Among the questions that can be asked from primary school all the way to university, but are not easily answered well, “does 0.999... actually equal 1?” is certainly a classic one. However, answering this question clearly is not easy. Often, the person being questioned becomes inadvertently confused, and some “amateur researchers” have even used this question to “create new mathematics.”
This article attempts to provide a relatively popular yet rigorous answer to this question.
What is Equality?
To answer whether 0.999... equals 1, we must first define “equality”! What counts as being equal? Does it really have to be written identically 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 both rigorous and generally accepted before we can judge it. Obviously, the following definition is acceptable to many:
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 that easy; we also need to define what 0 is! It is the same question as before: what is 0? Does it really have to be written identically to be 0?
This brings us to the foundation of mathematical analysis—0 is the non-negative number that is smaller than any positive number.
Indeed, one of the foundations of rigorous mathematical analysis is the definition of 0: 0 is the non-negative number that is smaller than any positive number. Readers might recall that the so-called limit theory is just a variation of this statement. Of course, this statement can have other forms of expression, but essentially, this is what it means. (Naturally, there are other topics like the completeness of real numbers, which we have not explicitly emphasized but have accepted by default.)
Now that we have the definition of 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....
Measure, Norm, and Isomorphism
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 have been 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, in many cases, conditions weaker than this are already “sufficient.” Therefore, by appropriately relaxing the definition of equality, different disciplines were born, such as the theory of functions of a real variable 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. What is the difference between these two functions in practice? In physics, integration is frequently used (since many problems involve solving differential equations, which involves integration). Obviously, the integration results for these two functions over the same interval are the same—a difference at a single point does not affect the integral result. In this sense, we consider f(t) and g(t) to be equal. In the theory of functions of a real variable, there is a more accurate term for this, called “equal almost everywhere.” One of the main contents of real analysis is the study of things that are equal almost everywhere. So-called equality almost everywhere means that if the “area” (more accurately, the measure) of the part where two functions differ is 0, then the two are considered equal (from a physical perspective, because such a small difference basically does not affect physical results).
By the time 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 completely abstract. It could be Euclidean distance, an integral, a limit, etc. For two “things” to be equal, it only requires the defined “distance” to be 0.
In algebra, the concept of “isomorphism” appears, which is a broader concept than equality. Isomorphism tells us that isomorphic things possess similar (algebraic) properties, so one only needs to study one of them. In this sense, it is also essentially a concept of equality, because it effectively says: peeling off the surface layer, the inner essence is the same. Of course, isomorphism is not a concept unique to algebra; it also exists in analysis, such as isometric isomorphism.
Back to the Starting Point
We have wandered quite far, from a primary school problem to functional analysis and algebra. The reason for expanding this topic is mainly to let interested readers know that the reason the doubt “does 0.999... equal 1” arises is primarily due to the lack of a clear definition of “equality” or an insufficient understanding of the definition. Once the concept of “equality” is well-defined, the question can be answered clearly. Otherwise, obsessing too much over this problem without finding the root cause will hinder 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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