@boomzilla said:
@rad131304 said:@theheadofabroom said:@boomzilla said:Technically, no it isn't 1, it just becomes indistinguishable from 1 since the space mapped by the points is the right-open interval of [0.9-1). They are, however, two distinct points on a continuous number line since I can always define an distance ε between them such that {ε | 0 < |ε| }.Did you know that 0.9 equals 1?Well of course, as 0.9 can be thought of as notation for Σ(9E-x) for x between 1 and infinity, which is trivially 1
It's always interesting to see novel proofs. theheadofabroom seems to be into limits (even if he's a bit sloppy). But the 0.9 equals 1 proof as I originally learned it has always amused me, so I'll repeat again...
x = 0.9 10x = 9.9 10x - x = 9.9 - x 9x = 9.9 - 0.9 9x = 9 x = 1Infinities are weird, but fun. I wonder what it means to be indistinguishable but distinct.
Another one is that 0.9 can be thought of as 1 - 0.01, which can be thought of as the value of y as x approaches infinity in the series y = 1 - E-x. Seeing as E-infinity is 0, that makes y == 1.
I realise I was a little sloppy in some of my previous posts, but let's put the onus on those who think that 0.9 != 1. We've given several proofs that it isn't, some even without glaringly obvious holes. If you can prove that it isn't, we'll believe you.