I would suggest that people reading this keep in mind that by .999~ it is meant the convergence point of the infinite series in question. Lest they think there are two distinct Real numbers that are equal.
You have essentially stated in laymen terms the definition of convergence …
If the sequence s[n] has a limit, that is, if there is some s such that for all e > 0 there exists some N > 0 such that |s[n] - s| < e, then the series is called convergent.
Great minds think alike eh?
It has always been my dream to “Be One With Cecil”, but somehow I’ve always fallen a little short. This column has given me renewed hope.
[nitpick]
If the sequence s[n] has a limit, that is, if there is some s such that for all e > 0 there exists some N > 0 such that |s[n] - s| < e whenever n > N, then the series is called convergent.
[/nitpick]
I much prefer the ISO 8601 format for displaying the date (YYYY-MM-DD). I’ve been using it at least since 1997. It avoids favoring either of the two dominant standards and facilitates communication between international business partners.
A fact that mmay have been overlooked is that all numbers are infinitely repeating. The number 1 has an endless number of zeros behind its decimal point. The paradox in this mod is that we are comparing a real number 0.9999~ to an integer 1. All of our matematical rules tell us to compare inequalities with the same units, and we omited that step when we first looked at this mod. Convert all the units into real numbers and then consider our question.
is 0.9 = 1.0? No
is 0.99 = 1.00? No.
is 0.999 = 1.000? No.
as the limit of decimal digits approaces infinity, the 0.999~ approaches the value of one, but it never reaches the value of one.
I for one prefer to conider Rebecca Romjin as opposed to this stuff.
Later People
Ficer67
Sigh. By convention 0.999~ does not mean approaching infinity but with an infinite string of nines.
Therefore it does reach the value of one and 0.999~ is precisely the same number as 1.000~.
This page has a good overview of various ways of showing this, including my favorite way (as the sum of an infinite series):
http://aah.ryan-usa.com/node30.html
Note the trick at the end to find a ratio for any repeating decimal.
OK, accepting that the argument about limits within the realm of various numbering systems could go on “infinitely”, I would like to know if the picture associated with this thread was an intentional knock on Lint (the original poser of the question) or if it was simply an oversight…
Reviewing the equaiton in the picture…
.333
+.333
+.333
-
6
1.005 ???
I get 6.999, or assuming the .333s were meant to be non-terminating, ~7
yes mapcase, I am sorta sure I typed that
The original poser was Lint6, so the six comes from there. Clearly not an oversight.
I’m sorta sure you did not type anything whatsoever that said that 0.999~ = 1.000~
You typed:
Integers are real numbers, and 0.999~ is an integer, so I’m sorta sure that I don’t know what this is supposed to mean.
I’m sorta sure this is absolutely wrong. It does reach the value of one. That’s what the infinite string means.
Hint. Sarcasm only works if you are right.
In the last thread on this, one math-savvy poster explained that there was a difference between real numbers and integers. So, just to be safe, I think we should say that we are comparing the real number 0.999… with the real number 1.0. 
The distinction between the real numbers and the integers appears to be more of an issue in programming than in the world of math.
We progress from http://www.eskimo.com/~scs/cclass/mathintro/sx1.html:
and http://www.astro.warwick.ac.uk/~tda/F90notes/node21.html
to http://www.rbjones.com/rbjpub/maths/math007.htm
But note that last. We are saying - indeed defining - that 0.999~ = 1 is the limit of a convergent series of rationals. But 1 is an integer that is obviously also equal to 1. and 1.0 and 1.000~.
So in the case of our problem, we are talking about the same thing, the integers being a subset of the reals. There’s no need at the level at which we are speaking to get into Dedekind, except to slam the door shut on any and all efforts to save the inequality by saying things like, 0.999~ is the largest number that is not equal to one and similar nonsense.
It appears Slug meant the 6 to mean 0.006, designated by it’s placement under the column of .333s.
.333
.333
.333
6
-----
1.005
Neither Cecil nor anyone else (I think; I must admit I have not read every post in this thread) has mentioned the real source of the confusion. There is a number 1. Where it exists is for the philosophers to debate, but I like to think of it as in some sort of platonic reality of which our world is just the shadow. We have various notations for this number. One is 1, another is 1.0, a third is 1.00, …, then there is 1.000… and another is .999… . That’s all there is to it. Then we have various rules of arithmetic. One of its properties (rather non-trivial, when it comes to adding infinite decimals) is that if we have two notations for the same number and we add them to a third, the results will be two notations for the same number. For example, if you add 1 + 1, you get the same result as adding 1 + .999… . The rules for adding infinite decimals are anything but obvious. To see why, explain how to add .555… to itself. It can be done, but is not self-evident.
Why is it that .999… = 1 seems to bother so many people, but no one asks about 1/2 = 2/4? Yet it is the same question. Different ways of denoting the same number, both shadows of a real half.
