# ? about .999~=1

by raven1
Tags: 9991
 P: 8 i came across am argument about does .999~=1 and someone used this webpage to show they are not equal http://www.math.fau.edu/Richman/HTML/999.htm this page seems somehow wrong to me but i havent gone far enough in math to disprove it , i just started to learn calclus so when it involes any in depth discussion of limits i try to be real carful what i say (Edited by HallsofIvy so that the url could be directly used.)
 Sci Advisor HW Helper P: 1,996 The conclusion is perfectly correct. I can't be bothered to read that page in any detail, but it looks alright at a first look. This has come up countless times before, so there's lots of nonsense to wade through: http://www.physicsforums.com/showthr...eating+equal+1 http://www.physicsforums.com/showthr...eating+equal+1 http://www.physicsforums.com/showthr...eating+equal+1 etc. probably more than you would care to read.
 Emeritus Sci Advisor PF Gold P: 16,101 shmoe: the guy's arguing that $0.\bar{9} \neq 1$, not that $0.\bar{9} = 1$. Basically, he's just arguing semantics. He doesn't like how things are named. He knows full well that, in the decimals, $0.\bar{9} = 1$. However, he wants to call some other number system the "decimal numbers". He also wants to allow the technical term "real number" to refer to something other than its technical definition.
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## ? about .999~=1

thhhbbbbpt!

.99999... = the smallest real number not smaller than any finite decimal of form .9999....9

therefore it equals 1. case closed.
 Math Emeritus Sci Advisor Thanks PF Gold P: 38,902 I notice the website includes "A skeptic who accepts the series interpretation could say that 0.999... converges to 1, or that it is equal to 1 in the limit, but is not equal to one. There is an ambiguity in standard usage as to whether the expression on the right stands for the series or the limit." That is completely wrong. It is not necessary to "accept the series interpretation". It is not an "interpretation", it is the definition of "base 10 representation" that $0.abc...= \Sigma a/10 + b/100+ c/1000+ \cdot\cdot\cdot$. There is no "ambiguity" except for people who simply do not know the definition of "series". Any calculus book will tell you that $\Sigma_{n=0}^\infty a_n$ is defined as the limit of the sequence of partial sums. In particular, the notation 0.999... means, by definition, the series $$\Sigma_{n=0}^\infty \left(0.9\right)\left(\frac{1}{10}\right)^n[/itex] That's a simple geometric series which has a simple formula: its sum (limit of the partial sums) is [tex]0.9\frac{1}{1- \frac{1}{10}}= 0.9\frac{1}{\frac{9}{10}}= 0.9\frac{10}{9}= 1$$. End of discussion!
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 Quote by Hurkyl shmoe: the guy's arguing that $0.\bar{9} \neq 1$, not that $0.\bar{9} = 1$.
That's what I get for not reading carefully! Somebody give me a beating.
 P: 113 It's listed on his website that he recieved his Phd from the University of Chicago I thought that was supposed to be a great school for math. Did this guy blow a fuse?
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Oh, and I love this quote
 Quote by Fred Richman Dedekind cuts are usually defined in the ring of rational numbers, but if we are interested in decimal numbers, we will want to work with a different ring.
Because "decimal numbers" are some kind of numbers other than "rational numbers"?

