One problem that strikes me as I read through these posts (and I think maybe this is the problem Archenar refers to) is the consistent use by some of the phrase “an infinite number of…” This is a phrase that has some intuitive feel colloquially but has all kinds of problems in a rigorous mathematical sense.
Basically, there is no “infinite number”. There are cardinal numbers, which have their own arithmetic, but it is an error to attempt to treat these as “normal” numbers.
When you say that 0.999… is “zero dot with an infinite number of nines”, you’re really using mathematical shorthand for the concept of an infinite sum, which comes back to the point that ultrafilter, Archernar et al have been making all along. That’s why you can define the limit to the sum and that’s why it makes sense to define 0.999… in terms of that limit.
(As an aside, RealityChuck, therein also lies the problem with the “10x - x = 9” proof. Because 9.999… is an infinite sum, you have to prove that it converges before you’re allowed to do things like multiply and subtract it… but once you’ve proved it converges you’ve already shown that it equals its limit and don’t need to do the multiplication-and-sutraction thing anyway! Therefore whilst the “proof” is definitely useful for giving an intuitive feel for the reason the solution works, it is actually begging the question.
Sadly, it isn’t as simple as saying “multiplication just moves the decimal place” because multiplciation on the reals is a field operation that doesn’t work on infinite sums unless they converge, leading you back once again to the same starting place)
Back to the point: next time you refer to an “infinite number” of something, stop and ask yourself what you really mean by that and remember that infinity is not a number, at least in the sense that you are using the word.
One != 1?
I’ve come back into the debate too late to address the 1[sub]Z[/sub] != 1[sub]R[/sub] issue, so I’ll just say that my education concurs exactly with ultrafilter’s explanation. The set of reals is a completely different species to the set of integers and it is only by clever mathematical construction that we are able to shift between the two.
Think of it in programming terms: you’ve defined parameter s as a string and parameter v as a variable. You set s = “one” and v = 1. Now what will the program do if you ask it to compute s + v?
So what does “1” mean?
In set-theoretic terms, “1” is the identity, defined such that under the multiplication operation 1.a = a. (Under a ring structure such as the reals we then use the addition operation in order to define things like “2” = “1” + “1”)
Because the identity is such a universal concept in sets with the multiplication operation defined, it looks very similar in the various sets where it is defined. It’s the same idea in each, right? But you’re still comparing apples and oranges, or s and v.
That’s why group, ring and field theory should be left to those who have mad hair, smell slightly and live in basements. It’s too annoying for the rest of us.
A digression on the nature of mathematicians
In my university the pure mathematics department shared a building with the statistics department. It was noteworthy that the statisticians had the top two floors, which were air-conditioned and had a great view whilst the pure mathematicians had the bottom floor and the basement, where the car fumes entered and it was noisy.
Also the statisticians had a car park with Aston Martins and Jaguars parked there whilst the pure mathematicians had a bicycle shed.
I went into statistics.
pan