Example 4.1.1: Zeno Paradox
(Achilles and the Tortoise)
|
Achilles is racing against a tortoise. Achilles can run
10 meters per second, the tortoise only 5 meter per second. The track
is 100 meters long. Achilles, being a fair sportsman, gives the tortoise
10 meter advantage. Who will win ?
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Let us look at the difference between Achilles and the tortoise:
| Time |
Difference |
| t = 0 |
10 meters |
| t = 1 |
5 = 10 / 2 meters |
| t = 1 + 1/2 |
2.5 = 10 / 4 meters |
| t = 1 + 1/2 + 1/4 |
1.25 = 10 / 8 meters |
| t = 1 + 1/2 + 1/4 + 1/8 |
0.625 = 10 / 16 meters |
and so on. In general we have:
| Time |
Difference |
| t = 1 + 1 / 2 + 1 / 2 2 + 1 / 2
3 + ... + 1 / 2 n |
10 / 2 n meters |
Now we want to take the limit as n goes to infinity to find out when
the distance between Achilles and the tortoise is zero. But that involves
adding infinitely many numbers in the above expression for the time, and
we don't know how to do that. However, if we define
S n = 1 + 1 / 2 + 1 / 2 2
+ 1 / 2 3 + ... + 1 / 2 n
then, dividing by 2 and subtracting the two expressions:
S n - 1/2 S n = 1 - 1 /
2 n+1
or equivalently, solving for S n:
S n = 2 ( 1 - 1 / 2 n+1)
But now S n is a simple sequence, for which we know
how to take limits. In fact, from the last expression it is clear that
lim S n = 2
as n approaches infinity. Hence, we have - mathematically correct
- computed that Achilles reaches the tortoise after exactly 2 seconds,
and then, of course passes it and wins the race.
A much simpler calculation not involving infinitely
many numbers gives the same result:
- Achilles runs 10 meters per second, so he covers 20 meters in
2 seconds
- The tortoise runs 5 meters per second, and has an advantage
of 10 meters. Therefore, it also reaches the 20 meter mark after 2 seconds
- Therefore, both are even after 2 seconds
Of course, Achilles will finish the race after 10 seconds, while the
tortoise needs 18 seconds to finish, and Achilles will clearly win.
The problem with Zeno's paradox is that Zeno was uncomfortable
with adding infinitely many numbers. In fact, his basic argument was:
if you add infinitely many numbers, then - no matter what those numbers
are - you must get infinity. If that was true, it would take Achilles infinitely
long to reach the tortoise, and he would loose the race. However, reducing
the infinite addition to the limit of a sequence, we have seen that this
argument is false.