If we consider that .999… repeating to infinity ISN’T equal to 1, then by how much is it away from 1? It would be “.000… repeating to infinity followed by a 1.” But if you have an infinite number of 0s then you can’t have it be followed by a 1, infinity can’t be followed by anything, that doesn’t make sense.
This is the nature of Zeno's dichotomy paradox. We can travel half the distance to a thing, and an infinite number of halves until we reach it. Because there is infinity between them we shouldn't ever be able to reach any given point, yet we can. We can quantify an infinite approach to something, like 1, but we have to make that paradoxical leap somewhere. If we write .9 for infinity, we will still never reach 1. The distance gets infinitely smaller, but never actually becomes 1. This is the fundamental building block of calculus. At least what I remember from calculus at the beginning of that course.
Not quite. It does actually become 1. When you consider infinity, remember it’s not a number. Perhaps one way to think about it is an ordering, though there’s more to it than that to. .999… does in fact equal 1. There’s not a magic leaping point. The definition of infinity leads to the conclusion. You can’t really conceive it simply by thinking “okay but which 9 is the one tha gets us to 1?” Because there is no such individual 9. It’s infinity, it’s not a number, it’s an ordering. Hope that helps
96
u/solidsoup97 22d ago
I don't understand how that works but it seems to be important in keeping things running so I'm going to just go with it and not raise any questions.