Mathematical Journeys: Zeno's Paradox

Suppose I place you at one end of a long, empty room. Your task is to get to the door at the other end of the room. Simple, right? But what if I told you that this simple task is actually mathematically impossible?

Think about it – in order to traverse the whole room, you first have to get to the halfway point, right? You'll have to travel one-half of the way there. And before you can get to that halfway point, you have to travel one-quarter of the way there (halfway to the halfway point). And before you can get to the one-quarter point, you have to travel one-eighth of the way there (halfway to the quarter-way point). Since you have to go half of each distance before you can go the full distance, you'll never actually get anywhere. The task requires an infinite number of steps, and you can never complete an infinite number of steps since there will always be another one. Furthermore, in order to even start your journey you would need to travel a specific distance, and even the smallest of specific distances can be divided in half, giving you another step before that one. So you'd have to traverse infinity in order to go anywhere.

Make your brain hurt? Don't be ashamed; it's supposed to.

In fact, this is a very famous thought experiment called Zeno's paradox. Zeno's conclusion was that all motion must be an illusion, since travel over any finite distance can never be either completed or begun.

In practice, of course, this is not really a paradox – at some point your remaining distance will be so small that you cannot practically traverse half of it. Perhaps it would be down to a distance that is smaller than the length of your foot, so you are in effect already standing at both ends. The point of Zeno's paradox, though, is to illustrate the elusive nature of mathematical infinity.

Infinity tends to cause problems for math students when they expect it to behave like a number. The truth is, infinity is a slippery concept, one that can only really be comprehended obliquely because if you ever try to stare directly at it it will squirm away. The way to understand infinity is to acknowledge that it is a concept that theoretically exists, but that you will never personally see or pin down. Very much like the imaginary number i, infinity has a definition but it cannot be evaluated as a number. This is why we can only talk about a series approaching infinity – we cannot say that it reaches infinity, because infinity cannot ever really be arrived at. Wherever you are on the number line, infinity is always out of your reach.


However, whereas infinity/infinities are difficult to conceptualize, say vs. any countable set, i is easy to visualize on the complex plane. It is, however, both a position (at 0, i) (corresponding also to a vertical movement on the complex plane for addition & subtraction), a rotation (through pi/2) (corresponding to a multiplication operation by i), and a few other things depending on the mathematical operation.
That's such a neat concept, I've even had students spontaneously bring other students up to the board in free time during classes, so that they could see it too (I substitute teach in high school locally, also). Talk about making my day!
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