Zeno’s Paradox

1. Zeno’s Paradox By: Deborah Lowe and Vickie Bledsoe

2. Zeno of Elea Zeno was a famous mathematician who was known for posing puzzling paradoxes that seemed impossible to solve. One of his most famous was his paradox of Achilles and the Tortoise.

3. Zeno’s Paradox involves a race between the mighty warrior Achilles and a tortoise. Achilles can run 10 times as fast as the tortoise and therefore gives the tortoise a ten meter head start.

4. If the tortoise has a ten meter head start can Achilles ever catch him? By the time Achilles reaches the ten meter mark, the tortoise will be at 11 meters. By the time Achilles reaches 11 meters, the tortoise will be at 11.1 meters and so on.

5. Each moment Achilles catches up the distance between them, the tortoise will be adding a new distance. The tortoise claims Achilles will never catch up. But will he? In other words, why ever move if we won’t ever get anywhere?

6. To rephrase that: Suppose I want to cover a specified distance. First, I must cover half the distance. Then I must cover half of half the remaining distance. Then I must cover half of half of half the remaining distance … and so on forever. In other words, 1=1/2+1/4+1/8...

7. At first this may seem impossible but adding up an infinite number of positive distances can add up to a finite sum. All of these distances add up to ONE!

8.      An infinite sum such as this is an infinite series. When such a sum adds up to a finite number, it is calledsummable.              

9. The solution is easy! Say it takes 2 seconds to walk 1/2 meter. It would only take 1 second to walk 1/4 meter, 1/2 second to walk 1/8 meter and so on.

10. It takes Achilles an infinite number of time intervals for Achilles to catch the tortoise, but the sum of these time intervals is a finite amount of time.

11. And poor old Achilles would have won his race!

12. The End