What’s in a name? that which we call a rose
By any other name would smell as sweet…—William Shakespeare
In our daily lives, we are accustomed to giving multiple names to a single object. My wife calls me Xander, while my brothers and sisters call me Alex, and my students call me Mr. H. The titles, honorifics, and diminutives that people use to address me do not alter my fundamental self; they do not imply that multiple people inhabit my body, nor that there are multiple mes. Somehow, we are able to tolerate the ambiguity of calling the same person by multiple names without too much confusion.
Moreover, we seem to recognize that changing an object’s name does not change the underlying object. We seem to understand that people and objects have a kind of Platonic ideal associated with them, no matter what we call them. Romeo is Juliet’s lover, no matter his family name; and a rose has a sweet smelling blossom, no matter how we label it.
Even in mathematics, many labels are often used to describe a single object. Take, for example, the number one. We can right a Hindu-Arabic numeral to represent this number: 1. The Romans gave it a different name: I. If we had a very precise scale, we might use another symbol: 1.000. Or we could get the quantity by combining four quarters: \(\frac{4}{4}\). All of these symbols are different, by they equate to the same abstract mathematical concept: one.
So far, I have said nothing controversial. The various symbols and names for the number one that I listed above should be familiar to most people, and we should all be able to agree that they are equivalent. Now consider the following symbol: \(0.99\overline{9}\).1The bar over the trailing 9 indicates that we keep writing 9s forever. Thus the symbol is a zero, followed by a decimal point, followed by an infinite number of 9s. The symbol is read “zero point nine nine nine bar.”
When confronted with this string of numerals, most mathematicians would instantly recognize it as yet another name for one. But, for some reason, many laypeople refuse to accept the equality \(0.99\overline{9}=1\).
As is customary when making a claim in mathematics, I will prove that these two symbols are, in fact, equal. In fact, I will provide not one, but three moderately rigorous proofs of this identity. Now, these proofs are going to be a bit mathy. There is some notation that you may have to mull over, and if you are not a terribly mathy person, then they may not be all that convincing. If this is the case, please simply try to keep in mind that many symbols can refer to the same thing. In short, a one by any other name…
Proof 1
Consider the quantity \(\frac{1}{3}\). When written as a decimal, we get \(\frac{1}{3} = 0.33\overline{3}\). Now consider multiplying both sides of this equation by 3. That is, consider that \(3\times\frac{1}{3} = 3\times 0.33\overline{3}\). Evaluating the left side of the equation, we have \(3\times\frac{1}{3} = \frac{3}{3} = 1\). Now consider the right side of the equation. Recall that when we multiply by 3, we are taking our original quantity, and adding it to itself three times. Using the notation that we learned in elementary school, it would look like this:
Clearly, in every decimal place, we would have \(3+3+3 = 9\). Hence \(3\times 0.33\overline{3} = 0.99\overline{9}\). Since we still have equality between the right and left sides of the equation, this gives us \(1 = 0.99\overline{9}\).
Proof 2
Let \(x = 0.99\overline{9}\). If we multiply both sides of this identity by 10, we will end up with the same string of digits on the right, but the decimal will move one place to the right. Thus \(10x = 9.99\overline{9}\). From each side of the equation, we will subtract \(x\): \(10x – x = 9.99\overline{9} – 0.99\overline{9}\). This gives us \(9x = 9\). Solving for \(x\), we have \(x = 1\). Substituting this value into the original identity gives us the desired result, which is \(1 = 0.99\overline{9}\).
Proof 3
This final proof is the most notationally intensive, but is also (in some ways) the most illuminating. It uses some properties of limits in order to get at the same identity. Note that \(0.9 = 1-\frac{1}{10}\). Similarly, \(0.99 = 1-\frac{1}{100}\), and so on. This means that we can rewrite \(0.99\overline{9}\) as
\(0.99\overline{9} = 1-\frac{1}{100\ldots} = 1-\frac{1}{10^\infty}.\)
While I think that the meaning of the last expression is clear, let me try to explain it. The idea is that the denominator of the fraction is a one followed by an infinite number of zeros. That being said, no mathematician would ever actually write something like $10^\infty$. Infinity (i.e. \(\infty\)) is not actually a number—it is really more of a concept. We cannot actually raise a number to the power of infinity. Normally, a mathematician would use limit notation to convey the idea. A mathematician would write
\(0.99\overline{9}=\lim_{n\to\infty}\left(1-\frac{1}{10^n}\right)\).
What this means is that, in some sense, we are going to let \(n\) get bigger and bigger and bigger. Eventually, \(n\) is going to be bigger than any number that we could possibly describe. Once it gets to be that big—that is, once we have let \(n\) grow to infinity—we are going to examine what happens.
First off, one of the nice things about limits is that we can break up sums and differences without changing the result. So, we end up with
\(\lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = \left(\lim_{n\to\infty} 1\right) – \left(\lim_{n\to\infty}\frac{1}{10^n}\right)\).
In the first term of the expression, there are no \(n\)s. This makes life easy. When \(n=1\), the first term will be 1. When \(n=100\), the first term will be 1. When \(n=10^{100}\), the first term will still be 1. No matter how big \(n\) gets to be, the first term will always be 1. So we can simplify the expression a little bit:
\(\left(\lim_{n\to\infty} 1\right) – \left(\lim_{n\to\infty}\frac{1}{10^n}\right) = 1-\left(\lim_{n\to\infty}\frac{1}{10^n}\right)\).
Now all we have to do is deal with the second term in the expression. Basically, as \(n\) gets larger and larger, the fraction with \(10^n\) in the denominator will get smaller and smaller. If \(n\) gets so large as to be infinite, the denominator will be infinite, and the fraction will simplify to zero. That is, \(\lim_{n\to\infty}\frac{1}{10^n} = 0\). This gives us the final identity
\(1-\left(\lim_{n\to\infty}\frac{1}{10^n}\right) = 1-0 = 1\).
Therefore \(0.99\overline{9} = 1\).