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.”
In: 87
As already said, plank length stuff is only a physical thing because our theorem of physics stop to make sense at this point. However mathematics don’t have this kind of limitation.
In mathematics (especially in field like set theory) we can define real numbers by a sequence of natural numbers (signed to include negative numbers), like for example 2.67 is the finite sequence (2 , 6 , 7). You can always create a new sequence that is longer by adding a new natural number like let’s say in our case the sequence (2, 6 , 7, 3) that would correspond to 2.673 and that correspond to a new real numbers between 2 already existing numbers and you can continue that way to find new numbers infinitely anywhere on the real numbers.
People are sometimes confused with this and the notion of measure. Measure is a different thing, while there is indeed an infinite number of “numbers” ( called real numbers ) between each of the number we use everyday, we do have a notion of measure (you can see it as a distance) that actually shows that any 2 reals are at a finite distance from each others despite having an infinite number of other reals between them.
EDT: I may as well add another thing. I didn’t really want to talk about it at first because it’s probably a notion hard to grasp but I also realize it somehow also match your question. When you create such an infinite sequence as mentioned earlier, we assume that the sequence is equal to its limit from a mathematical series standpoint. Mathematics do not make really a difference between 2 and the sequence that converge to 2 if the sequence keeps going infinitely, at least not when working with real numbers.
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