Hello Daniel,

I think you are looking at some of this the wrong way.

First of all, there is a matter of notation. You start out with the idea that the ellipsis and the vinculum are different. But these are just different notations for a repeated decimal. Whatever 0.999... and 0.9̅ mean, they mean the same thing.

You also start out with the idea that 0.999... means something in itself, and we must discover what it means. But this isn't true. 0.999... is a sequence of symbols, and we

*define* it to mean something. And what we define it to mean is the value of the series 9/10 + 9/100 + 9/1000 + ...

So the question is, what is the value of this series? It is the limit of the partial sums, as you yourself say here:

There is no denying that the limit of .999... (as n approaches infinity) is 1. In fact it is, as proven above.

As Kreuzritter points out, this value of a series is

*defined* to be this limit.

My problem with that is, "defining" the sum as the limit seems to violate the principle of non-contradiction. Because a limit and a sum are two separate things. The limit is the value that the partial sum approaches as you make i really big, whereas the sum is the value you would get were you to actually add them all up. Clearly the limit is 1. But the series has no sum, since it's impossible to add up all the terms (there will always be another term to add).

What is 2 x 3? 6, obviously. But why? Because multiplication is repeated addition, and 2 added to itself 3 times is 6. Now, can you multiply 2.5 and 3.5? It would seem not, because you cannot add 2.5 to itself 3.5 times. You cannot do anything 3.5 times. And yet, you know this problem has an answer. You learned to do problems like this in elementary school. And the way to do it is so easy that you didn't even notice they changed the definition of multiplication on you to make it possible. They changed the definition of multiplication from repeated addition to a kind of scaling. But the great thing about it is that when one of your factors in a natural number, this scaling corresponds exactly to repeated addition. So it wasn't really that the definition of multiplication was

*changed* so much as it was

*generalized* in a useful way, with the special case of it you knew at first still perfectly valid.

So here it is not that the definition of a sum has been changed to something contradictory; it has been generalized to something useful that still includes the old understanding of a sum. You are right to point out that this is not a "sum" in the simple sense of repeated counting - it is something more general that includes that simple sense.

And even theoretically if you could somehow add them all up, it would still be less than 1. Because if you move only halfway across the room, and halfway again, and halfway again, you're never going to reach the other side.

And yet you do reach the other side. You can get up and do it right now. You really can cross the room, even though at some point during your trip you were half way across the room. And then another quarter across the room. And then another eighth across the room. And every other fraction in that series.

It does seem to make sense. But I am thinking there must be a trick to it. Looks a lot like this similar (yet completely unrelated) proof I once saw:

Start with 1. Subtract 1 from it. Then add 1. Then subtract 1. Keep doing that forever. What do you end up with?

In reality, it all depends on whether the series ends on +1 or on -1. Because suppose the series was not infinite. We start with 1 which is 1. Then 1-1, which is 0. But 1-1+1 is 1. And 1-1+1-1 is 0 again. And 1-1+1-1+1 is 1. So it basically keeps bouncing back and forth between 0 and 1.

But the series is infinite. Which means it has no answer. I mean, really, how can there be an answer? No answer could possibly make any sense, because no matter how many iterations you have, the series does not approach anything.

But moderns don't like that.

And some of them think that all religions are equal, so they need to come up with a compromise. "How about 1/2?", they say. Because 1/2 is half way between 0 and 1.

1/2 is not the value of this series because someone shrugged his shoulders and came up with a compromise between 0 and 1. Certainly not a modernist attempting to do the same the thing with religions. This is Grandi's series, and Grandi was an Italian monk and priest.

This is again a matter of context, like with the multiplication before. Does 2.5 x 3.5 exist? It depends on the context. In the context of multiplication as repeated addition, then absolutely not. And that's a perfectly fine answer. But in the context of multiplication as a more general kind of scaling, then it does exist, and that's a perfectly fine answer too. Does the sum of an infinite series exist? In the context of sums as repeated counting, absolutely not. And again, that's a perfectly fine answer. But there is also a more general context, with sums as the limit of the partial sums, where the sum of an infinite series does exist.

So, can we find an even more general context than that, where we can assign a value to a divergent series like Grandi's series, where there is no limit of the partial sums? Again, yes. One example of this is Cesàro summation, where the value of a series is the limit of the

*average* of the partial sums. Let's work that out for Grandi's series:

The series:

1 -1 + 1 -1 + 1 - ...

The partial sums:

1, 0, 1, 0, 1...

The averages of the partial sums:

1, 1/2, 2/3, 2/4, 3/5...

And the limit of these is 1/2. And like before, the more general case also captures the special cases we had before. In the case of a convergent series, Cesàro summation just gives the limit of the partial sums.

And then they go using this proof to further prove that the sum of all positive integers is -1/12. Because they are completely out of their mind. I mean really, why would it be -1/12?

Remember, this is

*not* the sum in the simple context of repeated counting. In that context, there is no sum. But is there a context in which we can assign a value to this divergent series? Yes. Cesàro summation will not help us here, but there are other methods that will. Analytic continuation and Ramanujan summation, for example, allow us give this divergent series a value of -1/12.

But is this actually useful for anything except playing mathematical games? It is. In fact, assigning values to divergent series is done all the time in physics.

But they don't care. They say it's -1/12. Because that's the answer they get when they do the math. And they then go using that -1/12 number to get their string theory to work, which they then pass off as "science" even though all these hypothetical dimensions have never been observed...

It's true that -1/12 is where the 26 dimensions of string theory comes from, but this isn't just used in theories as speculative as string theory. This is used all over the place in quantum field theories, such as quantum electrodynamics, which is not only testable via experiment, it is arguably the most precisely tested theory in the history of science. These kinds of divergent sums come up in that theory, and when we actually run the experiments in real life, we don't get infinite results, we get results like -1/12, as the math predicts.

To finish up, I want to return to the original question of 0.999... being equal to 1. I think a lot of people have trouble with this because they have an intuition that 0.999... should mean something like "infinitely close to 1 but not 1." Is something like that possible? Again, this depends on the context. In the context of the real numbers, absolutely not, because there are no numbers that are infinitely close to other numbers. But is it possible to extend the real numbers to include infinitesimal numbers like this? Actually, yes, there are hyperreal systems that do this. And in those contexts you can have numbers that are infinitely close to 1 (though you have to specify

*which* infinity). But even in these hyperreal contexts, 0.999... still normally just stands for the same limit we defined above, so it is still equal to 1. But hey, you are perfectly free to define your own hyperreal system of numbers where 0.999... is defined to be something else, one of those numbers infinitely close to 1. If it turns out to be useful, maybe they'll even name it after you someday!