Here’s my problem (my disconnect);
The difference between 1 and .9 is .1, between 1 and .99 is .01, 1 and .999 is .001, and on and on, right?
Those are finite.
Why can’t the difference between 1 and 0.999… be 0.000…? It can’t be 0.000…1 because that equals 0. because of limits.
Write it this way, 0.diff…
Then;
1=0.999… + 0.diff…, or 1=x+y. x’s increase is opposite and proportional and opposite to y’s decrease.
Why must we limit the difference, but not the 9’s?
I’m not argueing, I’m yearnin’ for learnin’. 
Peace,
mangeorge
The limit of the difference is zero, and the limit of the 9’s is one.
It would be more proper to say that 1 = x[sub]n[/sub] + y[sub]n[/sub], where x[sub]n[/sub] is the sum of 9 * (1/10)[sup]i[/sup] from 1 to n. y[sub]n[/sub] is equal to 1 - [sub]n[/sub].
This is difficult to discuss informally. I might get around to posting a primer on series, limits, and whatnot tomorrow.
Let’s change that so it reads correctly:
y[sub]n[/sub] is equal to 1 - x[sub]n[/sub].
So both these limits occur, or are, at the same place on the number line, which is at 1. Therefore 0.999… does indeed equal 1.
Right?
If it’s that damn simple I’m gonna…
Peace,
mangeorge
Well, strictly speaking, they are the same number because they fall on the same spot on a number line. They’re just written differently. Like how “hi” and “hello” mean just about the same thing.
There is no law of place value that says 0.d must be less than 1.
If we are dealing with the standard definitions of real numbers and the standard definitions of decimal representations then 1 = 0.999… I have no problem with hyper-reals, pseudo-real, infinitesimals and such, but I would strongly advise anyone to get a firm grasp on the standard definitions first (where 1=0.999…) or you will be completely lost.
I’m not sure, but I think you might be thinking about complex solutions. If you are dealing with the reals, and the solution of your quadratic involves the square root of a negative number, then you throw it out, because taking the square root of a negative number is not allowed over the reals. It all depends on what your definition of number is – real or complex.
I want to point out that according to the standard definition of decimal representation and limit, even though the phrase “approaches infinity” is used, there is no “approaching” nor is there any actual infinity. As long as you know what every decimal position is, you can calculate the value, and you do not actually approach anything, and you don’t have to write down an infinite number of decimal digits or perform an infinite number of additions. “Approaches infinity” is a formal phrase. You are not approaching anything and there is no infinity here. 0.999… does not approach one. 0.999… is one.
This is not a debate. There is a correct answer. A debate might be “Should we consider the standard definitions of real numbers or should we allow infinitesimal numbers?”
Virtually yours,
DrMatrix — Why do I feel like a moth drawn to a flame?
Yeah, if I understand what you’re saying. Don’t sweat it, though–it’s taken me longer to figure out simpler things.
Like I said earlier, this is something that doesn’t lend itself well to informal discussion. I might get around to posting that primer I talked about, but it would be later this afternoon, after I have a chance to synthesize it.
All right, sequences, series, and limits…here we go!
A sequence is a list of numbers, nothing more. We denote the nth element of a sequence by x[sub]n[/sub]. The sequence itself is denoted by <x[sub]n[/sub]>. Some sequences have the property that they get closer and closer to some number l as n increases. We call these sequences convergent.
Formally, we say that the sequence <x[sub]n[/sub]> converges to the limit l if, for any e > 0, there is a number N such that |x[sub]n[/sub]| < e whenever n > N. This might look confusing, so let’s think of it as a game. You give me some number e that’s greater than 0, and I have to find a number N. I’m bound by the rule that all the numbers in the sequence past the Nth number have to be close to l. Specifically, the distance between x[sub]n[/sub] and l must be less than e. Note that a particular sequence can have only one limit.
It may be helpful to think of a sequence as a list of approximations to l. Then e represents the error in approximating l from x[sub]n[/sub]. So the above definition is just a way of asserting that the error of approximation gets smaller and smaller as we take more approximations. This is just an analogy, though; it should not be taken to mean that x[sub]n+1[/sub] is defined in terms of x[sub]n[/sub]. Is everything clear so far?
Let’s define a sequence <s[sub]n[/sub]>. The nth number in the sequence, s[sub]N[/sub], is equal to the sum of the elements of some other sequence (call it <a[sub]n[/sub]>) from 1 to N. If this sequence converges, we call the limit S and denote it as the sum over n from one to infinity of a[sub]n[/sub]. This limit is called a series, and can be denoted as either S or the infinite sum above. If the sequence s[sub]n[/sub] doesn’t converge, there’s no series to talk of.
