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| Saturday, November 18, 2000 |
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Lest men suspect your tale untrue, — John Gay The time is a year before the Trojan War. Old and wise men of Athens are sitting in the town hall. They are worried about Achilles
because he thinks that he is invincible. “Achilles is proud, which is not good for him. He is the best fighter in all of Greece, but has become lazy in the past few months. Gentlemen, we must do something to get him back on track,” says the eldest citizen. The best Greek mathematicians are also part of the assembly. “We know a tortoise who humbled a hare some days ago. We should set up a race between Achilles and this tortoise,” they say. Laughter travels through the hall. “Don’t judge our suggestion so cruelly wise men. We can prove that Achilles will lose this race,” they say. They present a paper to the eldest citizen and whisper something in his ear. The paper travels through the hall. A 200-yard race is set up between Achilles, who can run at a speed of 10 yards per second, and the tortoise, who can run at one yard per second (his best time from the previous race). To give the tortoise a chance, he is given a 100-yard head start. The mathematicians are nowhere to be seen at the start line. They are in a cafe, imagining the progress of the race. The eldest mathematician is talking. He says, “When Achilles covers that first 100 yards, to get to where the tortoise will be, the tortoise will be 10 yards ahead. When Achilles covers those 10 yards, the tortoise will be 1 yard ahead. When Achilles covers that 1 yard, the tortoise will be 1/10 yard ahead.” “There is no end to this sequence. We can go on forever dividing by 10. Therefore Achilles will have to cover an infinite number of smaller and smaller intervals before he catches up with the tortoise. However, to do an infinite number of things, he will take an infinitely long time, so, he’ll never catch up. He has no chance,” says another old man. “Something is wrong with this argument, but what?” a voice interrupts the old man. “Wrong? how can we be wrong?” the rest of them say. Suddenly, they realise what the dissident has said and rush out towards the start line. |
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Most of them have figured out how long will it take Achilles to cover the sequence of smaller and smaller intervals. They watch anxiously as Achilles takes 10 seconds to cover the first 100 yards, 1 second to cover the next 10 yards, 1/10 second for the next yard, 1/100 second for the next 1/10 of a yard, and so on. They write down running totals of time elapsed in covering each of these points. They write: 10 seconds, 11 seconds, 11.1 seconds, 11.11 seconds and so on. The total time elapsed in covering all the infinite number of smaller and smaller intervals is going to be 11.1111111…, with the 1s going on forever. However, this recurring decimal, 0.111111…, is just 1/9. It is possible to add together an infinite number of time intervals and still get a finite result. It means that there is a definite time — 11 1/9 seconds — in which Achilles will catch up with the tortoise, and after that instant, he’ll pass the tortoise. As expected, Achilles wins the race, removes his shoes and cools his heels. It is the only weak point that he shows. Centuries later, Zeno of Elea (495-435 BC) narrates this tale and some other problems of paradoxes to Parmenides and Socrates. Later, Eudoxus, Euclid, Archimedes and Newton gradually find ways to deal with these small quantities. Thus, the tale of Achilles and Zeno’s discussions give birth to a new branch of mathematics — calculus. Many centuries later, wise men narrate the tale of Achilles and recall these words of Jesus: I returned and saw under the sun that the race is not to the swift, nor the battle to the strong, neither yet bread to the wise, nor yet riches to men of understanding, nor yet favour to men of skill; but time and chance happeneth to them all. Mathematics prevails.
— Aditya Rishi |
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