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- Lancelot Hogben, Mathematics, the Mirror of Civilisation
Achilles runs a race with the tortoise. He runs ten times as fast as the tortoise. The tortoise has 100 yards start. Now, says Zeno, Achilles runs100 yards and reaches the place where the tortoise started. Meanwhile the tortoise has gone a tenth as far as Achilles, and is therefore 10 yards ahead of Achilles. Achilles runs this 10 yards. Meanwhile the tortoise has run a tenth as far as Achilles, and is therefore1 yard in front of him. Achilles runs this 1 yard. Meanwhile the tortoise has run a tenth of a yard and is therefore a tenth of a yard in front of Achilles. Achilles runs this tenth of a yard. Meanwhile the tortoise goes a tenth of a tenth of a yard. He is now a hundredth of a yard in front of Achilles. When Achilles has caught up this hundredth of a yard, the tortoise is a thousandth of a yard in front. So, argued Zeno, Achilles is always getting nearer the tortoise, but can never quite catch him up.
You must not imagine that Zeno and all the wise men who argued the point failed to recognize that Achilles really did get past the tortoise. What troubled them was, where is the catch? The Greeks ... found any problem involving division very much more difficult than a problem involving multiplication. They had no way of doing division to any order of accuracy, because they relied for calculation on the mechanical aid of the counting frame or abacus. They could not do sums on paper. For all these and other reasons which we shall meet again and again, the Greek mathematician was unable to see something that we see without taking the trouble to worry about whether we see it or not.
If we go on piling up bigger and bigger quantities, the pile goes on growing more rapidly without any end as long as we go on adding more. If we can go on adding larger and larger quantities indefinitely without coming to a stop, it seemed to Zeno's contemporaries that we ought to be able to go on adding smaller and still smaller quantities indefinitely without reaching a limit. They thought that in one case the pile goes on for ever, growing more rapidly, and in the other it goes on for ever, growing more slowly.
There was nothing in their number language to suggest that when the engine slows beyond a certain point, it chokes off... To see this clearly we will first put down in numbers the distance which the tortoise traverses at different stages of the race after Achilles starts. As we have described it above, the tortoise moves 10 yards in stage 1, 1 yard in stage 2, one-tenth of a yard in stage 3, one-hundredth of a yard in stage 4, etc... That is to say:
10 + 1 + 0.1 + 0.01 + 0.001 + 0.0001 + 0.00001 + 0.000001 and so on.
We have only to use the reformed spelling to remind ourselves that this can be put in a more snappy form:
or still better:
We recognize the fraction 0.i as a quantity that is less than 2/10 and more than 1/10. If we have not forgotten the arithmetic we learnt at school, we may even remember that 0.i corresponds with the fraction 1/9. This means that the longer we make the sum, 0.1 + 0.01 + 0.001, etc., the nearer it gets to 1/9, and it never grows bigger than 1/9. The total of all the yards the tortoise moves till there is no distance between himself and Achilles makes up just 11 1/9 yards, and no more. You will now begin to see what was meant by saying that the riddle presents no mathematical difficulty to you. You have a number language constructed so that it can take into account a possibility which mathematicians describe by a very impressive name. They call it the convergence of an infinite series to a limiting value.
Put in plain words, this only means that, if you go on piling up smaller and smaller quantities as long as you can, you may get a pile of which the size is not made measurably larger by adding any more. The immense difficulty which the mathematicians of the ancient world experienced when they dealt with a process of division carried on indefinitely, or with what modern mathematicians call infinite series, limits, transcendental numbers, irrational quantities, and so forth, provides an example of a great social truth borne out by the whole history of human knowledge.
- Isaac Asimov, The Realm of Numbers
CLOSER AND CLOSER AND CLOSER . . .
Consider a series of fractions like this: 1/2, 1/4, 1/8, 1/16 ... and so on endlessly.
Notice that each fraction is one-half the size of the preceding fraction, since the denominator doubles each time. (After all, if you take any of the fractions in the series, say 1/128, and divide it by 2, that is the same as multiplying it by 1/2, and 1/128 x 1/2 = 256, the denominator doubling.)
Although the fractions get continually smaller, the series can be considered endless because no matter how small the fractions get, it is always possible to multiply the denominator by 2 and get a still smaller fraction and the next in the series. Furthermore, the fractions never quite reach zero because the denominator can get larger endlessly and it is only if an end could be reached (which it can't) that the fraction could reach zero.
