I have cut a cube into two, then again, again, and again.
Finally, I'm down to a google-plex sub-atomic level. So, in theory, I
should be able to divide something forever and always have a half
remaining. Which brings me to my idea that there is no such thing as
nothing! Can math prove me wrong?
There is an entire branch of mathematics devoted to problems of the type
you mention -- that branch is infinite series, and as the name suggests it
has to do with infinite sums. Some sums converge to a finite number
[convergent]. Others just keep getting bigger and bigger as you add more
terms. The sum you used as an example: 1+1/2+1/4 + ... + 1/2n +... has no
finite limit [divergent]. It just keeps getting bigger and bigger.
In contrast, it is possible to prove mathematically that the sum of the
series converges: 1 + 1/4 + 1/16 + ... + 1/n^2 + ... = (pi^2)/6.
Within the realm of mathematics, many things are possible. What you have
found is that tere is no such thing as a smallest positive number. You have
discovered that numbers are "continuous": between any two numbers there is
another number. Nothing, the value zero, does exist. However, you can
always get closer to it without actually reaching it. This is also the
basis of limits: getting closer and closer to something without ever
actually reaching it.
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Update: June 2012