Zeno

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Zenos Paradox

• By Deborah Lowe
• and
• Vickie Bledsoe

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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.

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• Zenos 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.

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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.

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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 wont ever
  get anywhere?

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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,
• 11/21/41/8...

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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!

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

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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.
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And poor old Achilles would have won his race!
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The End