The problem of representing fractions is not trivial. We still commonly use at least two completely different representations. We have the fraction 1/3, for example. And then we have the decimal fraction 0.3333333... to represent the same number! The second way looks very awkward, because the "..." means that the 3's keep on going FOREVER. We can't actually write the number down explicitly, although we see the pattern, and we know what the digits are supposed to be which haven't actually been written (all those omitted 3's). The representation as 1/3 definitely looks simpler! The decimal fraction representation has, however, unexpected advantages.
First of all, it is easy to get from one representation to the other, and back. To turn 1/3 into its decimal representation, just divide 3 into 1 by long division (the process never terminates). To turn 0.333333... into its p/q form, you can argue as follows: if x = 0.333333..., then 10x = 3.333333... (multiplying by 10 just moves the decimal point over). Now, subtracting, 9x = 3.00000.... Notice how, in the subtraction, all but one of the 3's cancel! Since 9x = 3, x = 3/9 = 1/3. Try this yourself for the decimal fraction 0.142857142857142857..., in which the sequence 142857 repeats infinitely. You will have to multiply by 1000000, not 10, in order to move the decimal point over 6 places. Then, when you subtract, all the 142857's cancel except the first one, etc. Check that this decimal fraction is in fact 1/7 when you put it in lowest terms (as we did with 3/9 above).
Fractions of the form p/q always have a decimal representation which repeats something infinitely, like the examples above. (We will prove this later, when it will tie into some other topics, but if you play with long division, maybe you can see already why this is.) Sometimes what it repeats is just '0', as in 1/2 = 0.5000000..., in which case we don't usually indicate it. These p/q fractions are formally called "rational numbers." So what are the irrational numbers? You probably can guess now, even if you didn't know -- they are all the decimal fractions which DON'T repeat. The digits go on forever but without a repeating pattern. The square root of 2, for example, as a decimal fraction, is represented as 1.414213562373... In this case, the "..." stands for the infinitely many digits we didn't write. Unlike the case of rational numbers, we don't know the pattern, and we can't immediately supply the digits which aren't explicitly there. If someone insisted, we could laboriously find more of them. Of course we could never find ALL of them, because we don't have infinite time, and there is no simple pattern. So have we represented the square root of 2 or not? The Greeks, in effect, said "no." We say "yes" -- this decimal fraction, for which we have NOT supplied all the digits, but COULD in principle supply any FINITE number of digits, IS the decimal representation of the square root of 2. Does this make you feel slightly uneasy? That feeling is what is left of the "problem of the irrationals."
The decimal system of arithmetic came to Europe from India via the Arabs around 1200, when Leonardo of Pisa, also called Fibonacci, introduced it in Italy, having learned it himself in North Africa. Textbooks based on the books of Fibonacci were used, with basically no new development of any kind, for the next 300 years in "abacus schools," where little boys of the merchant class learned to do arithmetic (basically the way you also learned it). The name "abacus school" is a peculiar misnomer, because these schools were actually teaching a method which did away with the abacus. After all, an abacus is a system of movable counters for doing arithmetic without notation. This was necessary because the Roman numerals were clearly not designed for computation, but only to record the results. With Hindu-Arabic numerals, however, you can actually do the computation on paper, and you don't need counters, i.e., after you have been to abacus school, you don't need an abacus. As late as the 15th century you can find arguments that the results should still be recorded in Roman numerals, however!
Long after the decimal system was established, old units of measurement continued to be used which are not tied to the decimal system, and which are rather awkward in the decimal system. The English system of length measurement, for example, is
| 3 barleycorns | = 1 inch |
| 12 inches | = 1 foot |
| 3 feet | = 1 yard |
| 5-1/2 yards | = 1 rod |
| 4 rods | = 1 chain |
| 10 chains | = 1 furlong |
| 8 furlongs | = 1 mile |
If you wonder how many yards there are in 0.2 chains, for example, it's messy, even though 0.2 is a nice decimal fraction. These units are not adapted to decimal arithmetic. On the other hand you do see, particularly in the smaller units, that you can easily divide by 3, something which is awkward in decimal arithmetic, so the old units actually have some advantages too.