I, for one, would claim, that they are a bit different questions.
Arithmetic on repeating fractions is simply not well defined.
If
1 times 1/3 = 0.333…
then
2 times 1/3 = 0.666…
3 times 1/3 = 0.999…
4 times 1/3 = 1.222…
5 times 1/3 = 1.555…
6 times 1/3 = 1.888…
but 6 times 1/3 = 2, so you would say 2 = 1.888…?
Heh, better check your arithmetic on that… it’s 1.999… But you might be right to say it’s not totally legitimate to perform operations on a repeating decimal like that.
Try this: Reverse the perspective of infinity. Since nothing is bigger than infinity, why not set a finite number and say that’s the most you can have, therefore that will be our infinity. Then say infinity is eleven. Nothing is bigger than eleven. So you would have eleven 9’s after the decimal point.
Now picture your 11 nines on a number line between 0 and 1.0. The distance on the number line between .999999999 and 1.0 will be zero, because there is no way to make a larger number without using additional decimals. The number of decimals in .9999999991 (for example) is twelve - beyond our “infinity” barrier of eleven. If the distance between two numbers is 0, they are the same number, so .999… = 1.0 using our mini-inifinity, and thus with the real one (so to speak).
Oops … I see where I went wrong. If we all agree
1/3 = 0.333…
2/3 = 0.666…
3/3 = 0.999…
and we continue the progression, we continue to believe adding 0.333… to the previous value represents the next 1/3, then
4/3 = 1.332…
5/3 = 1.665…
6/3 = 1.998…
If you can’t agree with that, then you can’t agree that 3/3 = 0.999…
Isn’t this fun? LOL
This is a long post, consider yourself warned.
It bothers so many people simply because it is quite nonintuitive (at least to those who do not understand what the notation means). It is a little unfair, IMHO, to say this is the same question as why 1/2=2/4. In this latter equation, it is quite intuitive that two forths and one half are the same number. Plus, both sides of the equation are in rational form, unlike .99…=1. This is a quite nonintuitive result to most lay people, because .9, .99, .999, .999999, and in fact .999…9 (that is, if there is a terminating 9 there no matter how far out it is) are all quite intuitively less than 1, even to those who are relative novices in mathematics.
Hence the 'non’intuitiveness of something like .99…=1. Even some people with a fair amount of mathematical training will look at that and quickly declare, “nope, it’s not 1. It’s limit is 1, but that’s not the same thing.” …until they stop and think a little more about it, to include refreshing their memory of the meaning of the notation if necessary.
I have perused this thread (as I have many threads in several forums on this topic) and I must say, it never ceases to amaze me how much confusion can be avoided (but somehow never is avoided) by simply understanding the notation, what it implies (ie what definitions are implied from the notation) and simply applying this information to conclude the truth of this equation.
Instead, some will throw around terms like series, convergence, limits, approaching, etc., without really understanding how these concepts are actually applied in the equation .99…=1.
A good deal has been mentioned about a particular series’ “convergence” to 1, and how this is allegedly not quite the same thing as “equals” 1, so I’ll address this first. Many who argue in that fashion indeed understand what a limit is, how a limit is defined, and to some extent what a series is.
The confusion seems to be rooted in not understanding what convergence is, and how this relates to the sum of a series. Certain remarks have left me the impression that some are making a distinction between the concept of the value of a series (ie can it be equated with a number and, if so, what is that number) and the value a series “converges” to.
To those who make such a distinction, please consider:
1 + 1 …this is an addition problem
2 …this is a number, which is
also the SUM of the above addition problem.
Questions: Does 1+1 = 2? IOW, does the SUM of the addition problem 1+1 EQUAL 2? Is it safe to say that “2” and “1+1” denote the same real number? That is, is there any difference between the VALUE of the expression “1+1” (ie the real number it equals) and the value of the SUM of 1 and 1?
Conclusion: There is no real functional difference between 1+1 and 2 in the sense that they both clearly denote the same real number. Thus, the VALUE of the expression 1+1 and the value of the actual sum (the number 2) are indeed the SAME number.
Confusion arises when misunderstandings exist regarding the infinite analog to the above, ie it is not as “clear” to some that the SUM of an infinite series and the VALUE of the series (ie the real number that it equals) are indeed the same thing in much the same way that they are the same in the finite example.
To say a series CONVERGES, simply means it has a SUM. Saying “this series converges to 1” and “the sum of this series is 1” are effectively saying the exact same thing. IOW, being redundant. How we go about finding this sum, if indeed we even can, is a little different from the finite case, but the point here is simply that “converges” and “has a sum” mean the exact same thing in the context of series.
