On Fri, Nov 16, 2012 at 12:52 AM, Marc LeBrun <mlb@well.com> 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?
This isn't post-modern, it's pre-classical. This is exactly the way Euler reasoned, and he came up with the same sort of conclusions. But the reasons I, and a modern mathematician, say that .99999.... = 1 not just because of the proof of the formula for the sum of a geometric series (which, if you examine it closely, shows only that *if* the series has a limit, *then* the limit is 1), but because of the epsilon-definition of a limit, and a proof I can provide, using that definition, that this series indeed does sum to 1. If you try to provide a similar epsilon-delta proof that the series ...9999.0 sums to -1, you will fail, and I can provide an epsilon-delta proof that this series does *not* have a limit.
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,
Nonsense. 1 is a string, and so is .999999... And for that matter, so is 1 = .9999999..., a string which represents a proposition, that says that two of its substrings represent the same number.
(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...).
There's no need to invoke Platonism here; in a formalist view, I can provide a nice, finite, proof from finitely many axioms that the string 1 = .99999.... is a theorem.
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.
Exactly; just like 2 + 3 = 5. An agreeably shared delusion, not shared by those who make the equally good choice to use the squiggle "3" to represent the integer between 7 and 9. I also think the reference to infinite strings is a complete blind alley and confusion here. I'm using .9999..... to mean exactly the same thing as the finite string while on the right-hand we are placing real numbers named by those strings, sum_{i=1}^{i=infinity} 9 * (10)^(-i). The latter is just an imprecise shorthand for the latter. If someone asked me about what number was denoted by .12345....., I'd have to say "I don't know; you might mean sum_{i=1}^{i=infinity} rem(i, 10) * (10)^(-i). (that is, .1234567890123456789...) or you might mean sum_{i=1}^{i=infinity} i * (10)^(-i). (that is, .12345679012345..., where you put the next integer in each position and carry), or even .12345678910111213141516..., which I could write in summation notation, but it would be a bit more complex. If someone said about .9999..... "I don't understand; does this continue for 25 more 9's, or 100, or what? You haven't told me want the 734th digit is", I'd use the summation notation, which is better because it unambiguously gives the value for every decimal place, rather than playing the mug's game of "guess the pattern". Or to put it another way, I know of no sensible system that attaches different meanings to .9999.... and sum_{i=1}^{i=infinity} 9 * (10)^(-i), and the latter is a finite string, so any paradox or difficulty isn't related to the finiteness or infiniteness of strings.
There are also perfectly good arithmetics where this postulate is negated.
Depends on what you mean by "perfectly good". The arithmetic where "9" is used to mean the the number between 4 and 6 is of course a perfectly good arithmetic where te postulate is negated. But if you look at the proof of the sum of a geometric series (not the epsilon-delta part, just the proof that "if there is a sum, the sum is 1", then any system in which this statement is not a theorem (I know of no system where it is a 'postulate') must give up one of the axioms used in the proof, which makes it in my mind less than "perfectly good". By contrast, the same proof shows that ......999999.0 = -1, meaning that if the series has a sum, the sum is -1, but I know of no perfectly good arithmetic where this is in fact true. But the same proof also shows that in base 7, ....66666666.0 = -1, and there *is* a perfectly good arithmetic (unless your definition of "perfectly good" includes "Archimedean" where this is a theorem, namely the 7-adics.
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