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> a (rigorous) definition for what 0.999... means But 0.999... is not "defined" to be equal to 1, it is equal to 1 as a matter of fact. (In other words, both a
by Koshkin 4y ago
> a (rigorous) definition for what 0.999... means
But 0.999... is not "defined" to be equal to 1, it is equal to 1 as a matter of fact. (In other words, both are just different spellings of the same number arising, automatically, from the construction of the decimal representation of the reals.)
- patrickthebold 4y agoI'm not familiar with the construction of the reals, but somewhere in there you'd have to say what 0.999... means for it to be equal to 1. I'd probably just define it as an infinite series, where it's straight forward to show it is 1. But on second thought, kids run into decimals before constructing the reals, I'd imagine you could define rational numbers and repeating decimals and end up with a simpler argument.
- deleted 4y ago[deleted]
- pdpi 4y ago0.99… doesn’t equal 1 in, say, base 16. 0.11… can be 1/9 in decimal, or 1 in binary. Those equalities aren’t a “matter of fact”, they’re provable artefacts of positional number systems.
- macrolocal 4y agoSee ~ page 16 here for the (historic) definition: https://personal.math.ubc.ca/~cass/courses/m446-05b/dedekind-book.pdf https://personal.math.ubc.ca/~cass/courses/m446-05b/dedekind... It’s possible to work with both {x : x < r} and {x : x <= r} and maintain the distinction between 0.999… and 1, but then eg. 1 = 0.999… + x has no solution.