EDIT:11 May 2008. Reopening and hoping for the best :)

Other axioms will lead to other results, viz. 0.9 periodic != 1 (you just have to introduce infinitesimals)

For a good and accessible introduction as well as a cognitive science view on this subject see:

Lakoff and Nunez: Where Mathematics comes from: How the embodied mind brings mathematics into being. 2001

Your proof has two issues with it. First, you say that 1/infinity=.00000000001… . This is not true; in fact, 1/infinity=0. The issue people have with this fact is the same as the issue people take with the fact that 1=.999… — that of a lack of understanding of the infinite process. Infinite processes are very touchy things; something that holds true at every single finite step need not hold at the “infinite” step — whatever that means.

The second problem with your proof is the line

0 = 1/infinity = 2(1/infinity) = 3(1/infinity) = infinity(1/infinity) = 1.

The last two equalities are not true. In fact, infinity/infinity is what’s called an “indeterminate form,” meaning that it is not feasible to assign it any value whatsoever. Consider the following problem. Choose any number x, and multiply it by three. Divide the result by your original number x. You will get an answer of 3. In other words, we have that 3x/x=3, a fact that you will no doubt concede. Thus, you might think it natural to assign “3*infinity/infinity” the value 3, and you’d be correct. But then 3*infinity=infinity, so what you are really saying is that infinity/infinity=3. Of course, there is nothing special about 3 at all; you could repeat the process with any finite number in place of 3, including 1 or even 0. So in this sense it is “natural” to assign any value you want to infinity/infinity. But this doesn’t make any sense — any symbol we write down and want to do computation with must have a precise meaning. It can’t mean a bunch of different things at the same time.

Like I said, infinite processes are tricky things.

In fact, we have that n(1/infinity)=0 for any finite value of n, but infinity/infinity is indeterminate (has no meaning or intrinsic value; is left undetermined). Other indeterminate forms include 0*infinity, 0/0, 0^0, infinity^0, and 0^infinity (where “^” means “to the power of”).

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The real issue with the decimal expansion system is that the representation of a number is not unique. A ‘number’ does not change based on its representation. The concept of ‘5′ is something that we all understand; we all know what it means to have ‘5′ of something. The fact the the Romans called it ‘V’ and the Japanese call it ‘rocking chair with a roof’ (or ‘go’) have nothing to do with what it is.

Our decimal system is an extension of the usual base-10 representation of natural numbers (positive integers) that is meant to represent values between the integers. This is done in the most natural, innocent of ways. We know the number 365 does not mean 3+6+5 (as an alien might think upon learning the digits), but 3*100+6*10+5*1. It is a shortcut, so that we do not have to come up with an infinite set of symbols; instead, we write arbirarily long “words” of a finite set of symbols (namely, 0123456789). What we gain by doing this is that we can represent arbitrarily large integers without having to create a new symbol each time.

In the same vein, then, we attempt to go the other way by letting numbers of the form 36.576 mean 3*10+6*1+5*(1/10)+7*(1/100)+6*(1/1000). You may be suprised to notice that using numbers of finite length is not sufficient to represent all of the numbers we want to. Obviously numbers such as sqrt(2) (the square root of 2) and pi have issues (the latter is actually not simple to prove), but even such “normal” (rational) numbers such as 1/3 an 1/7 cannot be represented finitely by such sums. It is not because these values don’t exist; rather it is the fault of our choice of a system of representation. Choosing another base (meaning choosing another number to serve the role of 10) does not help; while 1/3=.3=3*(1/9) in base-9, we now have the new issue that 1/2 cannot be respresented by a finite “word.”

The only thing we can do is allow these “words” to be infinite. Fortunately, we can make a precise definition of what these infinite words are; the infinite sums that we get “converge” in the nicest way possible (what we call “absolute convergence”). Even more fortunately, we now have that every real value can be assigned a number (a decimal representation). This includes 1/3, sqrt(2) and pi (both rational and “irrational” numbers). Unfortunately, we get the “problem” that is the root of this discussion, namely that we can assign multiple “numbers” to the same value. 1 (or 1.0000000…) is not the only exception either. In fact, ANY number with a finite decimal representation (except 0) can be represented with an infinite representation that ends in an infinite string of 9’s. This is because

1/(10^n)=9/(10^n+1)+9/(10^n+2)+9/(10^n+3)+… for every positive integer n.

