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
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.
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.
Yes, the infinity between real numbers is infinite. It’s “more infinite” even than the number of integers for example. The real numbers are said to be “dense” which basically means the same thing—there cannot be two real numbers where there aren’t also numbers in between.
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.
The planck length doesn’t really have anything to do with math itself. Planck length, time, etc. have to do with the fact that light is measurably quantized, and there is a max speed limit to the universe through space (speed of light). Because “speed” is defined in terms of distance and time, a max speed turns into the idea that there’s a minimum distance and minimum time in which anything can “happen.” But if the speed of light was different, or perhaps in a universe that worked a different way, there would be different values. Math itself does not imply any limit, however.
Mathematically, numbers have no “true” meaning in the real world, so numbers can be infinitely small or infinitely big. When it comes to science, though, like physics, that’s where the Planck length comes into play. It’s the theorized limit as to how small something can be in the universe. The Planck length is measured in Planck units, you can use any other units, like cm or inches, which will give you different numbers which is what I mean by math having no true meaning, it’s more of a way to consistently count things and it, by itself, has no limits.
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.
Not only is it infinite, but it’s provable that there are more numbers in between numbers than there are numbers.
To be more precise, the set of real numbers is a larger infinite set than the infinite set of integers.
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