(When I am feeling really hard-nosed, I point out the "decimal numbers" is itself a mis-nomer. It should be "decimal numerals".)
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 I thought that was supposed to be a great school for math. Did this guy blow a fuse?
The thing is -- the guy is actually talking about reasonable and interesting stuff. It's just that, for some inexplicable reason, he's decided to graft it onto the 0.9~ vs 1 "debate".
P: 499
 Quote by HallsofIvy Because "decimal numbers" are some kind of numbers other than rational numbers"?
Yes, as I read it, his 'decimal numbers' are infinite strings of digits. The ring he is starting with isn't the 'decimal numbers' or the rational numbers, but the terminating decimals. What he is doing is adding some extra numbers x- to the real number system, so that 0.9~ can be taken to represent a different object to 1.0 . Now in mathematics you can invent whatever system takes your fancy, but whether it is of any interest is a different matter. It's not entirely clear whether he's adding a whole new copy of the reals R-, (in which case some of his numbers are no longer representable as infinite decimals, rather defeating the point) or just adding a copy of the terminating decimals. In either case the resulting object is no longer a field, or even an additive group - I'd prefer to be able to do subtraction.
 Sci Advisor HW Helper P: 9,428 the fact that .9999... = 1.0000 .... is about as deep as the fact that f. delano roosevelt = franklin d. roosevelt
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 Quote by chronon Yes, as I read it, his 'decimal numbers' are infinite strings of digits. The ring he is starting with isn't the 'decimal numbers' or the rational numbers, but the terminating decimals. What he is doing is adding some extra numbers x- to the real number system, so that 0.9~ can be taken to represent a different object to 1.0.
Well, that's a fine thing to do, but it brings up an important question to me. If 0.9~ != 1 in his system, then either 1/3 != 0.3~ or 0.3~ x 3 != 0.9~. Can he even do arithmetic on (the equivilent of) basic rational numbers?
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 Quote by mathwonk the fact that .9999... = 1.0000 .... is about as deep as the fact that f. delano roosevelt = franklin d. roosevelt
The way my father used it, that "f." was an obscenity!
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 Quote by mathwonk the fact that .9999... = 1.0000 .... is about as deep as the fact that f. delano roosevelt = franklin d. roosevelt
Might I disagree?

I believe the question has profound philosophical consequences:

It concerns the concept of a limit, that of the one Hall describes above. The limit exhibits a profound property of the Real numbers: they are dense. It is this simple property of the Reals which I beleive is responsible for Mathematics working so well in describing Nature. Nature too appears dense: no smallest small nor largest large. This synergy between math and nature emerges (my opinion) as a survival strategy by life as it seeks to live in a massively non-linear world: when in New York, act like a New Yorker. Thus evolves a likewise massively non-linear brain that creates a non-linear geometry we call Mathematics that enables life to ponder this question.
 Math Emeritus Sci Advisor Thanks PF Gold P: 38,902 Then I will jump in and disagree completely- this does not involve any property of the real numbers- it is entirely a matter of how we represent the real numbers in a "base 10 positional notation". If we were to use, say, base 3, then "0.9999....= 1.0" would not be true (although, I imagine that "0.2222...= 1.0" would be). If we used some representation that was not a positional notation, the question would never arise. This is a question about representation only, not about the real numbers. Indeed, 1 (and 0.9999....) are integers so, in particular, this has nothing to do with the "density" of the real numbers. (Whatever that means. The only "densities" I know of are relative to some set. Do you mean the fact that the rational numbers are dense in the real numbers?)
 Sci Advisor HW Helper P: 9,398 I think saltydog is attempting to describe the 'continuum' property. Which nature does not necessarily follow, or use, at all, saltydog. Lots of parts of nature behave in a quantized manner. The rest just seems to belong in philosophy, not mathematics, though I have no idea what geometry has to do with any of this, nor have I ever come across the term 'linear geometry' before.
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 Quote by HallsofIvy Then I will jump in and disagree completely- this does not involve any property of the real numbers-
Let me attempt a defense then:

The sum:

$$0.9\sum_{n=0}^{\infty}\frac{1}{10}^n$$

converges to 1 because between any two real number lies another real number (no holes). In this way the reals are "dense". My argument was not in regards to notation but to its relation to this property of the number system we create which bears a striking similarity to the apparently infinitely divisible nature of the Universe. Discussions about "0.99...=1" in my opinion reflect this beautiful connection between the geometry of real numbers and the geometry of nature.
 Quote by saltydog Let me attempt a defense then: The sum: $$0.9\sum_{n=0}^{\infty}\frac{1}{10}^n$$ converges to 1