[split] 0.999... equals 1

[fr0d0]

And limitlessly 'gappy' - otherwise it'd be 1

To infinity the gap gets smaller, but never ever converges. Or are you saying that it does converge?

It's not limitlessly gappy, that's the point. If it were limitlessly "gappy", it wouldn't be an infinite string of 9s.

So does it converge?

I understand infinity perfectly well I think. I don't understand this for the reasons given. I find it interesting. the fat lady hasn't sung yet (unless I missed it! )

Evidently you don't. You think that there can be a finite gap between 0.9r and 1, when 9 is the highest digit in the decimal system, and 0.9r has an infinite string of them. There is no number you can add to 0.9r that turns it into 1. Saerules said 0.0r1 is that number, but 0.0r1 is a mathematical impossibility because it is an infinite (i.e. endless) string of 0s with a 1 on the end. An endless string with something on the end. You might not see the contradiction mathematically, but formally you can. An endless string cannot (by definition) have an end. There is no end for Saerules' magic 1 to go.

I agree (how can't I?) I can see nothing would go on the end. Only 9's are possible. But this is surely illustrating the weakness of the digital representation of a whole. 1 is simple. 1/3 is simple. 0.999... & 0.333... are inaccurate representations of those.

"can't always" allows for "is more accurate" on occasion.

Yet they provably do. There is no value that cannot be expressed as a decimal number, be it irrational or rational. The decimal might be infinitely long, but it is still a valid number.

I get that it's a valid number and works for maths. I think it's the convergence thing I have a problem with logically. If 0.999... & 1 never converge how are they the same?