Just how big is it?

[HalationEffect]

OK, how about this:

The formula for finding the sum of an infinite geometric sequence is S = a1 / (1 - r), where S is the sum, a1 is the first term in the sequence, and r is the common ratio.

In the case of 0.999... we get S = 0.9 / (1 - 1/10) = 0.9 / 0.9 = 1

Source: http://www.bungie.net/it-it/Forum/Post?id=61117593

Also (from the same source) let's say that x = 0.999...
10x = 9.999...
10x - x = 9x = 9.000...

If 9x = 9.000... then x must equal 1.000..., even though we stipulated at the beginning that x = 0.999... therefore 0.999... and 1.000... must be the same.

From what I've been reading, amongst top mathematicians the notion that 0.999... equals 1 isn't even controversial; it's broadly accepted. I know that's the logical fallacy of 'argument from authority', but it'll do for me.

[Qqmike]

I know I said I had to go for now, but ...

HalationEffect, my friend, you are playing with words, with symbols, with shorthand notations, and citing other people playing with words. Stop. Think for yourself, quit quoting stuff like this. When you read something, you must take it in the context of which it was written, you can't just apply it to ANY context. Trust me, the number 1 is not defined to be that limit. It is true that the value of that limit happens to be the number 1. Hell, it could have been the number 4.276; but no, it is 1. The number 1 is altogether something else having NOTHING to do with that limit (i.e., with 0.99999.... and that expression IS nothing more than shorthand for an infinite summation which IS nothing more than shorthand for an infinite limit of finite summands). If you want to push this, OK, the number 1, in THIS context, is the multiplicative identity of the field of real numbers (every field, axiomatically must have a multiplicative identity; i.e., there exists an element, call it "1," such that for every element x of the real field, 1*x = x*1 = x). It is ridiculous to say that we define the multiplicative identity of the field of real numbers to be the limit represented by the shorthand symbol 0.9999.

That did it! :-)
Now I AM done and excuse myself from this nonsense!

[HalationEffect]

I guess you need to go and edit the Wikipedia page then - especially the bit that says:

The equality 0.999... = 1 has long been accepted by mathematicians and is part of general mathematical education

But in order for such an edit not to be immediately reverted, you'd also have to show that all the proofs listed on that page are wrong.

(Edit) Your mathematical knowledge is greater than mine, of that I have no doubt. You'll most assuredly be able to win any mathematical argument with me. However, there are those whose mathematical knowledge is greater than either of ours, and they aren't in agreement with you on this matter. I'm afraid that I'll be deferring to them rather than to you.

On one thing we do agree though: neither of us are interested in continuing this debate, as it will lead nowhere.

[Qqmike]

I just lost a post, just a second ...

[jlittle]

I think what Qqmike is missing is that a real number is defined to be the limit of such sequences. Maths was stuck on "what is real number" till someone came up with that. 0.9999... equals 1 because it matches the definition of 1.

Regards, John Little

[Qqmike]

Yes, the wiki needs editing, it's ridiculous. It's a good example of not believing everything you read, esp on the Internet (vs professional, peer-reviewed, published sources).
It is true that 1 equals the infinite summation represented by the shorthand 0.999..... But 1 is not defined to be that summation! There are infinitely many equations where the number 1 is on the left or the right side. Infinitely many of them can be rigged up to involve various infinite summations. So what! Are there infinitely many definitions of the number 1 then? Let's define 1 to be
1 + limit (as x --> infinity of 1/x)
That's one of infinitely many ways. In that case, we are coming down on 1 (from numbers greater than 1) as opposed to increasing up to 1 (as we are with the infinite summation represented by the shorthand notation 0.999....).
Quit quoting this stuff and think about it, and you'll get it. To be clear, 1 is not defined to be 0.999 ... Rather, 1 is the (unique) multiplicative identity of the real field, the unique real number that has the property that for every real number x, 1*x = x*1 = x. I implore you, quit quoting this stuff where either the author is mistaken, or the author is quoted out of context, or where you may not fully understand the intent or context of the author. Here, do this for yourself: go get a mathematician! Ask him or her to review this thread. Get a personal, second opinion. It's not that serious, of course, but it would settle this. It is a neat thing that the series limit (as n --> infinity) (9*(10^(-1) + 10^(-2) + 10^(-3) + ... 10^(-n))) equals the real number 1, but that is not how "1" is defined in the real field. (The term "field" is meant technically, http://en.wikipedia.org/wiki/Field_%28mathematics%29) I'm trying to be helpful here as I think you can get this if you'd try. But I agree, I don't think it's going to happen. Mods: not to worry on this one; I do now finally and formally cease and desist! ;-)

[Qqmike]

Sorry, but this might be worth it, a post to keep peace here :-)

jlittle, your post kind of triggered this for me in a helpful way ...
Snowhog's post, #19, the first paragraph, is good, and I had to read it carefully.
Key, from that paragraph: the symbols "0.999..." and "1" represent the same number.
That is good!

I think maybe we are all basically on the same page here, HallationEffect, but it has been a matter of terminology, and maybe even more so, perspective. The culprits inciting riot have been these words: equal, represent, same as, and importantly (for me) "is defined to be." By 0.999... is meant an infinite string of symbols, which is meant to be that infinite summation we have hammered to death, which is actually a limit (as the number of summands approaches infinity of a series of finite sums), which by modern methods of limit theory can be shown to be EQUAL to the real number 1 (KEY: it does not define the element 1, but the limit is defined to be "1"). As a consequence, in particular, we may use the symbols 0.999.... and 1 interchangeably, at least in certain contexts. In fact, we may write 0.999.... = 1, for example. My focus here has been on viewing the reals as an algebraic field where we have agreed to represent the unique multiplicative identity by the symbol "1." This is a modern view. But there is the important historical view, too, which has its roots in computation; where early mathematicians worked to understand, define, and compute infinite summations (of which 0.999.... is an example).

