An infinite question: Why doesn’t .999~ = 1?

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Dear Cecil: Someone asked this question on another message board I go to. I’m asking it here because the other message board is mostly about video games and I figured someone here would have a better idea. 1/3 = .333~ 2/3 = .666~ 3/3 = .999~ But: 3/3 = 1 So why doesn’t .999~ = 1? Lint6, via the Straight Dope Message Board


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Cecil replies:

I won’t be the first to say this, Lint (boy, there’s a screen name that bespeaks an ambivalent self-image), but:

It does.

Before we get into the reasoning, some preliminaries:

Q: What does .333~ mean?

A: The squiggly line represents a repeating decimal, in this case an infinite string of 3s following the decimal point. The more common notation is to put a horizontal bar above a single 3 or to use an ellipsis, like so: .333 … The idea is to express fractions such as 1/3 that don’t divide out evenly, or “terminate,” when expressed as decimals.

Q: Why are we talking about this incredibly esoteric topic when there are so many more interesting things to discuss, like whether Rebecca Romijn-Stamos is wearing anything besides blue body paint in the new X-Men movie?

A: Because it’s cool, you wanker. Here, have a banana while the rest of us discuss.

Now then. Lint has already provided proof that .999~ = 1. From grade school math we know that .333~ = 1/3, .666~ = 2/3, and 1/3 + 2/3 = 1. Clearly .333~ + .666~ = .999~. Ergo, .999~ = 1.

The mind (yes, even mine) instinctively rebels at this conclusion. We readily concede that .999~ gets infinitely close to 1 — to put it in mathematical terms, 1 is the sum of the converging infinite series .9 + .09 + .009 + … But, we protest, .999~ never quite reaches that limit. If at any step we halt the progression to infinity to take a sum, we find that we remain separated from 1 by some infinitesimal amount.

But that’s just the point, the mathematicians say. When a decimal repeats ad infinitum, you never stop.

The lower primate in us still resists, saying: .999~ doesn’t really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense. The fraction 1/3 is an ordinary number, and .333~ is the same ordinary number; an infinite series of 3s simply happens to be the only way to express said number given the limitations of decimals. Granted, decimals let us express the quantity 1 without difficulty, but the process of infinite repetition produces the same result; .999~ is merely another way of saying 1. Likewise, pi is an ordinary number; it’s just a quirk of the real number system that we have to express it as 3.14159 etc (without ever repeating or stopping). Rational numbers, which by definition can be expressed as fractions, translate to repeating or terminating decimals; irrational numbers (like pi) never repeat or terminate in their decimal form.

If you’re still having trouble, consider another example involving a converging infinite series: Zeno’s paradox, proposed by the Greek philosopher Zeno in the fifth century BC. Suppose Achilles and a tortoise have a footrace. Achilles is ten times faster than the tortoise, but the tortoise has a ten-meter head start. In the time Achilles runs those ten meters, the tortoise crawls one meter. In the time Achilles runs that one meter, the tortoise plods another .1 meter. In the time Achilles runs that .1 meter, the tortoise lumbers ahead .01 meter. You get the picture. We seem to be reasoning ourselves to the conclusion that Achilles can never pass the tortoise.

But common sense says he does, and common sense is right. The expression 10 + 1 + .1 + .01 … is a converging infinite series whose sum is 11.111~ (or, to express it as a mixed number, 11 1/9). Common sense also tells us that Achilles does not merely approach this limit (as Zeno’s paradox would have us believe), but reaches and then passes it — i.e., that Achilles overtakes the tortoise at 11 1/9 meters. We thus see (I hope) that there’s nothing magical and unattainable about limits, and so no barrier to grasping that .999~ = 1.

Doesn’t that enhance your quality of life? Of course it does. Not that body paint doesn’t have its place, but there’s just no substitute for the pleasures of an infinite series.

Cecil Adams

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