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.)
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: https://www.physicsforums.com/showthread.php?t=125474&highlight=repeating+equal+1 https://www.physicsforums.com/showthread.php?t=110434&highlight=repeating+equal+1 https://www.physicsforums.com/showthread.php?t=106212&highlight=repeating+equal+1 etc. probably more than you would care to read.
shmoe: the guy's arguing that [itex]0.\bar{9} \neq 1[/itex], not that [itex]0.\bar{9} = 1[/itex]. Basically, he's just arguing semantics. He doesn't like how things are named. He knows full well that, in the decimals, [itex]0.\bar{9} = 1[/itex]. 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.
thhhbbbbpt! .99999... = the smallest real number not smaller than any finite decimal of form .9999....9 therefore it equals 1. case closed.
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 [itex]0.abc...= \Sigma a/10 + b/100+ c/1000+ \cdot\cdot\cdot[/itex]. There is no "ambiguity" except for people who simply do not know the definition of "series". Any calculus book will tell you that [itex]\Sigma_{n=0}^\infty a_n[/itex] is defined as the limit of the sequence of partial sums. In particular, the notation 0.999... means, by definition, the series [tex]\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[/tex]. End of discussion!
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?
Oh, and I love this quote 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".)
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".
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.
the fact that .9999... = 1.0000 .... is about as deep as the fact that f. delano roosevelt = franklin d. roosevelt
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?
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.
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?)
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.
Let me attempt a defense then: The sum: [tex]0.9\sum_{n=0}^{\infty}\frac{1}{10}^n[/tex] 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.
red alarm light comes on: that is 1. It's partial sums converge to 1. the rational numbers also have the property that 'between two rationals there is another rational', and they do not posses limits of sequences. However, they are dense in the reals in the proper meaning of the word (a set is dense in itself tautologically, if the notion of denseness makes sense at all.)
Very well Matt. I am struck by the similarities between the properties of non-linear systems and the geometry of math itself. Not non-linear geometry but the very geometry of mathematics itself: nested, fractal, and ergodic (the last property explaning why we can get to the same result from so many ways). But Philosophy it should be then.
What saltydog stated was the definition of a "dense ordering" -- an order is dense iff for any two elements, you can find a third between them.