| To Infinity and
Beyond! Zeno's Paradox  Zeno
of Elea (490 BC - 425 BC) is credited with several interesting
paradoxes about infinity. For example, he claimed that motion is
impossible. Suppose we want to move one meter forward. Then first we
need to travel half the distance (1/2 a meter), and then half of the
remaining distance (1/4 meter) and then half of the remaining distance
(1/8 meter) and so on, for ever. Since one needs to take infinitely
many steps to move one meter, then movement itself cannot be possible.
This paradox is related to a very difficult concept, the sum of an
infinite amount of numbers (or infinite series). The fact is that
1/2+1/4+1/8+...=1.A more refined version of this paradox, also due to Zeno, is the paradox of Achilles and the tortoise. You may find a downloadable version here. Thompson's
Lamp
Consider
a lamp, with a switch. Hit the switch
once, it turns it on. Hit it again, it turns it off. Let us imagine
there is somebody who likes to play with this
lamp as follows. First, he turns it on. At the end of one minute, he
turns it off. At the end of half a minute, he turns it on again. At the
end of a quarter of a minute, he turns it off. In one eighth of a
minute, he turns it on again. And so on, hitting the switch each time
after waiting exactly one-half the time he waited before hitting it the
last time. Applying the above discussion, it is easy to see that all
these infinitely many time intervals add up to exactly two minutes.
QUESTION: At the end of two minutes, is the lamp on, or off? A strange answer: the first time we hit the switch the lamp turns on, the next time the lamp turns off. Let us write +1 when the light goes on, and -1 when the light goes off. Then, the process which is carried away before can be represented as: +1-1+1-1+1-1+1-1+1-1...
A
partial sum of the first 5 terms, say, tells you that after the first 5
rounds the lamp is on (+1-1+1-1+1=1). Thus, if we want to know if the
lamp is on or off, we need to add up all the +1's and -1's. Let's do it
this way, let S=1-1+1-1+1-1...But then:
S=1-1+1-1+1...=1-(1-1+1-1...)=1-S
Thus 2S=1 and S=1/2 !!!
So, is the lamp on or off??
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