A brief and incomplete exploration of the 1-ness of an infinite string of 9s for .9~
(1)One can see that carrying this operation out forever will lead to an infinite string of 9s. We’ll just forget about the 1
because numbers disappear in Infinity. Infinity is a magical realm. Numbers go in but they never come out. Like
the Roach Hotel of Mathematics.
How about if we square .9 and square 1.
We can not do the operation out because we can’t see what happens at the right end of the string where it exists
in Infinity. So, let’s try it in the finite realm and see if we can tell what is happening in Infinity.
.9 x .9 = .81 No 9s in the resulting string of numbers and .81 is obviously not equal to .9.
How about:
$.99^{2} = .81 + .081 + .0891 = .8 + .17 + .010 +.0001 = .9801$ One 9 out of 4 digits but it’s a leading 9.
(I wanted to do this out to show all values are present and accounted for).
Ok, next problem: (Assume the same process of calculation for all that follows)
.999^2 = .998001 More leading 9s. However there are still less 9s then other numbers.
Again:
.9999^2 = .99980001 Wow, I see a pattern here. Now let’s assume this process goes on forever.
There will be just as many 0s as 9s and the 9s will be 1 less then the original
number of nines. Also, there are always 2 more digits that are neither 9s nor
0s then making the string of digits of non-9s two longer then there are 9s.
Now here’s the thing about magic realms. All travelers arrive at once. None who leaves for a magic realm can get there before another as long as they left at the same time. There is no closer or farther away. Not only will the length of the string of numbers reach infinity but the 9s will get there at the same time and so will the 0s. You can’t get half way to infinity. You are either there or not. So, there are not ½ - 2 to Infinity of 9s. And, as I already stated about the magic realms including Infinity, once you go in, like magic, you disappear. Only the 9s can be found because they are behind in the journey to infinity. Although, they arrived there at the same time as the 1s, they are still last and push all the numbers ahead of them through the gates where they are lost forever, like magic. The nines have a foot hold in the finite realm and unless another traveler on its way to Infinity, and I might point out that they have to also be on the same path, starts later then the 9s, they will remain in the Finite realm. There is no indication that anyone else wants to travel this path to Infinity so the 9s are all that are left. Does that make sense? Magic!
Wait, wait, How do I know the 9s as well as the other numbers actually make it to Infinity? It’s a long long long way. They can’t make it on their own. Don’t I or someone have to keep doing to calculations never endingly. I got it! Let’s not and say we did. Even still. Any teacher would know I was lying. “You never did the calculations out!”
I would say, “Well… Ah, I did do the calculations, teach.”
“No, I don’t thinks so. I don’t know how you got those 9s to Infinity but I know you didn’t do all the calculations. Now did you?”
“Well… No, not all of them. But I did most of ‘um. Really, I did nearly all of ‘um.”
“I am unconvinced. You couldn’t have done that many in the time you had.”
“Ah… um, at least half of them. I did them ‘till the 8 got to infinity, then I had to go to bed. That‘s 1 more then half.”
“Son, if you did that many calculations then you would know that the 8 as well as the 1 never get to Infinity.”
I defend myself, “Yeah they do. They are gone, aren’t they? They disappeared once they got to Infinity.”
The teacher is really angry now, “No no no no, you didn’t do all the calculations and they can’t just disappear. What did you do with the 8 and the 1. There’s only one of each so you must have stashed them somewhere. I think you need to go into the corner and think about where those missing numbers are.”
What could I do but sit in the corner. I didn’t know where they went. I still don’t.