When Infinity Has an End

Somewhere out in the Utah desert, among outcrops of piñon pines and juniper trees, a cottontail rabbit twitches its nose under a sagebrush. The rabbit is a short run, only a second or two, from its rabbit hole. Every other muscle of the rabbit is sandstone-still. It knows a coyote is near, that the coyote’s golden eyes are watching. The rabbit can hear and smell – yes, even see – that hunter with pointy ears and teeth, whipcord muscles and bushy tail. The rabbit’s plan is to bolt, bolt, bolt to the hole. But only at the last panicky moment will it run, only when it’s sure it has been seen.

That moment happens, and as we expect, the rabbit sprints away, flashing its cottony tail. It runs in a line so straight it would make Archimedes proud. It streaks away as fast as it possibly can, grey-flecked sides heaving. But the coyote, of course, is speedy too.

Does the rabbit make it, or does the coyote get some fast food?

If I told you the rabbit made it, would you believe me? It’s reasonable. How, then, would you explain this puzzle? In the process of running between the sagebrush and the rabbit hole, the rabbit passes a point exactly halfway between the two. And then it passes another point exactly half of the remaining distance. Then half again, again, again. Wouldn’t this process of cutting the remaining distance in half go on forever because there is always a distance, no matter how small, to cut in half? Wouldn’t that mean the rabbit never reaches the hole?

So the coyote must catch the rabbit. It’s reasonable. Unfortunately, for the same reason, the coyote can’t reach the rabbit either: there is always a distance between the coyote and the rabbit to divide in half. So how can anything ever go anywhere in this world and actually arrive there???

What if the rabbit was heading for a point twice the distance to its rabbit hole? In this case it would drop down the hole at the first halfway point, and it would get away for sure! All anything would have to do is aim for twice the distance it actually means to cover. But this isn’t reasonable. No rabbit really does this. Besides, no matter how you double or triple things, there is still that distance between the sagebrush and the rabbit hole that can be divided in half infinitely.

In this puzzle, the cutting in half is an infinite process, and infinity has no end. But what if there are some kinds of infinity that actually do “end”? What if the coyote and rabbit puzzle is this kind of infinity?

Take a look at this classic math proof: as you may know, one way to express nine tenths is 0.9; ninety-nine one-hundredths is 0.99. The more nines you add, the closer the value gets to 1. If the 9’s were to continue forever, we would call it “.9 repeating”, which, we may suppose, is still almost, but not quite, equal to 1.

But we would suppose wrong. It’s not almost equal to 1; it’s exactly equal to 1.

If we started with ten chocolate kisses and took away one chocolate kiss, we would be left with nine chocolate kisses, right? In the same way, if we had ten “.9 repeatings” and took away one , we would be left with nine “.9 repeatings”. If we could show that those nine “.9 repeatings” (all taken together) were exactly equal to 9, wouldn’t each “.9 repeating” have to be exactly equal to 1?

If you like to play with numbers, here is the proof:

10  x  “.9 repeating” = 9.9 …

– 1  x   “.9 repeating”= 0.9 …

9   x   “.9 repeating”  =  9

Therefore, “.9 repeating” = 1

So, why do some infinite series of numbers have exact values like this one? It’s because the digits repeat. If the digits didn’t repeat in a regular pattern infinitely, they wouldn’t have an exact value. Oddly enough, these infinitely repeating numbers are called “rational,” even though they seem to defy reason. (Infinite numbers that don’t repeat are called irrational; it’s ironic, because you would expect an infinite number not to have an exact value, wouldn’t you?)

If you don’t like to play with numbers, think about this:  “.9 repeating” is exactly equal to 1 precisely because the 9’s are infinite, precisely because the nines don’t end at any point short of one. And no matter how much we divide the distance, the rabbit actually doesn’t stop at any point short of its rabbit hole either.

So, when is there an end to infinity? It turns out to be pretty often!

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