As everyone knows, the leaders of the French Revolution, in an attempt to reorder the world on rational lines, set up new units of measure which are adapted to our decimal arithmetic. Those units, millimeter, centimeter, meter, kilometer, etc. are changed by moving only the decimal point, without changing the digits at all. This seems like a very good idea, even if the base 10, which we are stuck with, has its limitations. It has been adopted everywhere in the world, with one notable exception.
It is instructive to see an early notation which was cumbersome, but which worked well enough to be used for over 1000 years, the Egyptian system. Our main sources for how it was used are two large papyrii, the Rhind Papyrus in the British Museum, and the Moscow Papyrus in Moscow. They describe methods in use in the Egyptian Middle Kingdom, around 2000 BC, but which were still in use in historic times, when the Greeks also used them, around 400 BC.[1] The Egyptian system was base 10, but without the modern place-holding system. We can use the same numeral '1' to mean different things depending what "place" it is in. It means "1" if it is in the ones-place, but "10" if it is in the tens-place, etc. The place is indicated, where necessary, by a zero. The Egyptian system, however, had different,unrelated, symbols for 1, 10, 100, etc. Here are the first few Egyptian numbers:
Since Egyptian script evolved from pictures, it is tempting to think that the symbol for 10 might have been some kind of container, perhaps inverted to hide the "10 things inside," and the symbol for 100 is a long tape measure.
Adding two numbers is easy in this system: you can start by collecting all the "ones" together, replacing 10 of them by a "ten" if necessary, etc. All such notations are suitable for adding. To multiply, the Egyptians used a system of doubling, which is just adding a thing to itself, so it reduces to adding. To multiply 28 x 13, for example, according to the papyrii, they would form the following table:
| 1 * | 28 * |
| 2 | 56 |
| 4 * | 112 * |
| 8 * | 224 * |
where in the second column 28 has been doubled, doubled again, etc. Then you say to yourself
28 x 13 = 28 x (8 + 4 + 1) = 224 + 112 + 28 = 364.
That is, you just add the "starred" entries in the 2nd column. Because of the repeated doublings, this is a kind of base-2 (binary) multiplication system. Some modern computer multiplication algorithms do this: it is quite efficient. The operation of halving, i.e., dividing by 2, was similarly used as part of a (more complicated) division process.
So far we have only seen integers. The Egyptian notation for fraction was essentially a notation for 1/n, where n is an integer. In particular, the numerator, as we would call it, had to be 1. The only exception was 2/3, which had its own symbol. The notation is systematic from 1/4 on, but the first few entries seem to be a holdover from a more primitive system [2]:
Thus the symbols for the fractions were mostly just the symbols for the integers, but with that football-shaped fraction sign over them. To halve a fraction, you just double the integer part (what we would call the denominator of the fraction), and that was already part of their standard method of computing, as we said above, so that part of the arithmetic works very well. The trickiest operation was to double a fraction. If the denominator is an even integer, you just halve it: no problem there. But if the denominator is an odd integer, there are tables that tell you what to do. They include lines like 2 x 1/7 = 1/4 + 1/28. As you see, the result is expressed in terms of fractions of the form 1/n, but it is certainly not obvious. For some reason, they would not write the result as 1/7 + 1/7, which seems both obvious and natural. Thus, with every doubling of a fraction, it was transformed into something that looked quite different! Here are a few more lines of the standard table [3]:
| 2 x 1/9 = 1/6 + 1/18 |
| 2 x 1/11 = 1/6 + 1/66 |
| 2 x 1/13 = 1/8 + 1/52 + 1/104 |
| ... |
| 2 x 1/89 = 1/60 + 1/356 + 1/534 + 1/890 |
| 2 x 1/91 = 1/70 + 1/130 |
As you see, even things that ought to be fairly simple look complicated! It is instructive to try checking the first line of the table, which supposedly represents 2/9. If we multiply it by 9 we should get 2. Do we?? Multiplying by 9, in the Egyptian scheme, means doubling, doubling, and doubling again (so that we have multiplied by 8) and then adding the original number. Here it is:
| 1 * | 1/6+1/18 * |
| 2 | 1/3 + 1/9 |
| 4 | 2/3 + 1/6 + 1/18 |
| 8 * | 1 + 2/3 + 1/9 * |
Thus 9 x 2/9 = 1 + 2/3 + 1/6 + 1/9 + 1/18. That is right, but it is hard to recognize it as the number 2! In order to clean up the mess that this system produced, there were also methods for recognizing and combining certain sums of fractions. One would certainly want to recognize 1/2 + 1/3 + 1/6 = 1, for example. Thus an Egyptian scribe would get the right answer to our computation, namely 2, but think of the effort required! The papyrii seem to be more sophisticated than they are. They describe how to solve problems of allocating resources in various ways, which require the solutions of various kinds of equations, but in fact they are entirely consumed just with doing the arithmetic to get the right number. There is no possibility of thinking more generally, about general ideas and principles, and letting the arithmetic take care of itself. Perhaps the amazing capacity of the Greeks to generalize and abstract, to go to the heart of the matter and not get lost in the details, was a reaction to this wretched system. Their geometric algebra was perhaps an overreaction, though: it eliminated notation entirely!
