Unity and Infinity

In recent weeks, Yahoo! News, that worthy and reliable news source, ran an article purporting to list the eleven most beautiful mathematical equations (ten having been too few). Alas, of the list, five are not mathematical, and one is not an equation. However, among the remaining five is the following:


where the 9’s are understood to keep going forever. In my experience, this equation prompts varied responses. The most common is sheer disbelief. “How could these two be equal?” people ask. The answer to this is that, if they aren’t equal, then we should be able to find their difference:

1 - .999\ldots = \;?

But these doubters of the mysteries of unity and infinity can’t state the answer. Generally they’ll admit that it must be a number smaller than every other number, but they still insist that it can’t be zero. What’s interesting is that they’re generally comfortable with the equation

\frac{1}{3} = .333\ldots

After all, a calculator tells us it’s so. But if we multiply both sides by 3, then don’t we get the original?

It isn’t surprising that our equation seems a little dubious. What this really stems from is not understanding what infinite decimal expansions like .333… and .999… actually mean. Now, terminating decimal expansions are easy to understand. For instance,

.9 = \frac{9}{10} \;\;\;\;\;\; .99 = \frac{9}{10} + \frac{9}{100} \;\;\;\;\;\; .999 = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000}

But by what right can we write down something like

.999\ldots = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots

where the sum just keeps going and going? How can we add infinitely many numbers? Does it even make sense to talk about something like that?

To answer this, let’s consider the sequence of partial sums:

.9 = \frac{9}{10} \;\;\;\;\;\; .99 = \frac{9}{10} + \frac{9}{100} \;\;\;\;\;\; .999 = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000}

and so on. Notice that each term in the sequence is a bit larger than the preceding, and that all are less than 1:

.9 < .99 < .999 < .9999 < \cdots < 1

Modern mathematics defines the meaning of the infinite decimal expansion as follows: the value of the expansion is the number x such that, for any positive number ϵ, no matter how small, our sequence of partial sums is eventually in between x – ϵ and x. Put in another way, the value is x if, for any small number ϵ, the difference between x and the nth partial sum is smaller than ϵ if n is large enough.

So, you see, the admission that the difference between 1 and .999… is a number “smaller than every other number” means that .999… is in fact equal to 1.