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1/9 has after the 0. infinite number of 1's. 9/9 has after the 0. infinite number of 9's. 9/9 is 1

It's because of the period. 0.999999...999.. is 1. That's because 1/3=0.333...33.. and 1/3×3=1. Or 1/9=0.1111...111... 2/9=0.2222...222... etc 8/9=0.8888...888.. 9/9=1

Yes, I understood that it's because of the period. But even then it would still be 0.000...1 off and thus approximately equal. Excuse me if I'm wrong, but I see numbers as each occupying it's own, imaginary, indefinitely small space, but still not touching the numbers next to itself. And this "barrier" is what gives each number one exact value. If you just say, that because it's so indefinitely thin, one number can be another number, what's stopping you from just saying no number has its own definite value, which is, in my understanding, correct me if I am wrong, the basics of all of mathematics?

0.999999... equals 1. The ellipsis (...) at the end means the decimal expansion goes on forever. Aen explained it correctly.

Yes, I know about the eliipsis, but isn't there still a 0.000...1 difference to 1? You can't just round numbers up like this. That would defie the basic principle of mathematics that each number has exactly one value assigned to it, as I tried explaining in my comment above.

It would be better expressed as a recurring or repeating decimal. I can't type the dot notation I would have used, but they can be written with parethesis around the repeating part, so 0.(3) times 3 is 0.(9) which is 1. What was written above only took the representation of a third to three decimal places, but it is not, it is a recurring decimal which goes on forever, hence the error.

Yes, but then it's still not 1. There still is some difference left, that distinguishes the two numbers. Hence my attempt with 0.(0)1, where the 1 at the end after the infinite 0 represents the indefinitely small rest, indefinitely small yes, but nethertheless a rest, that distinguishes 0.(9) from 1, because the two are still, as explained above two different numbers, because each number can only have one value assigned to. I understand the general concept, but I don't understand how two different numbers can have the same value, which would, in my understanding, violate the basic principle of mathematics. ( Also, please don't downvote me for not understanding difficult mathematics and just wanting it explained! Being downvoted on BP can get you banned from commenting. 🥺)

Party Poison, there is no "1 at the end" because there is no "end". The 9's go on forever, so writing 0.(0)1, as you did, makes no sense. Perhaps this Wiki article will help you understand: https://en.wikipedia.org/wiki/0.999...