Can numbers get so small (or so large) that there is kind of a “planck length” effect where you just can’t get any smaller? Or is it really possible to have 1.000000…(infinite)1
EDIT: I know planck length is not a mathmatical function, I just used it as an anology for “smallest thing technically mesurable,” hence the quotation marks and “kind of.”
It’s not possible to get actually infinite number of zeroes before the final one, because the presence of that final one would inevitably make the preceding sequence of zeroes finite. It is, however, always possible to add another zero to any finite sequence of zeroes, making the number of possible sequences infinite.
They really are infinite because you can always add another decimal place. Take the gap between 1 and 2.
Halfway is 1.5.
Another fractional step towards 2 would be 1.51.
Another would be 1.511.
Another would be 1.5111.
Another would be 1.51111.
There’s nothing stopping you from adding yet another “1” to the end of the number. Sure, it’s such a small piece of a number that most people would ignore it and round, but that doesn’t mean it doesn’t exist.
So yes, it’s infinite.
I’m a bit confused about your question, however, yes there are infinitely many numbers between any two numbers, but what you’ve written is not a well defined thing. You can certainly pick any two numbers, like 10.1 and 10.2 and find infinitely many numbers between them by just putting more decimal points, like 10.11, 10.11, 10.111, etc.
Math is useful for approximating reality, but math can do its own thing too and not necessarily correspond to something physical.
Real quick, the planck length is not what you seem to think it is.
Anyways, there is no reason mathematically that we can’t infinitely divide numbers. *However*, there is no difference between 1.000000000000… and 1. It’s a bizarre quirk of infinitesimals.
They really are infinite, and the Planck scale isn’t some physical limit, it’s just where our current theories stop making useful predictions about physics.