This post will focus on the workings of two of Zeno’s
paradoxes, and how they relate to mathematics. First, the two paradoxes will be
covered, and then the mathematical continuation into limits of infinite sums. Secondly,
this post will cover how irrational numbers like Pi, e, and the square root of
two, fit into the paradox.
Zeno was a
Greek philosopher that lived roughly around 500 B.C. He proposed a set of paradoxes,
and the focus will be on two of those paradoxes, for their similarity. The
first paradox starts out with a race between Achilles, and a tortoise. Since a
tortoise is much slower than Achilles, the tortoise is given a hundred-meter
head start of Achilles. After this head start, Achilles sprints a hundred
meters to catch the tortoise. Now during the time it has taken Achilles to
reach the hundred meters, the tortoise has move forward 10 meters. Again
Achilles sprints ten meters to catch the tortoise, but again the tortoise has
moved again another meter. This pattern will continue on forever. The paradox
is that Achilles can run faster than the turtle, but cannot beat the turtle.
The second
paradox is similar, but it has to do with shooting an arrow. If an archer
shoots an arrow at a target, surely it will travel the full distance to the
target. Another way of thinking about it is the arrow must first travel halfway
to the target. After traveling halfway, it must travel halfway of the new
distance to the target. This process of halving the distance will go on forever,
the arrow will just go another half the distance infinitely many times. The
Paradox is that if there is an infinite number of halving, then how does the
arrow ever hit the target? Surely it is known that the archer can hit the target,
but then what happened to the infinite series?
There is
only one explanation that satisfies the two paradoxes. The sum of the infinite series
must equal the finite distance traveled. That may seem impossible, but it is completely
possible. This process is used heavily in understanding calculus, the study of
rates of change. To explain this answer we will start with partial sums, and
then move into limits. To start the arrow travels half the distance, and then
it travels half of the remaining distance, which would be a fourth of the total
distance. So the sum of the fractional values would look like 1/2 + 1/4 +1/8
+1/16 . . . The sum of the first four terms .94 the total distance. So the more
terms that are added to the series the closer the series approaches 1. This
series, if summed all the way, with an infinite number of terms would equal one
exactly. This idea is the introduction to infinite limits. A limit is a way of
saying what happens as something approaches something else. In this case the
limit as the number of terms approaches infinity would equal one. Many
individuals struggle with the idea that there is a sum associated with an
infinite series. If someone tried to put the series into a calculator, they
would not ever be able to press enter, because then it would not be all of the
infinite number of terms in the series. This idea is hard to understand, but at
the same time it must be true because an archer can hit a target, people can
clap their hands, people can drive from point A to point B. Every kind of movement
requires an infinite sum of fractional parts of that distance, and yet everyone
does it.
This
paradox can be seen again in irrational constants such as, pi, or the square
root of the square root of two. Looking at pi, if a circle is draw with a
compass, having radius of one, then the area of the circle is pi. This is
concerning because pi goes on forever, so how can a finite area be an endless
decimal? The same goes for the square root of two. Imagine a right triangle
with two legs of length one. The hypotenuse is equal to the square root of two,
a never ending decimal. Except at the same time the length of the hypotenuse
does not go on forever. In both of these cases, it shows that some infinite
series can be expressed as a finite number. In general, the numbers of the
terms in these series must be getting smaller over time, or else they would
just shoot off towards infinity.
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