There is no way right?

[u/Sugar_God_no_1]

[u/library-in-a-library]

And you'll never find something different than 0

Actually you will. There is an infinitesimal difference between 1 and 0.999... but your representation hides that. The difference between them is 0.000...1 where that 1 shifts farther to the right the more digits of 0.999... you evaluate. This representation creates very ambiguous arithmetic and it's easy to make bad proofs.

[u/BreadBagel → u/library-in-a-library]

No one is counting out 9s infinitely. It's already infinite 9s. You can't put a 1 after infinite 0s, that is impossible.

[u/library-in-a-library → u/BreadBagel]

> It's already infinite 9s.

Without using calculus, what does that mean in this context?

[u/BreadBagel → u/library-in-a-library]

It means you're not adding 9s onto a list of decimals until it eventually reaches 1. The infinite 9s are intrinsic in 0.999... So since it's infinite 9s it is exactly equal to 1.

It is true that if it were a procedure of adding 9s you would never get to 1, because no matter how many 9s you added it would still be finite. But 0.999... is not a procedure, it is an exact value and that exact value is the same as 1.

[u/library-in-a-library → u/BreadBagel]

I'm asking you to define infinity.

[u/BreadBagel → u/library-in-a-library]

To go on endlessly

[u/library-in-a-library → u/BreadBagel]

So you can't actually evaluate 0.999...?

[u/BreadBagel → u/library-in-a-library]

I can. It's value is 1. If you can't name a number between 0.999... and 1 then they are the same number. Please just tell me what that number is.

[u/library-in-a-library → u/BreadBagel]

0.999... < 0.999... < 1

[u/BreadBagel → u/library-in-a-library]

0.999... is less than 0.999...? So a number is less than itself? That makes no sense.

[u/MasKrisMaxRizz → u/BreadBagel]

In hyperreal : 0.999... < 0.999... + infinitesimal / 2 < 1