So, I do now see what you guys may have been trying to say, and from what perspective. You are stuck on the (perhaps partly historical) definition and numerical value (the limit or convergence) of that infinite summation; I am stuck on viewing the reals as an algebraic field (where the symbol 1 is clearly--though arbitrarily--set aside, defined, for a special purpose, the multiplicative identity). Sorry for missing what I think may have been driving your perspective on this. I think we'd all agree on this stuff ... once we knew what it is we are each talking about! Don't tell me now that this is about apples and oranges; you might be right.

[kubicle]

I won't touch the mathematics (much), but philosophically (and logically) for two numbers (or the representations of two numbers) to be different, one should be able to put something "between" them, in other words:

There should be a non-zero X that would make 1 - X = 0.999... true.

If there is, the numbers are different.
if there isn't, the numbers are the same.

Side note, I'm sensing a logical problem with the first "simple proof" on the wiki:
1/9 = 0.111...
9 x 1/9 = 0.999...
1 = 0.999...
However, 1/9 = 0.111... is true only if 1 = 0.999... which is the thing we're trying to prove, so we're using the conclusion as a premise in circular logic (it's true if it's true).

[Qqmike]

kubicle, yes, kind of cutesy regarding the little "9 times" proof, it begs the question.
But also, the argument 1 - X = 0.999 ... normally makes sense, but not this time as we are abusing notation when we write the symbol 0.999.... . Familiar example to many people is e, which we know to be a certain well-known limit, this limit:

e = limit (n --> infinity) (1 + 1/n)^n

But the compact notation, 0.999.... gives no clue to most people that it is actually a limit, the limit of a series, the value of that limit is 1, or the limit equals 1, or the series (represented here) converges to a limit and that limit is 1, or the limit of the series (represented here) equals 1, and so on.

[HalationEffect]

I just knew it was going to be a problem of semantics in the end!

I was expressing it as a layman, you were expressing it as a mathematician... and it took a while for our differing explanatory methods to asymptotically approach each other (rimshot)

[Qqmike]

Yes, Halation, I completely agree, and I don't think either of us meant to be (too) huffy ;-)


If I may have a little technical aside with kubicle ... :-)

kublicle, actually, your 1 - X argument is insightful. It is true that

limit (as n --> infinity) (9*(10^(-1) + 10^(-2) + 10^(-3) + ... 10^(-n))) EQUALS the real number 1.

We can make the sums as close to 1 as we wish, arbitrarily close, by choosing enough summands, i.e., by choosing n large enough. How close do you want to represent 1 by a summand? Let's say by X units. Your X is my epsilon (Greek letter) in my (several) posts above.
Choose, for example, X = epsilon = 0.0001.
You could easily figure out an n large enough so the difference between

9*(10^(-1) + 10^(-2) + 10^(-3) + ... 10^(-n))

and the number 1

is less than 0.0001. Similarly, we could represent 1 by a sum to within an "error" (= X = epsilon) of 0.0000000001 by choosing enough summands, and so on.

THAT is a verbal description of the true mathematical proof that
limit (as n --> infinity) (9*(10^(-1) + 10^(-2) + 10^(-3) + ... 10^(-n))) EQUALS 1

and it is what you are thinking of with your X and it is what I've been trying to explain all along! Basically, you have given an intuitive, verbal definition of "limit." This what math teachers try to get math students to do on their own prior to all the fancy notation and symbols.

[Qqmike]

I like your rimshot, HalationEffect. And, of course, you do know what I mean by not being "too" huffy, right? Like, just huffy enough (where "enough" shall remain undefined), it's clear, right?

[kubicle]

But the compact notation, 0.999.... gives no clue to most people that it is actually a limit, the limit of a series, the value of that limit is 1, or the limit equals 1, or the series (represented here) converges to a limit and that limit is 1, or the limit of the series (represented here) equals 1, and so on.

Does it really matter (honest question, my mathematics days are long gone ) other than metaphysically that it is a limit?
I mean most people would say:
1 - X = 1
is true, when
X = limit ( n --> infinity) (1/10^n)
so
X = 0 (even though it's just a limit with a value of 0)

[Detonate]

You guys are way over my head, but I could turn you into great artillery fire direction officers pretty fast with your knowledge of math.

[Qqmike]

Yeah, you have a point there, kubicle.

The term "limit" is nothing more than the mathematician's (perhaps clumsy) way of saying, "it can be made arbitrarily close to."
In our case, when we write
0.999 ...
with the repeating periods (ellipsis) and 9's, that right there is a tip-off that something's up, that we are dealing not with 0.9 but with some other animal, so
1 = 0.999 ...
works in SOME understood contexts or discussions as a shorthand.
However, for some who might be thinking of the reals and an algebraic field with multiplicative identity symbolized by "1" OUCH to the eyes! And even more ouch if one were to even hint that we are "defining" 1 by this equation. However, we can define the limit (or the symbol "0.9999....") to be 1 by this equation
0.9999.... = 1
and it is OK.
Semantic, precision in, as Halation said above. Or, maybe even, personal "style" preference for rigor. And when you are drunk (which I was not last night, by chance), it may all be different.