Of course ...9999.0 = -1. That's just ten's complement arithmetic, as any old programmer would know. --ms On 2012-11-16 00:52, Marc LeBrun wrote:
Since this was originally an historical question perhaps as an arithmetical post-modernist punk I should hold my piece, but perhaps those that hold that
...0.999... = 1
will also equably agree that
...999.0... = -1?
After all, are we not told (presented with precisely the same corroboration as given for 0.999...=1), that the geometrical series with term ratio r=10:
9 10^0 + 9 10^1 + 9 10^2 + ...
must be equal to 9/(1-r) = 9/(1-10) = 9/-9 = -1?
And also therefore, surely, the apparently maximally heavy-weight
...999.999... = -1 + 1 = 0?
Of course, as those correspondents here who've surrounded 0.999... with "'s implicitly intimate, the real issue is around what '=' means ("what your definition of 'is' is" as someone-or-other put it).
As every type-whipped programmer would note, on the left-hand of the = sign we are placing strings, while on the right-hand we are placing real numbers named by those strings, that is "=" doesn't mean "is identical with" but rather should be read as "is an alias name for" the same (Platonic) entity.
(More precisely, '"0.999..."' is our finite name for an infinite string, whilst '1' is our discrete name for a real number (neither of which are ultimately finitely "knowable"). So I guess we are merely declaring that the referents of these names is the same (Platonic) entity--but anyway, we shave barbers...).
What the string 0.999... "means" depends on what interpreter we feed these (infinite) objects into.
Agreeing that 0.999... = 1 is just confessing to an agreeably shared delusion. Fine, but it's only "true" with respect to certain models.
There are also perfectly good arithmetics where this postulate is negated.
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