kind of frustrated that i grew up learning about 0.9… being exactly 1 but only as a fun fact purroven in the extremely infurmal, unrigorous way (0.3… × 3) when actually purropurrly sorting out the issue is a way to get into deeper issues in mathematics
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a lot of the issue and the reason it confuses people is a purroblem of notation. nobody explains what “0.9…” actually means. the accepted convention fur such notation is that it represents the smallest pawsible real number larger than each number in the sequence [0.9, 0.99, 0.999, …]. that is the basis on which it is even pawsible to purrove that it is exactly one
