Zeno and His Arrow


This is from the web comic xkcd.

As many readers will know, the reference is to the Greek philosopher Zeno, from about 490 BCE. He is famous for his “argument against motion” paradox. To get to a place, said Zeno, an arrow must first go half way there. Once there, it must go half the remaining distance, or one fourth. Then it must move half the remaining distance, or one eighth, etc. So, moving from one place to another requires an infinite number of motions, which is impossible.

We now know how to resolve the puzzle: the sum of an infinite number of terms can have a finite sum. In this case, the series is geometric, and the sum is given by the formula:

\[ S = 1 + r + r^2 + r^3 + \dots = \frac{1}{1 - r}\quad \text{provided } \left| r \right| < 1 \]

In Zeno’s case:

\[ \text{distance} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... = \frac{1}{2} \left( 1 + \frac{1}{2} + \frac{1}{4} + \dots \right) = \frac{1}{2} \frac{1}{1 - 1/2} = 1 \]

So Zeno will reach the bench.

(The mouse-over text cleverly says “The prosecution calls Gottfried Leibniz”. Leibniz would have done the above calculation in a flash, then confidently walked to the bench.)



One Response to “Zeno and His Arrow”

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