So what’s a decimal representation? It’s a sequence of numbers d[sub]n[/sub] that are each between 0 and 9 (inclusive; that is, d[sub]n[/sub] may equal either 0 or 9). This sequence represents the value of a number in a very specific way. For simplicity, let’s concentrate on numbers that are between 0 and 1. If <d[sub]n[/sub]> is the decimal representation of x, x is equal to the sum over n from 0 to infinity of d[sub]n[/sub]*(1/10)[sup]n[/sup].
So what’s the value of the number represented by 1.0… (which we normally just write as 1)? Well, d[sub]0[/sub] is 1, and all the other d’s are 0, so 1.0… represents 1 (remember that (1/10)[sup]0[/sup] is equal to 1). That’s comforting, isn’t it?
Now we address the value of .9… d[sub]0[/sub] is equal to zero, so .9… can be thought of as the sum over n from 1 to infinity of 9*(1/10)[sup]n[/sup]. Using a little algebra and some properties of limits, we can show that this is equal to 9/10 * 1/(1-1/10). 1/(1-1/10) is the same thing as 1/(9/10), which is equal to 10/9. So we have that .9… is equal to 9/10 * 10/9, which is equal to 1.
A good reference is K. Binmore’s Mathematical Analysis: a Straightforward Approach. The material on sequences and series can be found in chapters 4 and 6.
Wow! The only thing left to show is that every representation 0.d[sub]1[/sub]d[sub]2[/sub]d[sub]3[/sub]… is a convergent series and converges to a number in the closed interval [0,1]. We will leave that as an exercise for the reader. 
It is worth noting that even though the phrases “approaches infinity” and “goes to infinity” are used, infinity is used as a formal symbol here, and nothing actually goes to infinity. You don’t have to perform an infinite number of additions to verify that the value of 0.999… is 1.
That and the closed form of a geometric series, but who’s counting?
Looks like I got a new .sig.
And that’s just where I was stuck, seeking infinity.
Thanks, guys.
Peace,
mangeorge
okay, I think I followed all of that…I also think I see why I didn’t want to take any more calculus – forgetting everything you learned for some reason seems counterintuitive for me here (not that I ever have to do it for medicine, but that’s another issue)
So, does 0.999… equal one? Well the math says so, it seems counterintuitive to me, but the math still says so
Yeah, being counterintuitive is one of the things that math is really good at.
Okay, we’re settling out now, the thread is cooling off.
Ultrafilter–as you are the only one to respond to my last post–I may have expressed myself poorly.
I am saying that this debate is really about whether some given numeral (the symbol for a number) is NO MORE THAN a quantity, or whether it is BOTH a quantity and a concept…perhaps an ambiguous concept.
There is often a difference between how something functions in a well-defined process, and how we “think about it” (for lack of a better term).
It is absolutely true that two-thirds is the same quantity as 34-51ths. But the former has a sort of palpable meaning to me in addition to its raw quantity–the idea of two things in a group of three. The latter is just a numerical abstraction.
I suggest again: The underlying issue here is not whether .9rep is the same QUANTITY as 1 (that is ACCEPTED), but whether “what we think of as the meaning of” .9rep is the same as “what we think of as the meaning of” 1.
As .9rep suggests an endless ongoing process, and 1 suggests something finished and “there,” the two numerals signify different concepts.
Again: Frege’s distinction between sense and reference is relevant here.
:: eyes glaze over from all the math ::: since i never got past 8th grade math why dont we do this the easy way and ask unca cecil ? or one of his flunkies ?
We are his flunkies.
What, nobody took my hint?
1/3 in the base 12 is 0.4. No decimal repeats.
So it is, in your terms, representable on the number line - I simply place my pen exactly on 0.4.
Nobody ever said that the number line had to be in base 10. There is nothing “fundamental” about the base 10.
And as for (base 10) 0.99… = 0.33… + 0.33… + 0.33…
well in the base 12 that sum just becomes 0.4 + 0.4 + 0.4 = 1
Precisely 1. No converging limits to worry about.
So it comes down to this: do you believe that 0.333… (base 10) is the same thing as 0.4 (base 12)? If not, why not?
pan
That’s very subjective, though. I view 2/3 and 34/51 as equally abstract concepts.
Yes, but that’s only because we omit all the zeroes after the 1. If I wrote it as 1.0…, would they seem so different?
I’ll definitely give you that they appear different on the surface, but I don’t see any difference between the two concepts. If you want to discuss this further, you should probably start a thread in GD.
Never seen this one before. It’s quite elegant in its simplicity.
Why thankyou. And for once it is something I thought of myself, rather than transcribed from elsewhere.
Always nice to be appreciated
pan