0.999… / 3 = 0.333… 1 / 3 = 0.333… Ergo 1 = 0.999…
(Or see algebraic proof by @Valthorn@feddit.nu)
If the difference between two numbers is so infinitesimally small they are in essence mathematically equal, then I see no reason to not address then as such.
If you tried to make a plank of wood 0.999…m long (and had the tools to do so), you’d soon find out the universe won’t let you arbitrarily go on to infinity. You’d find that when you got to the planck length, you’d have to either round up the previous digit, resolving to 1, or stop at the last 9.
Any real world implementation of maths (such as the length of an object) would definitely be constricted to real world parameters, and the lowest length you can go to is the Planck length.
But that point wasn’t just to talk about a plank of wood, it was to show how little difference the infinite 9s in 0.999… make.
Afaik, the Planck Length is not a “real-world pixel” in the way that many people think it is. Two lengths can differ by an amount smaller than the Planck Length. The remarkable thing is that it’s impossible to measure anything smaller than that size, so you simply couldn’t tell those two lengths apart. This is also ignoring how you’d create an object with such a precisely defined length in the first place.
Anyways of course the theoretical world of mathematics doesn’t work when you attempt to recreate it in our physical reality, because our reality has fundamental limitations that you’re ignoring when you make that conversion that make the conversion invalid. See for example the Banach-Tarski paradox, which is utter nonsense in physical reality. It’s not a coincidence that that phenomenon also relies heavily on infinities.
In the 0.999… case, the infinite 9s make all the difference. That’s literally the whole point of having an infinite number of them. “Infinity” isn’t (usually) defined as a number; it’s more like a limit or a process. Any very high but finite number of 9s is not 1. There will always be a very small difference. But as soon as there are infinite 9s, that number is 1 (assuming you’re working in the standard mathematical model, of course).
You are right that there’s “something” left behind between 0.999… and 1. Imagine a number line between 0 and 1. Each 9 adds 90% of the remaining number line to the growing number 0.999… as it approaches one. If you pick any point on this number line, after some number of 9s it will be part of the 0.999… region, no matter how close to 1 it is… except for 1 itself. The exact point where 1 is will never be added to the 0.999… fraction. But let’s see how long that 0.999… region now is. It’s exactly 1 unit long, minus a single 0-dimensional point… so still 1-0=1 units long. If you took the 0.999… region and manually added the “1” point back to it, it would stay the exact same length. This is the difference that the infinite 9s make-- only with a truly infinite number of 9s can we find this property.
Except it isn’t infinitesimally smaller at all. 0.999… is exactly 1, not at all less than 1. That’s the power of infinity. If you wanted to make a wooden board exactly 0.999… m long, you would need to make a board exactly 1 m long (which presents its own challenges).
It is mathematically equal to one, but it isn’t physically one. If you wrote out 0.999… out to infinity, it’d never just suddenly round up to 1.
But the point I was trying to make is that I agree with the interpretation of the meme in that the above distinction literally doesn’t matter - you could use either in a calculation and the answer wouldn’t (or at least shouldn’t) change.
That’s pretty much the point I was trying to make in proving how little the difference makes in reality - that the universe wouldn’t let you explore the infinity between the two, so at some point you would have to round to 1m, or go to a number 1x planck length below 1m.
It is physically equal to 1. Infinity goes on forever, and so there is no physical difference.
It’s not that it makes almost no difference. There is no difference because the values are identical. There is no infinity between the two values.