…Which raises the question (and even more confusion) regarding the actual definition of convergence. First, a little about sequences and what it means to say one converges. A sequence is essentially a list of expressions, that’s all. The terms of a sequence are not combined in any way. It’s just a “list” of expressions. An infinite sequence (one with infinitely many terms) is said to converge if the limit of its nth term exists. That’s what “converge” means in the context of sequences.
An series is, plain and simply, an addition problem. IOW, some sequence can be taken and the terms of it added together and you have a series. It’s JUST an addition problem, not really any different from the addition problem 1+1. But unlike finite addition problems (like 1+1), not all infinite addition problems (a.k.a. infinite series) have a sum. Those that do are said to converge, while those that do not are said to diverge.
Hence the need to understand the definition (the instructions for) adding an infinite number of terms. IOW, what is the SUM of an infinite number of terms? Can we know the sum exists or not? If it does can we find the sum? As stated before, to say 1)a series converges, and 2)a series has a sum, is really being redundant, since they imply each other. The SUM of a series, and the number a series CONVERGES TO, if any, are two ways of saying the exact same thing. Much confusion can be avoided by simply accepting this, and not trying to formulate some argument against .99…=1 based on some alleged real “difference” between the two concepts “converges” and “has a sum.”
To summarize the definition of convergent and divergent series: An infinite series converges if it has a sum. If not, it diverges. A series converges (hence, has a sum) if the sequence of partial sums converges. As stated above, a sequence converges if its nth term has a limit (as n–>infty). Furthermore, if this limit exists, it is by definition the SUM of the series. Remember, the SUM (whatever that number is) can be interchanged freely with the actual series notation (ie the actual addition problem), in the same sense that the expression “2” can freely replace the expression “1+1” or vise-versa, because they both are just different names for the same number as we have shown.
.999… can be expressed as the series:
.9 + .09 + .009 + …
If you take the sequence of partial sums (that is, the sequence obtained by performing the additions .9+.09, then .9+.09+.009, then .9+.09+.009+.0009, and so on) you will get the infinite sequence:
{.99 , .999 , .9999 , …}
The limit of this sequence is 1. Even most of those against .99…=1 will not argue against this result.
However, for some strange reason, it will not click with some of those same people, that this result (1) is by definition the sum of the original infinite series (IOW, it’s what .99… is EQUAL to).
Those were just my words, but they are consistent with definitions you will find in calculus texts. The one I have handy at the moment is Larson, et al, 5th ed. which states the following:
"Definition of Convergent and Divergent Series
For the infinite series Sigma a_n, the nth partial sum is given by
S_n = a_1 + a_2 + … + a_n.
If the sequence of partial sums {S_n} converges to S, then the series Sigma a_n converges. The limit S is called the sum of the series.
S = a_1 + a_2 + … + a_n + …
If {S_n} diverges, then the series diverges."
This should (but for some strange reason often does not) dispell the notion that there is some functional difference between “converges to S” and “the sum is S” and this allegedly means that .99…<>1.
The actual sum of the series (ie the number the addition problem is ‘equal’ to, hence can be replaced with) is not approaching the limit S, close to the limit S, or anything at all like that. By definition, the sum IS the limit S. Look above at the line:
S = a_1 + a_2 + … + a_n + …
The left side is S, the limit of the sequence of partial sums. The right side is the actual addition problem (ie the ‘series’). As applied to our specific problem, this is very clearly telling you that 1 = .9+.09+.009+… (it’s already been made clear that .99… and .0+.09+.009+… are the same thing).
…Which leads to my second point (as if I haven’t been longwinded enough already). If they are not equal, then as one poster reminded us there is at least one number between the two. In fact, if they are not equal, there will be uncountably infinitely many numbers between the two. But no one can name a single one of them (and no one can, since one doesn’t exist.)
This argument may be slightly more appealing to a relative novice as opposed to the more formal argument above, since it is pretty intuitive that an expression such as .99…9 (that is, there is a string of 9’s no matter how long with a terminating 9 at the end), and this is indeed less than 1. Since a number indeed exists between .99(insert however many 9’s you want)9 and 1, then I can always find one greater than yours but still less than 1, by simply attaching more 9’s onto the string as appropriate.
However, one cannot find a number between .99… and 1 (the recurring pattern of 9’s, that is) because the string already has more 9’s there than yours (remember, “your” number has a terminating 9 at the end of some finite string.) It is not difficult, even for novices, to realize that if there is a number between .99… and 1, then it would be of the form .99…9 with a terminating 9 somewhere. Problem is, .99… has no such terminating position (the string goes on forever) which dispells any argument based on the existence of such a number between .99… and 1.
The other thing that amazes me is my bad habit of repeating things I have already documented. Those interested may wish to read my alt.algebra.help FAQ article on the subject:
http://aah.ryan-usa.com/node30.html
Regards,
D. Ryan