In 1857 H.C. Rawlinson, the decipherer of the Babylonian cuneiform script [4], found a small tablet fragment which he correctly identified as being one corner of a multiplication table. From this small piece he suspected, rightly, as it turned out, that the Babylonians used a base-60 arithmetic, with place-value notation. The piece said 59 x 59 = 58 1. Since it was a corner, Rawlinson must have suspected that 59 was the last line of the table, and in fact 59 x 59 = 58 x 60 + 1, so that the answer is given as a sexagesimal number: 58 in the 60's place and 1 in the 1's place.
The cuneiform characters were incised into clay with a stylus. The whole system of notation uses only two distinct characters, one for 1 and one for 10 (as you see, there is a base-10 notation underlying the base-60 arithmetic). Here is what they look like:
I think it is clear that this is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ..., 58, 59, 60, 61, 62. You may be a little worried about the last few entries. Why is that 60 and not 1? I would say, because it follows 59. Why is it 61 and not 2? Because there is a space between the two wedge-strokes, so the first one, being in a separate place from the second, represents a 60, not a 1. The slight vagueness about what place the digits are in must have been resolved by common sense in the minds of Babylonian scribes, since they did not, on the whole, introduce devices to make it any clearer, like our zero and decimal point.
Virtually complete sets of multiplication tables have been found. They do not go up to 60 x 60. Rather the base 10 system underlying the notation is used. There are tables for multiplication by 1, 2, ..., 9, 10, 20, 30, 40, and 50. To multiply by 28, for example, you would multiply by 8, multiply by 20, and add the two results. There are also tables for reciprocals with entries like
| 1/2 | = 30 |
| 1/3 | = 20 |
| 1/4 | = 15 |
| 1/5 | = 12 |
| 1/6 | = 10 |
| 1/8 | = 7 30 |
| 1/9 | = 6 40 |
| 1/10 | = 6 |
It is easy to understand this if we think of hours and minutes. 1/2 hour = 30 minutes, 1/3 hour = 20 minutes, ..., 1/8 hour = 7 minutes 30 seconds, etc. There should really be some indication that the second column is in a different unit, but that is left to common sense. The reciprocals are for the most part very simple numbers, because the base is 60, which has many factors. Notice that 1/7 is missing from the table, however. It would have an infinitely repeating sexagesimal representation, just as it has an infinitely repeating decimal representation.
As the examples of the reciprocals show, Babylonian arithmetic could handle fractions just as easily as we handle them today, one might even say it handles them better. What is more, their system of measures was adapted to their arithmetic: typically there were 10, 20, 30, or 60 small units to a large unit [5], conversions which are easy to do. In particular, multiplying or dividing by 60 means do nothing, because 60 is represented by 1. In the other cases, depending on whether you are converting up or down, you either multiply by 10, 20, or 30, or multiply by their reciprocals, 6, 3, or 2. Common sense takes care of the place value, just as if I were to express 0.2 meters = 20 centimeters by saying 2 = 2. As far as digits go, that is the statement. The Babylonian system reached this highly developed form already in the Old Babylonian period, by 1800 BC. It is arguably better than our modern decimal/metric system, in that more fractions can be expressed simply in it. It is certainly not worse, except in being vague about which place is which, i.e., in whether 30 means 30 or 1/2. But as far as mere computation goes, this does not matter, a rather sophisticated fact, built into this system and its tables. As proof that this is not a bad system, let me point out that it is still in use, and has been continuously in use, despite all the reforms of the intervening 4000 years. We use it for time, when we say hours, minutes, and seconds. That is why I could appeal so easily to something you know very well to explain the table of reciprocals. And we also use it in measuring angle: degrees, minutes, seconds of arc.
