By 苏剑林 | July 21, 2015
Among the questions that can be asked from primary school all the way to university, yet are not easy to answer well, "Does $0.999...$ actually equal $1$?" is certainly a classic. However, providing a clear answer to this question is no easy feat; often, the person being asked becomes inadvertently confused, and some "fringe scientists" have even used this problem to "create new mathematics." This article attempts to provide a relatively accessible yet rigorous answer to this question.
To answer whether $0.999...$ equals $1$, we must first define "equality"! What counts as being equal? Does it really have to be written exactly the same way to be called equal? If that were the case, then $2-1$ would not even equal $1$, because $2-1$ and $1$ look different.
Clearly, we need to provide a definition of "equality" that is mathematically rigorous yet universally accepted before we can judge equality. Obviously, the following definition is acceptable to many people:
$a = b$ if and only if $|a-b|=0$.
Based on this definition, we need to calculate whether $1-0.999...\stackrel{?}{=}0$. Answering this is still not quite simple; we must also define what $0$ is! Returning to the previous question: what counts as $0$? Does it really have to be written exactly as $0$ to be $0$?
At this point, we arrive at the foundations of mathematical analysis—$0$ is the non-negative number that is smaller than any positive number.
Indeed, one of the foundations of the entire rigorous system of mathematical analysis is the definition of $0$: $0$ is the non-negative number smaller than any positive number. Readers might recall that the so-called theory of limits is essentially a variation of this statement. Of course, this statement can be expressed in other forms, but its essence remains the same. (Naturally, there are other topics like the completeness of real numbers which we haven't explicitly emphasized, but we have implicitly accepted them.)
Now that we have a definition for $0$, we can calculate $1-0.999...$. Clearly, it must be smaller than any positive number we can provide, so it can only be $0$. Therefore, $1=0.999...$.
Above, we defined the equality of two numbers as the absolute value of their difference being $0$. From this, a series of definitions for equality are derived. For example, for two functions $f(t)$ and $g(t)$ on $\mathbb{R}$, equality is defined as:
$f(t)=g(t)$ if and only if $|f(t)-g(t)|=0\,(\forall t\in \mathbb{R})$.
However, this definition is generally too strict. On one hand, this condition is difficult to satisfy, especially in various physical phenomena where it is almost impossible to find two functions that are equal at every single point. On the other hand, many situations weaker than this condition are already "good enough." Consequently, appropriately relaxing the definition of equality gave birth to various different disciplines, such as real analysis and functional analysis.
Consider the following two functions:
$$ \begin{aligned} &f(t)=e^t,\,t\in [0,\infty)\\ &g(t)=\left\{ \begin{aligned} &e^t,\,t\in(0,\infty)\\ &0,\,t=0 \end{aligned} \right. \end{aligned} $$
Obviously, $f(t)$ and $g(t)$ differ only at $t=0$, and are identical everywhere else. What is the difference between these two functions in practical use? In physics, where integration is frequently used (since physics often involves solving differential equations, which in turn involves integration), these two functions clearly yield the same integral results over the same interval—a difference at a single point does not affect the outcome of an integral. In this sense, we consider $f(t)$ and $g(t)$ to be equal. In real analysis, there is a more precise term for this, called "equal almost everywhere." One of the primary subjects of real analysis is the study of things that are equal almost everywhere. So-called "almost everywhere equality" means that if the portion where two functions differ has an "area" (more accurately, a measure) of $0$, then the two are considered equal (from a physical perspective, such a tiny difference basically does not affect physical results).
When we reach functional analysis, the concept of equality is relaxed even further. In functional analysis, one can define a custom "distance" (norm); this distance may not even have geometric meaning and can be entirely abstract. It can be Euclidean distance, or given by an integral, or a limit, etc. For two "things" to be equal, we only require the defined "distance" between them to be $0$.
In algebra, we encounter the concept of "isomorphism," which is a much broader concept than equality. Isomorphism tells us that isomorphic things possess similar (algebraic) properties, and therefore one only needs to study one of them. In this sense, it is essentially also a concept of equality, because it effectively says: peeling away the surface layer, the inner essence is the same. Of course, isomorphism is not a concept exclusive to algebra; it also exists in analysis, such as isometric isomorphism.
We have wandered quite far, from a primary school problem all the way to functional analysis and algebra. The reason for branching out this topic is mainly to let interested readers know that the reason the doubt "whether $0.999...$ equals $1$" arises is primarily due to a lack of a clear definition of "equality" or an insufficient understanding of that definition. Once the concept of "equality" is well-defined, this question can be answered clearly. Otherwise, excessive obsession with this problem without finding the root of the issue is detrimental to our progress in learning mathematics.
Incidentally, we can see that mathematical analysis, real analysis, and even functional analysis are constantly reinterpreting the meaning of equality.
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,
author={Su Jianlin},
year={2015},
month={Jul},
url={\url{https://kexue.fm/archives/3402}},
}