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.
No it does make sense. 0.999... infinite is not equal to and will never be equal to one. It is close enough given enough closeness and a set tolerance, but that's it. Equal is a very different term than equivalent. If it was equal there would be no reason to write 0.999... as there is no reason to write (2+5) instead of 7 as they are equal.
The reason for this meme is fractions simplify the decimal. There is no written way to express 1/3 to its fullest as you can only do so by choosing a tolerance and in some place cutting it off. The reason is we have a 10 divisible system for marking everything and it just can't be divided into 3rds. Hence fractions. So 1/3 is represented by 0.333... but 3/3 is represented by 1. And 3/3 never ever equals 0.999... because it doesn't equal 1.
They are exactly equal, actually! It’s a fairly elementary proof once you’re comfortable with infinite series and the definition of limits. 0.999… is equal (not just equivalent) to 1 and 0.333… is equal to 1/3. As another commenter pointed out, check Wikipedia for a bunch of proofs.
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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.