or 3 '8 '60
If you know the diameter of a circle, multiply it by one of the above series, and you obtain the circumference.
Now I shall inform you about an easy way of finding many more handy
Me: Professor Ahmes? hello? professor Ahmes?
The connection was lost. We shall try again tomorrow.
Let me go on with my imaginary dialogue with Ahmes, scribe of the Rhind Mathematical Papyrus.
Me: Good morning, professor Ahmes. You promised to show me something ...
Ahmes: Oh yes, I remember. I told you how to proceed with an equation of mine. Now have a look at the previous ones:
"3 '15 plus ??? makes 1 "3 '15 plus '5 '15 makes 1
"3 '30 plus ??? makes 1 "3 '15 plus '5 '10 makes 1
Me: I recognize them (Rhind Mathemtical Papyrus nos. 21/22).
Ahmes: Well then. Multiply the first line by a factor of 135, and the second one by a factor of 50:
99 plus 36 makes 135 35 plus 15 makes 50
Now multiply the numbers by 3 '7 and round the results:
99 x 3 '7 makes 297 + 14 (rounded) makes 311
36 x 3 '7 makes 108 + 5 (rounded) makes 113
135 x 3 '7 makes 405 + 19 (rounded) makes 424
35 x 3 '7 makes 105 + 5 makes 110
15 x 3 '7 makes 45 + 2 (rounded) makes 47
50 x 3 '7 makes 150 + 7 (rounded) makes 157
Divide the last numbers by the first ones and give the results in unit fraction series:
'99 of 311 makes 3 '9 '33 '36 of 113 makes 3 '9 '36
'135 of 424 makes 3 '15 '27 '135 '35 of 110 makes 3 '7
'15 of 47 makes 3 '10 '30 '50 of 157 makes 3 '10 '25
Me: Five new values for the number of the circle?
Ahmes: Quite so. Do you remember the ones from yesterday?
Me: '600 of 191 '49 of 154 or '7 of 22 '120 of 377
Ahmes: Now let me come back on my promise and show you an easy way of how to find all the above and many more handy values for the secret number living in the circle.
Write 4 above 1 and add repeatedly 3 above 1:
4 (plus 3) 7 10 13 16 19 22 25 28
1 (plus 1) 2 3 4 5 6 7 8 9
Write 3 above 1 and add repeatedly 22 above 7:
3 (plus 22) 25 47 69 91 113 135 157 179 201 223 245
1 (plus 7) 8 15 22 29 36 43 50 57 64 71 78
267 289 311 333 355 377 399 421 443 465 487 509 531
85 92 99 106 113 120 127 134 141 148 155 162 169
Write 6 above 2 and add repeatedly 22 above 7:
6 (plus 22) 28 50 72 ... 424 ... 600
2 (plus 7) 9 16 23 ... 135 ... 191
Write 21 above 7 and add repeatedly 22 above 7:
21 (plus 22) 43 65 ... 2463
7 (plus 7) 14 21 ... 784 = 28x28
Write 9 above 3 and add repeatedly 19 above 6:
9 (plus 19) 28 47 66 ... 256
3 (plus 6) 9 15 21 ... 81
If you solve a problem that involves a circle or a sphere you may simply choose a value that comes handy. A few examples:
If a square measures 10 by 10 royal cubits or 70 by 70 palms, the diagonal measures practically 99 palms, and the circumference of the circumscribed circle measures practically 311 palms.
If the diameter of a circle measures 30 palms or 120 fingers, the circumference measures practically 377 fingers.
Let the radius of a circle measure 1 royal cubit or 28 fingers. The area measures practically 2463 square fingers.
Let the radius measure 13 palms. The area measures practically 531 square palms.
Let a rectangle measure 14 by 16 palms. The area measures practically 355 square palms.
Square a circle using the value '81 of 256 and you will find a formula for beginners: a circle of the diameter 9 fingers or palms or cubits and a square of the side length 8 fingers or palms or cubits have about the same area.
Me: Thank you, professor Ahmes. Next week I shall ask you what numbers are ...