Now, as it turns out, the values that have this exact problem are the only values that have multiple representations, and moreover these two are the only possibilities. Thus, the natural thing to do is to choose the finite representation in each case, as the whole point of this system is for book-keeping purposes, anyway. We do NOT choose the finite representation because the infinite one doesn’t work; you can in fact use strings that end in infinite 9’s in any of the usual (+-*/) calculations, and as long as you are careful you will still arrive at the correct answer. But since we’re only book-keeping, why bother?

The next natural question to ask is, “why don’t we choose a system where each number has one and only one representation?” As far as I know, a natural such system does not exist. In fact, it would not surprise me if it could be proven that so such system existed. Remember, we aren’t just writing numbers down, we want to calculate with them as well. Finding a system of representation in which it is easy to perform arithmetic calculations is not easy.

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Yes, I know this post is way too long. But hey, if can sit around for hours every day talking about math, you guys can read a long comment, can’tcha?

Infinity x 1 = that same number

But…

Infinity x .999 = A different result.
For example
4 x .999= 3.996

Hmmm… so if your telling me .999=1 then because of rounding any number is actually equal to any other number!? Whhooaaa i just blew my own mind.

I’m not sure that you are addressing anything that was said in the post, so let me point to it:
When you divide 1 on 3 parts in decimal system, you will get new decimals 3 over and over, and you will *never* be able to finish the process. So, we have those two things a)the decimal 3 will be repeating in every iteration and b)the process can’t be ended.

So, this repeating doesn’t signify some aggregation of “lots of lots” of ciphers 3, but the ones that are written, and the “power” which is generating the new ones.

As for your proof (1/infinity=2/infinity=…infinity/infinity), you make the mistake of thinking of infinity as an aggregate. It assumes that you can get to infinity by starting from 1, and then adding +1, +1, +1.
But it is impossible to get to infinity by adding +1. Let me put it like this: infinity-1=infinity. Namely, there is no number from which you can get to infinity, by adding 1.

Let’s assume for a moment that .999… is = 1.

Therefore, 0 = 1/infinity (if you don’t accept this premise, just think about it, 1/infinity = .0000000001… just as 1 = .999…)

Consequently: 0 = 1/infinity = 2(1/infinity) = 3(1/infinity) = infinity(1/infinity) = 1.

So if .999.. = 1, then 0 = 1.

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Have a good philosophical article? Post it on the the philosophy wiki (http://sophiasdialectic.com).

(1) If we are summing a finite series, we can permute the numbers in the series any way we wish and get the same result. In general, this is not true in the infinite case.

(2) The statistical concept of the expected value of a random event is based on an infinite sum. Statisticians (and gamblers) who are too quick to assume that this matches up with the finite case will lose a lot of money playing the St. Petersburg game. That game works something like this: “You’re gonna flip a coin. If it’s tails, you lose. If not, keep on flipping. If it’s tails, you get $1; if not, keep on flipping. If it’s tails, you get $2; if not, keep on flipping. If it’s tails, you get $4, and so on.” The expectated value of this game to the gambler is infinite, so it would seem a good deal to pay $5 a game to play. Try this, however, and you’ll rapidly lose all your money.

The fact is, in standard analysis the real numbers are defined as either the Cauchy sequences or Dedekind cuts of the equivalence classes of pairs of integers under the cross-product relation, endowed with a cross-product-based ordering. In other words, there’s really nothing intuitive about them. They are a mathematical construct designed to have certain properties, and beyond these properties they behave in manners which one might find bizarre.

To put it in more philosophical terms, a car and a horse can be made to serve the same purpose, but a man whose intuition tells him that he can buy a pair of cars and let nature take its course is, unfortunately, mistaken.

This particular line of reasoning only works, though, if we don’t have infinitesimals (numbers greater than zero but smaller than any finite number). My understanding is that in nonstandard analysis these can arise, even though they don’t in standard calculus; but I don’t know enough about nonstandard analysis to say much about them, or how they relate to this problem. And, in any case, I doubt that any of us have difficulty with 0.9999…=1 because we have developed a consistent system of infinitesimal numbers.