A small tablet in the Yale Babylonian Collection makes the point:
It is Old Babylonian, from around 1800 BC. It shows a square, with diagonals. The side of the square, in the upper left, is labeled by the number 30. Across the middle of the square (not easy to read in the picture!) is the number 1 24 51 10, and beneath it is the number 42 25 35, which is the product of the previous two numbers. Someone has multipled 30 by 1 24 51 10 and written the result along the diagonal of the square. From the positions of the numbers you may suspect that they refer to the length of the side and the length of the diagonal. If that is so, 1 24 51 10 should be the square root of 2, and in fact 1+24/60+51/3600+10/216000=1.4142130... is the square root of 2 to within a few parts in ten million! So there is no question about the meaning of this tablet. Otto Neugebauer says this same value for the square root of 2 was used by Ptolemy in his astronomical tables almost 2000 years later (after Alexander, the Greeks switched to Babylonian arithmetic). How is it that such sophistication was achieved so early? No one really knows.
We know how this value was probably found, though, because the method is described on other tablets. In more modern notation, if x is the square root of 2,
If x is the square root of 2, then it satisfies any of those relations. The so-called "Babylonian method" uses the last one to generate ever better (approximate) values for x. You start with a guess for x, and you put that into the right hand side of the last equation. If, when you evaluate it, the same x results, then you have found the answer, but of course this will not happen, because we already know that x, being the square root of 2, is irrational. Rather some different value will be generated, but this new value is a better approximation. Now you can repeat the process: put the new value into the right hand side, get a new (better) value out, etc. Let us try it with a starting guess x = 3/2. Then we get
x = 3/2 = 1.5000... (starting guess)
x = 1/2(3/2+4/3) =1/2 x 17/6 = 17/12 = 1.4166666... (new improved value)
x = 1/2(17/12+24/17) = 1/2 x 577/204 = 577/408 = 1.4142156... (even more improved)
etc.
This is already very close!
If you do exactly the same computation with sexagesimal arithmetic, keeping 3 sexagesimal places, you get
x = 1 30 00 00 (this is the starting guess, x=3/2)
x = 1 25 00 00 (improved value)
x = 1 24 51 10 (the value on the tablet!)
Use the Babylonian method to find a good approximation for the square root you proved irrational in Assignment 2. Be clear how you are generating better approximations, and what your starting guess is. In order to make your work different from everyone else's, choose an inventive starting guess. As an example, I could find the square root of 19 by evaluating
1/2 (x + 19/x)
for some suitable x, then putting the result back in as a new x, etc. If I start with the guess 4 (not very inventive, but no one else will be doing the square root of 19), then the next approximation is 35/8, which is already not bad (its square is 19.1406...) If I use the guess 35/8 and put that in, I get 2441/560, which is excellent: its square is 19.00025...
Try this also with one of the perfect squares, 4, 9, or 16. Find the square root! Check that if you put in the (known) square root for x, the same value comes back to you, and you are done. What happens if you put in a value which is not exactly right for a starting guess? Include that case in your answer. For these computations, borrow a friend's calculator if you do not have one.
1. Neugebauer, Otto, Vorlesungen ueber Geschichte der antiken mathematischen Wissenschaften
2. Ibid., p. 90.
3. Ibid. p. 153.
4. Gordon, Cyrus H., Forgotten Scripts, Dorset Press, New York, 1987.
5. Friberg, J., "Numbers and Measures in the Earliest Written Records," Scientific American, February, 1984, p. 110.