On "Does 0.999... Equal 1?"

从“0.999...等于1”说开来 苏剑林 · 2015-07-21 · Mathematics function analysis

Among the questions that might be asked from elementary school all the way through university—questions that are easy to ask but hard to answer well—"does 0.999... really equal 1?" is certainly a classic. Yet giving a clear answer is not so easy, and quite often the very act of asking the question ends up confusing people, to the point that some amateur mathematicians even claim to have "invented new mathematics" out of this issue.

This post attempts to give a reasonably accessible yet rigorous answer to this question.

What does "equal" mean?

To answer whether 0.999... equals 1, we first need to define "equal"! What counts as equal? Do two things really have to be written in exactly the same form to be called equal? If that were the case, then even 2−1 wouldn't equal 1, since "2−1" and "1" don't look the same at all.

Clearly we need a definition of "equal" that is both reasonably rigorous and widely accepted, so that we can actually judge equality. The following definition is one that most people would accept:

$a = b$ if and only if $|a-b|=0$.

Based on this definition, we now want to work out $1-0.999...\stackrel{?}{=}0$. But answering this isn't quite so easy either—we also need to define what 0 is! Again, the same question arises: what exactly is 0? Does it really have to be written in exactly the same form to count as 0?

This is where we arrive at the foundation of mathematical analysis: 0 is the non-negative number that is smaller than every positive number.

Indeed, one of the cornerstones of rigorous mathematical analysis is precisely this definition of 0: 0 is the non-negative number smaller than every positive number. Readers might pause and recall that the entire theory of limits is really just a variation on this one statement. Of course, this idea can be phrased in other ways, but in essence it always comes down to the same thing. (There is also, of course, the matter of the completeness of the real numbers and so on, which we haven't explicitly emphasized here but have implicitly taken for granted.)

Now that we have a definition of 0, we can go ahead and compute $1-0.999...$. Clearly, it must be smaller than any positive number we can name, so it can only be 0. Therefore $1=0.999...$.

Measure, norm, isomorphism

Above, we defined the equality of two numbers as the absolute value of their difference being 0. From this, a whole series of definitions of equality can be derived. For instance, for two functions $f(t)$ and $g(t)$ defined on $\mathbb{R}$, we say

$f(t)=g(t)$ if and only if $|f(t)-g(t)|=0\,(\forall t\in \mathbb{R})$.

However, this definition is usually too strict. On one hand, this condition is very hard to satisfy—especially in physical phenomena, where it's almost impossible to find two functions that agree at every single point. On the other hand, in many situations something weaker than this condition is already "good enough." As a result, relaxing the definition of equality in various appropriate ways has given rise to entire fields, such as real analysis and functional analysis.

Consider the following two functions:

$$\begin{aligned} &f(t)=e^t,\,t\in [0,\infty)\\ &g(t)=\left\{ \begin{aligned} &e^t,\,t\in(0,\infty)\\ &0,\,t=0 \end{aligned} \right. \end{aligned}$$

Clearly, $f(t)$ and $g(t)$ differ only at $t=0$, and are identical everywhere else. So what practical difference does this make? In physics, the operation used most often is integration (since solving differential equations, which is common in physics, involves integration). And clearly, these two functions have the same integral over the same interval—differing at a single point doesn't affect the result of integration at all. In this sense, we regard $f(t)$ and $g(t)$ as equal. In real analysis, this has a more precise name: "equal almost everywhere." One of the main themes of real analysis is precisely the study of things that are equal almost everywhere—meaning that if the "area" (more precisely, the measure) of the set where two functions differ is 0, then we consider them equal. (From a physical point of view, such a negligibly small difference essentially has no effect on the physical outcome.)

By the time we reach real analysis, the notion of equality has been loosened even further. In real analysis, one can define one's own notion of "distance" (a norm), and this distance doesn't even need to have any geometric meaning—it can be entirely abstract. It could be Euclidean distance, or it could be defined via an integral, or via a limit, and so on. And two "things" are considered equal as long as the "distance" defined between them is 0.

In algebra, meanwhile, we encounter the notion of "isomorphism," which is an even broader concept than equality. Isomorphism tells us that isomorphic objects share similar (algebraic) properties, so it suffices to study just one of them. In this sense, isomorphism is fundamentally still a notion of equality, because what it really says is: strip away the surface layer, and what's underneath is the same. Of course, isomorphism is not a concept unique to algebra—it also appears in analysis, for instance in the form of isometric isomorphism.

Back to the starting point

We've wandered quite far, from an elementary-school question all the way to functional analysis and algebra. The reason for taking this digression is mainly to let interested readers see that the puzzlement over "does 0.999... equal 1?" arises chiefly because "equal" was never given a sufficiently clear definition, or because the definition wasn't properly understood. Once we've clearly defined the notion of "equal," this question can be answered without ambiguity. Otherwise, getting overly tangled up in this question without identifying the real crux of the matter only ends up hindering our progress in learning mathematics.

As an aside, one can see that mathematical analysis, real analysis, and even functional analysis are, in their own ways, continually reinterpreting what "equal" actually means.