0.999... = 0.999...9
0.999...9 + 0.000...1 = 1
0.999...0 + 0.000..1 = 0.999..1
0.000...1 = 1/∞
0.999...9 = 1 - 1/∞
0.999...0 = 1 - 1/∞ - 9/∞ = 1 - 10/∞
If x/∞ = 0, then 0.999...x = 1.
If x/∞ ≠ 0, then 0.999...x ≠ 1.Ok, now you're saying that infinite decimals have final digits.
If Universe is infinite, then if we compare you to size of Universe, you are infinitely small, so you don't exists at all. Why I should waste my time?
If Universe is finite, then finite number of elements can make only finite number of combinations, thus this discussion is repeated infinite number of times again. Why I should waste my time again?
Yes, you're wasting your time if you compare my size to the size of an infinite universe. If you really want to waste your time that way, you don't want to use real numbers. On the real number line my size is exactly zero. You need to go into infinitesimals, which are out of place when you're looking at decimal notation, which is only for real numbers.
None of these, except 0.999… and 1 are well-known standard objects in this setting. You have to define what you mean.
I defined it:
0.000...1 = 1/10^∞ = 1/∞That's not a definition. Neither 10^∞ nor 1/∞ is defined in any standard system, so you'll have to define those too if you want to use them to define 0.000…1.
You're working with surreal numbers? This is not what people would expect unless it's explicitly stated. In addition, you're likely going to have a hard time explaining surreal numbers to someone who struggles to grasp that 0.999... = 1 in the ordinary reals.
Surreal numbers and infinistemal are simpler to work with when you need to work with infinite series.
Here John Conway explains them: https://www.youtube.com/watch?v=1eAmxgINXrE
It's 0.999... and not 0.999...0
In the same way, it's 0.000... and not 0.000...1.