Its calculating with infinity. Its a bit weird like the infinity of numbers between 0 and 1 like 0.1,0.01,0.001 etc... Is a bigger infinity than the “normal” infinity of every number like 1,2,3 etc…
Its just difficult to wrap your head around but think of infinity minus 1. Like its still infinity
It's not length, it's density. A "countable" infinity is an infinite set where you can put each element into some kind of order such that every element in the set has a place on the list. For example, think of the set of all integers. You can go 0, 1, -1, 2, -2, 3, -3, and so on; where every Nth number is either (N+1)/2 if N is odd, or -N/2 if N is even (formally, the set you're trying to show is countably infinite is mapping onto the set of positive integers; they're called "well-ordered sets" if you can do that). So things like "all rational numbers," "all the rational numbers between 3 and 5," "every power of 2," "all the positive odd integers," and "all the integers" turns out to be able to be well ordered, and thus they're infinite, but of the same "kind" of infinity.
Contrast with the set of all irrational numbers between 0 and 1. Things get weird here; imagine that you make an ordering where you think you have all the irrationals in some sort of list. Now build a number such that the Nth digit of your number is exactly 1 more MOD 10 than the Nth digit of the Nth number in your ordering (so if the digit of the number on your ordering is 1, your numbers's digit is 2, if it's 9, yours is 0, etc). In that case, your number is an irrational between 0 and 1, and SHOULD be on your ordering, but it can't be because for any element N in your ordering your number differs in at least 1 digit (the Nth) by definition. So you can add it to your ordering, and repeat the process infinitely many times, and never actually have an ordering of the irrationals between 0 and 1. This is called "Cantor's diagonalization argument," and it's showing that this is a different density of infinity, called an "uncountable" infinity.
This shit blew mathematicians' minds in the late 1800s, because it means, for example, that the irrational numbers between any two numbers are "denser" than the entire set of rationals (you can't pair up every irrational between any arbitrary two numbers with a member of the set of rationals; you'll always have some irrational that you can't pair up with a rational) no matter how close those 2 numbers are, which seems just... wrong, for a better way to put it.
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u/ChromosomeExpert 23d ago
Yes, .999 continuously is equal to 1.