The question is, What is the sum of all those fractions? It might seem that the sum of an endless series of numbers must be endlessly large ("it stands to reason!') but let's start adding, anyway.
First 1/2 plus 1/4 is 3/4. Add 1/8 and the sum is 7/8; add 1/16 and the sum is 15/16; add 1/32 and the sum is 31/32, and so on.
Notice that after the first two terms of the series are added, the sum is 3/4 which is only 1/4 short of 1. Addition of the third term gives a sum that is only 1/8 short of 1. The next term gives a sum that is only 1/16 short of 1 ... and so on.
In other words, as you sum up more and more terms of that series of fractions, you get closer and closer to 1, as close as you want, to within a millionth of one, a trillionth of one, a trillionth of a trillionth of one. You get closer and closer and closer and closer to 1, but you never quite reach 1.
Mathematicians express this by saying that the sum of the endless series of fractions 1/2, 1/4, 1/8 ... "approaches 1 as a limit."
This is an example of a "converging series," that is, a series with an endless number of members but with a total sum that approaches an ordinary number (a "finite" number) as a limit.
The Greeks discovered such converging series but were so impressed with the endlessness of the terms of the series that they did not realize that the sum might not be endless.
Consequently, a Greek named Zeno set up a number of problems called "paradoxes" which seemed to disprove things that were obviously true. He "disproved," for instance, that motion was possible. These paradoxes were famous for thousands of years, but all vanished as soon as the truth about converging series was realized.
Zeno's most famous paradox is called "Achilles and the Tortoise." Achilles was a Homeric hero renowned for his swiftness, and a tortoise is an animal renowned for its slowness. Nevertheless, Zeno set out to demonstrate that in a race in which the tortoise is given a head start, Achilles could never overtake the tortoise.
Suppose, for instance, that Achilles can run ten times as fast as the tortoise and that the tortoise is given a hundred-yard head start. In a few racing strides, Achilles wipes out that hundred-yard handicap, but in that time, the tortoise, traveling at one-tenth Achilles's speed (pretty darned fast for a tortoise), has moved on ten yards. Achilles next makes up that ten yards, but in that time the tortoise has moved one yard further. Achilles covers that one yard, and the tortoise has traveled an additional tenth of a yard. Achilles...
But you see how it is. Achilles keeps advancing, but so does the tortoise, and Achilles never catches up, Furthermore, since you could argue the same way, however small the tortoise's head start---one foot or one inch--Achilles could never make up any head start, however small. And this means that motion is impossible.
Of course, you know that Achilles could overtake the tortoise and motion is possible. Zeno's "proof" is therefore a paradox.
Now, then, what's wrong with Zeno's proof? Let's see. Suppose Achilles could run ten yards per second and the tortoise one yard per second. Achilles makes up the original hundred-yard head start in 10 seconds during which time the tortoise travels ten yards. Achilles makes up the ten yards in 1 second, during which time the tortoise travels one yard. Achilles makes up the one yard in 0.1 second during which time the tortoise travels a tenth of a yard.
In other words, the time taken for Achilles to cover each of the successive head starts of the turtle forms a series that looks like this: 10, 1, 0.1, 0.01, 0.001, 0.0001, 0.00001, and so on.
How much time does it take for Achilles to make up all the head starts? Since there are an endless number of terms in this Zeno series, Zeno assumed the total sum was infinite. He did not realize that some series of endless numbers of terms "converge" and have a finite sum.
For instance, the sum of the first two terms in the Zeno series above is 11; the sum of the first three is 11.1; of the first four, 11.11; of the first five, 11.111 and so on. As you see, if you add up all the endless series of terms, you get an endless decimal as the sum: 11.111111111111111111 ... and so on forever.
But if you work out the decimal equivalent of the number 11 1/9, you find that it also is the endlessly repeating decimal 11.111111111111111111111 ... and so on forever.
The sum of the Zeno series is therefore 11 1/9 seconds and that is the time in which Achilles will overtake and pass the tortoise even though he has to work his way through an endless series of continually smaller head starts that the tortoise maintains. He will overtake the tortoise after all; motion is possible, and we can all relax.