1 = 0.999..., the Conclusive Argument
Many students encounter the equation
$$ 0.999... = 1 $$
and find this surprising. Or confusing. Or some even think that it just must be wrong.
The internet has probably been filled with tens of thousands of pages of people arguing about whether this equation is true.
Here I hope to settle the argument conclusively.
And my conclusive point is this:
0.999... is undefined. Until you define it.
We have to confront the fact that we will never know whether 0.999... = 1 until we first address:
What is 0.999...? What is its technical, rigorous definition?
If we do not take the slow and careful time to precisely define it, then we will not make progress in understanding it. I believe that all of the vast misunderstanding about this question stems from a failure to be very careful about this definition.
First note that we don't run into problems with, say, 1+1 = 2. Because we already know what this means! We can make it sensible, even physical, to place one object next to another object, and then say that the group contains two objects. That is at least good enough of a demonstration of the truth of the equation for our purposes.
We don't even really need to stress too much about definitions, to understand 0.2 + 0.3 = 0.5. After all, we can understand this as saying "If we take 2 tenths, and 3 tenths, then all together we have five tenths."
The problem comes from an infinity of digits. The expression 0.999... is not inherently defined. We have to give it meaning by coming to an agreement about what it expresses.
I am about to give my definition of this expression. If you disagree with that definition, then feel free to comment with a superior definition -- BUT, be warned: For your definition to be superior it will have to be at least as rigorously defined as my definition, referring only to thoroughly understood mathematical concepts.
Definition: 0.999... means the number which the sequence 0.9, 0.99, 0.999, ... becomes eventually close to.
The limit
The definition may be confusing. It is essentially the same thing as the idea of the "limit" in calculus. That is to say, the definition of 0.999... is
$$ 0.999... = \lim_{n\to \infty} a_n $$
where \(a_1 = 0.9\) and \(a_2 = 0.99\), and so on.
Let's get clear on the idea of the limit by a simple example. Consider the sequence 0.1, 0.01, 0.001, and so on. The limit of this sequence is equal to 0.
That is because the sequence "gets closer and closer to 0". There is technically more to the idea of the limit, but at least for our current purposes this is good enough.
Well, in the same sort of way, the sequence 0.9, 0.99, 0.999, ... also gets closer and closer to 1. And therefore the limit of the sequence is 1.
Since 0.999... merely means the limit of that sequence, and since that limit is equal to 1, then this is reason why
$$ 0.999... = 1 $$