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