If God created the Universe, who created God? If the Big Bang gave rise to the Universe, who created the physical laws that caused the Big Bang? Life overflows with paradoxes. They make life confusing, but they also make life fascinating. Zeno's Paradox is no exception.

Zeno of Elea, Greece invented a paradox in about 500 B.C. According to this paradox, a turtle challenged a fellow, named Achilles, to a foot race. The turtle was given a head start of 10 meters. Achilles ran the ten meters, but as he did, the turtle advanced 1 meter. Then Achilles ran the 1 meter, but as he did, the turtle ran .1 meter. Achilles ran the .1 meter, but as he did, the turtle ran .01 meter. As a result, Achilles never caught up with the turtle.

The origin of this famous Zeno's Paradox is simply a result of a failure to recognize that the word "never" pertains to time, so it is time that must be measured. Qualitatively, each additional distance that Achilles must run gets smaller and smaller, so the time gets smaller and smaller. This time approaches zero as Achilles comes closer to the turtle. As a result, even though this time is the result of an infinite sum of times, this time is finite rather than infinite.

Let us now examine the situation quantitatively. If you are not a fan of math, I recommend that you read the remainder of this article soon before going to bed because it is more effective than a sleeping pill. But, if you like math, what follows will probably help you develop your algebra skills.

Suppose the tortoise gets a head start X, the human requires time, T, to travel distance, X, and the tortoise travels distance .1X in time T.

Then we would first need to show that the human catches up with the tortoise in finite time. The human catches up with the tortoise when the distance they both run is equal. The human thus has to run distance X, which takes time, T, followed by distance .1X, which the turtle will have traveled in the first time, T. Therefore, the human would have to run for time T+.1T to travel the distance of X+.1X. But in the time .1T, the turtle will travel distance .01X. The human must thus travel an additional distance of .01X, which takes the human a time, .01T. And, while the human is running the distance of .01X, the turtle is running the distance of .001X, which will require the human to run for an additional .001T. Following this same pattern, the time, t, that it takes the human to catch the turtle is given by

t=T+.1T+.01T+.001T+...

Subtracting T from both sides, multiplying both sides by 10, and noting that the result on the right equals t, we have

10(t-T)=t

or

t=(10/9)T

Therefore, the time it takes to catch the tortoise is finite. Now, all we need to show is that the total distance traveled at the time when the human catches up with the turtle is less than the distance from the starting line to the finish line. The human runs at speed, X/T. The product of this speed with the total time, t, yields the total distance traveled by both the human and the tortoise when the human catches up with the tortoise. This distance is

(X/T)(10/9)T=(10/9)X

So, as long as the distance from the starting line to the finishing line is greater than (10/9) times the head start of X, which is given to the tortoise, the human wins the race.

Gee, are you grateful that you finally understand Zeno's Paradox? Now you can tell all your friends, and I'm sure they will be happy for you. If you see them yawn a lot, it's probably because they did not get much sleep the night before.

Learn more about this author, Joey Benedict.
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