Suppose one day you wake up and decide to run a mile. To do so, first you must run half a mile. But to run a half a mile, first you’ve got to run a quarter mile. And to run a quarter mile, you’ve first got to run an eighth of a mile. To run an eighth of a mile, you must run a sixteenth of a mile, and on and on.
Those fractions can always be divided in half again, yet each of those segments takes some finite time to run. But we’re humans, and we don’t have an infinite amount of time, so we can never run an infinite number of tiny little mile parts. Therefore, if you ever tried to run a mile, you would die before finishing. Maybe you should try the elliptical machine.
Of course there’s something wrong with Zeno’s argument—after all, it is possible to run a mile—but his logic is so convincing. How do we deal with it?
It turns out that any finite quantity, like a mile or a minute, can be expressed as an infinite sum of ever smaller parts. We can see why with a little algebraic trick.
Consider the series created by adding up all the half-steps in our mile run. We’ll give it a convenient one-letter name—x.
Now look what happens when we multiply both sides of the equation by 2:
Notice that we have almost identical expressions for 2x and x. In fact, the only difference between the two infinite sums is that 2x contains a 1. Otherwise, every fraction that appears in our expression for 2x shows up in our expression for x. So when we subtract one from the other, our infinite series of fractions completely cancels out:
And suddenly it turns out that our infinite progression of half-steps is precisely the same as the whole mile! Now we all have one less excuse not to exercise.
(Note: For those interested, there are two things we left out here. We did not formally define what it means for an infinite series to equal some sum, and we assumed that x exists. We do both with a little bit of calculus.)
~Todd Bryant
Math Instructor
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