Why is 0.999... one? It's an ambiguous decimal expansion.
1 - 0.999... = 0.000... Some part of you might think that there "must" be a 1 at the end of all those zeros. The problem is that there is no end at which to put a 1.
And you'd end up with tiny holes everywhere if they're not equal. 1 = 1/3 + 1/3 + 1/3 = 0.333... + 0.333... + 0.333... = 0.999... oops. Where did our missing one go this time? Surely it's clear from this that we have nothing but 9s in that expansion and that three 3s can never be larger than 9, even if you repeat them over and over forever? And there can't be any funny business going on at the end, because infinite lists do not have ends by definition.
At some point you go through the rules and just accept that this is how they play out and that using other rules just leads to weirdness (AKA "nonstandard analysis").
1 - 0.999... = 0.000... Some part of you might think that there "must" be a 1 at the end of all those zeros. The problem is that there is no end at which to put a 1.
And you'd end up with tiny holes everywhere if they're not equal. 1 = 1/3 + 1/3 + 1/3 = 0.333... + 0.333... + 0.333... = 0.999... oops. Where did our missing one go this time? Surely it's clear from this that we have nothing but 9s in that expansion and that three 3s can never be larger than 9, even if you repeat them over and over forever? And there can't be any funny business going on at the end, because infinite lists do not have ends by definition.
At some point you go through the rules and just accept that this is how they play out and that using other rules just leads to weirdness (AKA "nonstandard analysis").