If we’re measuring a length of pi centimeters, why does it look finite but the number of digits is going on forever? It looks like that it’s moving small amount of time every time. Same with 0.5555555… or any number with infinite decimal places. I can’t wrap my head around it.
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The length of something that you write down doesn’t have to be related to what it looks like.
For example, the word “big” is shorter than the word “small”, but a big thing is bigger than a small thing.
In English, we just happen to need more letters to say the word “small” than we do “big”.
The number 0.5555555… represents a very specific size. It needs infinite digits to spell out, but just because you need to write a lot of digits doesn’t mean it’s a bigger number. It just means you need to use more pen ink to write about it and more breath to talk about it.
So, just like in English where some words need more letters to say, in math, some numbers need more digits to say. But there isn’t really a link between the number of letters in a word and the word’s meaning, and there isn’t really a link between the number of digits in a number and the number’s meaning.
(Actually, we don’t even need to write out all the digits; just as you have done, putting “…” at the end of it is the same as writing all the 5s out infinitely. So actually, 0.555… is exactly the same number as 0.5555… which is exactly the same number as 0.55555… which is exactly the same number as 5/9. We have many ways of writing a number that all have the same meaning!)
Anyway, the reason the 5s go on forever is because, well, it’s a different number than if you stop the 5s at some point. 0.55555…5 with one hundred 5s is a liiiiiiiittle smaller than 0.55555… infinitely.
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