Zeno’s Paradox

By: Deborah Lowe

and

Vickie Bledsoe

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.

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.

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.

Each moment Achilles catches up Each moment Achilles catches up the distance between them, the the distance between them, the

tortoise will be adding a new tortoise will be adding a new distance. The tortoise claims distance. The tortoise claims

Achilles will never catch up.Achilles will never catch up. But will he?But will he?

In other words, why ever In other words, why ever move if we won’t ever get move if we won’t ever get

anywhere?anywhere?

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...

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!

An infinite sum such as this is An infinite sum such as this is an infinite series. When such an infinite series. When such

a sum adds up to a finite a sum adds up to a finite number, it is callednumber, it is called

summable.summable.

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.

It takes Achilles an infinite It takes Achilles an infinite number of time intervals for number of time intervals for

Achilles to catch the tortoise, but Achilles to catch the tortoise, but the sum of these time intervals is the sum of these time intervals is

a finite amount of time.a finite amount of time.

And poor old Achilles would have won his race!

The EndThe End