## Does 0.999… Really Equal 1?

One of the most oft-asked questions in K-12 math is:

Does zero point nine repeating really equal one?

And here’s a longer answer, which the question really does deserve, because resolving the core issue that this question raises is, metaphorically, like moving mathematically from the 18th to the 19th Century.

The question asks whether two things are equal.  But, before we can answer whether two things are equal, we must know what exactly the two things are.  In this case, the two things are the number 1 and “zero point nine repeating,” which is also written “$0.\overline{9}$.”  The number 1 is not the source of confusion.  It’s the $0.\overline{9}$ that stirs debate, and it does so because it involves the concept of infinity.

Struggling with the meaning of $0.\overline{9}$ is grappling with the difference between the finite and the infinite. With the invention of Calculus, mathematicians began constructing arguments that used the concept of infinity in various ways, such as in the vague, undefined notion of “infinitesimal quantities.” Perhaps Euler’s argument that $\frac{1}{2}$ could reasonably be substituted for the expression

$1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + \dots$

gives us the most memorable quote of that era that is relevant to this post (see section 10 of Euler’s 1760 paper De Seriebus Divergentibus).

The desire to eliminate the unconvincing nature of such 18th century arguments gave rise to a new level of precision and rigor that set the stage for modern mathematics.  This history is summarized very nicely in Tom Archibald’s article The Development of Rigor in Mathematical Analysis, Section II.5 of the marvelous Princeton Companion to Mathematics.  Today, there is unanimous agreement on the relevant definitions pertaining to what $0.\overline{9}$ is.  From these definitions, the definitive answer of “yes” can be obtained, and here’s one way to go about it:

By definition, the decimal number $0.\overline{9}$ stands for the infinite series

$\frac{9}{10}+\frac{9}{10^2}+\frac{9}{10^3}+\ldots+\frac{9}{10^n}+\ldots.$

This is, in fact, an infinite geometric series, and if you are comfortable with those, you could satisfy yourself that this particular geometric series sums to 1. However, I’ll press on and establish this fact from first principles.

By definition, the value of the infinite series is equal to the limit, as $n$ tends to infinity, of the finite geometric series

$\frac{9}{10}+\frac{9}{10^2}+\frac{9}{10^3}+\ldots+\frac{9}{10^n}=\frac{9}{10} \frac{1 - (\frac{1}{10})^n}{1-\frac{1}{10}} = 1-(\frac{1}{10})^n.$

By definition, $\lim_{n \to \infty} a_n = L$ if and only if for all $\epsilon > 0$, there exists $N$ such that $\vert a_n - L \vert < \epsilon$ for all $n > N$.  So, if $\epsilon > 0$, let $N$ be the least integer greater than $1/\epsilon$.  (This choice of $N$ is overkill; you could take $N = \log_{10} \epsilon^{-1}$, but I don’t want to assume that the logarithm has been defined.) With this choice of $N$, we can see that if $n > N$, then $10^n > 10^N > N > 1/\epsilon$.  Therefore, $10^{-n} < \epsilon$. So, for $n > N$, our finite geometric series is within $\epsilon$ of $1$ which establishes, once and for all, that $0.\overline{9} = 1$.

There’s really no point in arguing about whether two things are the same before the things being compared are clearly defined. Understanding this can save a lot of time and not just in math.  I wish Girls’ Angle could have a penny for every hour that was lost by people arguing over things that were never clearly defined. Finally, one warning: while rigorous proof settles the question, this is not the same as saying that understanding is equivalent to rigorous proof.  For more, see Terence Tao’s There’s more to mathematics than rigour and proofs.

For another take on this topic, see Elisenda Grigsby’s WIM Video.

p.s. There are shorter, valid proofs of the fact that $0.\overline{9} = 1$, such as the one where you let $x = 0.\overline{9}$ and then observe that $10x - x = 9$ which has the unique solution $x = 1$.  However, such proofs assume that certain manipulations of infinite series are valid.  I wanted to give a proof that didn’t make such assumptions.

p.p.s. You could debate whether the definitions should be changed.  In this case, the definitions have been well-established, so if you think you have interesting non-equivalent definitions, my advice would be to introduce your idea as a new concept rather than try to replace the standard definitions.  If your definitions really are more suitable, they’ll replace the old definitions in time.