Take the number 0.999..., with the 9s repeating forever. Does it equal 1? Many mathematicians, myself included, would argue yes, even though it is a bit counterintuitive. I will attempt to take a deeper look into this, and list a few proofs of this, in ascending levels of depth.
Proof by Wishful Thinking
It's true because I want it to be true. Here, I'll wave my magic wand. Mwahaha, I'm right.
That doesn't convince you? Let's move to the next proof.
Proof by Unjustified Patterns
For any two distinct numbers, you can find a number between them. A number between 0 and 1 is 1/2; between 0 and 1/2 is 1/4; between 0 and 1/4 is 1/8; and so on. Is there a number between 0.999... and 1? I didn't think so. So they're equal. Mwahaha, I'm right.
But why does there need to be a number in between? Let's move to the next proof.
Proof by Implying Something Equivalent
You agree that 1/3 is equal to 0.333..., right? Right? Well since you do, just multiply both sides by 3. 1/3 times 3 is 1, and 0.333... times 3 is 0.999... Mwahaha, I'm right.
Saying that 1/3=0.333... is the same as that 1=0.999..., so the proof rests on previous assumptions? Let's move to the next proof.
Proof by Algebra
Let's do some algebra. Let 0.999...=x.
0.999...=x
We multiply both sides by 10.
9.999...=10x
We subtract 0.999 from the left and x from the right. From the first equation we know these are equal.
9=9x
We divide both sides by 9.
1=x
Mwahaha, I'm right.
You still don't agree? But you're not allowed to disagree. I did algebra, and you can't argue with algebra. Onward!
Logical Axioms
Before we formally start the proof, let's look at this imaginary dialogue:
Elizabeth Warren supporter: I believe that Elizabeth Warren's policy knowledge would make her the best president among the Democratic field.
Joe Biden supporter: I don't agree with that; Joe Biden has the best chance to defeat Donald Trump.*
They could go round and round arguing this point, but, chances are, neither will convince each other. Why is this? The answer rests in the fundamental logical axioms, the core beliefs, that each of these two people have.
The Warren supporter might have this axiom:
"I support the candidate that would make the best president."
Whereas the Biden supporter might have this axiom:
"I support the candidate that has the best chance to win the general election."
The Warren supporter is not disputing Biden's chances to defeat Trump, and the Biden supporter is not disputing who would make a better president. They're disputing the logical axiom that is relevant here, but they don't realize it.
And if this discussion doesn't tie back to these axioms, no one will be convinced of anything. Both sides will have internalized that their axiom is correct. However, if you grant one of these axioms as fact, then the other case is much harder to defend. If you grant the Warren axiom, then it's hard to make the case for Biden. (Polling suggests that if the candidates could bypass the general election and just become president, Warren leads. Also, electability is Biden's primary selling point in all of his ads.) If you grant the Biden axiom, it's hard to make the case for Warren. (Her favorable ratings, while good among Democrats, are underwater with the general electorate.)
A Second Example
As another example of this principle, an atheist might believe that there is no hard evidence of a god, so you shouldn't believe in one. On the other hand, an evangelical Christian** might say that the Bible tells us that there is a god, so we should have faith that one exists. This atheist and this Christian could argue about this for a while, but chances are, neither will convince each other. Why is this? Let's examine the axioms at work.
The atheist would have this axiom:
"I need to have hard evidence before I accept that there is a deity."
Whereas the Christian would have this axiom:
"It's okay for me to have faith in something without having hard evidence."
Again, if you grant one side's axiom, the other side's case looks much worse. The point is, it's important to debate the axioms, instead of acting like your axioms are true, period.
The Last Proof
So what are the core axioms in the discussion of whether 0.999...=1? Here's the axiom that I think people who believe that 0.999...=1*** have:
"It is meaningful to add up an infinite number of numbers."
And people who don't:
"It is not meaningful to add up an infinite number of numbers."
If you grant that it's meaningful to consider this, then you were probably convinced by the algebra proof. But if you don't grant this, then while you may not have a rebuttal for, say, the algebra proof, you might still not believe it.
Zeno's Paradox
So let me address the axioms at work here. I believe that it is meaningful to add up an infinite number of numbers. To demonstrate why, I will discuss Zeno's Paradox. Attributed to the Greek philosopher Zeno, who lived over two thousand years ago, it states something along the lines of:
"To walk across a room, I must first walk halfway. Then, once I'm there, I must go halfway again, walking a quarter of the way over all. Next, I must go halfway again, an eighth of the way overall; and so on. I can never actually get there, because I'm always going halfway first. Yet I am clearly capable of walking across a room. How can this be?"
So let's take a look at this paradox. I believe that there are three main statements it makes:
A. To walk across a room, I must walk halfway an infinite number of times.
B. Doing something an infinite number of times can't be done in a finite amount of time.
C. I can walk across a room.
All of these statements seem reasonable. But I hope you'll agree that they can't all be true, since they contradict each other. Statement C cannot be false. (I have empirical evidence.) So it's down to Statements A and B.
Statement A is true because, in the process of crossing a room, I have to go past every point in the room, so it's legitimate that I pass all of these halfway points. After all, however long the room is, you can divide it by 2. And continue this process. It's legitimate logic.
So we are forced to conclude the Statement B is false, because if it were true, either Statement A or Statement C would have to be false, since they contradict each other. Therefore, it doing something an infinite number of times can be possible.
Or, in other words, adding up an infinite number of numbers can get you somewhere.
You might still disagree with me, and that's OK. But if you disagree, can we at least talk about the axioms?
–beautifulthorns
*I would disagree with this person, as you can see in my post "Don't Settle for Biden."
**This isn't specific to Christianity; this is just an example. By the way, I'm certainly oversimplifying for this example, and my objective is not to offend. I apologize if I did.
***I think we can all agree that saying that 0.999...=1 is the same as saying that
9/10+9/100+9/1000+...=1, which is clearly an infinite sum.