<p>because [sum]n=1, INFINITY (9*10^-n)</p>
<p>how can you say that infinity does NOT relate to this problem if you use it in the proof? :)</p>
<p>Furthermore (and I know I am posting a lot to read), decimals and limits are not processes. According to this view, limits are nothing more than a “short-hand” for describing approximation schemes. I believe this idea comes from the descriptions used by math teachers to first introduce the idea of limits. Unfortunately, the student never moved beyond these original incomplete conceptions. The basic definition of the limit of a sequence (the particular type of limit needed here) is from Cauchy and involves epsilons and deltas - and if you understood this the proof would be obvious to you.</p>
<p>Similarly, 0.999… is not a process of continually adding more 9s, just as a limit is not a process. From the definitions for decimals and limits, it is evident that decimal expressions such as 0.999… as well as limits are defined to be particular numbers, not processes.</p>
<p>"because [sum]n=1, INFINITY (9*10^-n)</p>
<p>how can you say that infinity does NOT relate to this problem if you use it in the proof?"</p>
<p>Hriundeli, I sincerely hope that you are joking, because there is only one correct answer to this question. Infinity is just a concept, what I said was irrevelent was the notion that infinity does not exist in “real life”. Infinity in the context of limits is defined, rigorously, in terms of epsilon and delta.</p>
<p>ooook…
i gtg lol- was staying the whole day nearby
thanks River, Ben, Sagar for a nice argument, i really enjoyed it he-he</p>
<p>however, i’m still not fully convinced, and i’ll wait for the Ben’s proof</p>
<p>so i dont give up yet lol</p>
<p>btw
River, i wasnt joking, i sort of suck in math (i dont even fully know what;s integral)
so, pardon me he-he
i did NOT know that difinition of limit :P</p>
<p>I don’t like epilsons and delta’s. I know what they are, but they are very non intuitive for beginners, and almost made me give up calculus. I think I am going to give them another shot.</p>
<p>Hriundeli, if you don’t know what an integral is, than it might be that you haven’t covered the end of calc BC where they go indepth into things like this. I don’t know.</p>
<p>Have fun.</p>
<p>heey, heey
i didnt say i ever took calc!
i am self-studiing it though (now) and i just not up to the integrals yet
so…yeah hehe</p>
<p>only somewhat related (b/c i’ve hated discussions about this since 8th grade): “…6 is actually a surprisingly robust approximation for infinity.”
- a professor who will go unnamed, passed on to me by quakerboy2</p>
<p>
</p>
<p>Nope, that’s not correct. In the standard framework of mathematics that is universally accepted by professional mathematicians, .999… is a perfectly legitimate real number, equal to the geometric series \sum_{i=1}^\infty 9/10^k, which is equal to 1.</p>
<p>Secondly, have you taken calculus? Hopefully you have studied the fact that we can define infinite series, and evaluate them. For example \sum_{k=1}^\infty (1/2)^k = 1. We could define an alternate system in which we treated infinite sums differently and in which they didn’t come out to be real numbers, but this is a system that is inconsistent with the standard mathematical axioms.</p>
<p>You can manufacture axioms to give whatever result you want. For instance, River Phoenix gives a construction (perfectly consistent internally) of numbers that aren’t real, but can be quite well-defined. In this system, there are numbers like .000…1. (This is reminiscent of John Conway’s nonstandard analysis.) So, of course Hriundeli, you could define a system in which your claims hold. They’re just not the standard axioms of real analysis.</p>
<p>At this point I have to run… I’ll try to think of the shortest proof I can that no real number has more than two decimial expansions. On the other hand, Hriundeli, if you haven’t finished calculus, I doubt the technical arguments about infinite series that would be involved would make much sense… perhaps it would be better for you to learn a little more… otherwise it would just be wasted time for you and me.</p>
<p>Within the scope of highschool and undergraduate mathematics,</p>
<p>of course 0.9999… equals 1.</p>
<p>I don’t know why this stuff keeps coming up.</p>
<p>Or maybe this discussion is beyond my expertise?</p>
<p>My preferred proof (with nothing more than simple algebra):</p>
<p>Let 0.999… = x.</p>
<p>10x - x = 9.999… - 0.999…
9x = 9
x = 9/9
x = 1</p>
<p>All the proofs Ive seen thus far have infinite series and limits. I’m still skeptical. The dividing proof I’ve seen shown many times is not convincing</p>
<p>2007, just because you do not understand a proof does not mean it is invalid.</p>
<p>Agrophobic:EXCELLENT EXPLANATION!!!</p>
<p>you guys need to take a second and read Agro’s post :P</p>
<p>Agrophobic’s response is a good way to convince somebody, but it doesn’t really prove anything - actually it begs the question.</p>
<p>huh? why does it not prove anything??</p>
<p>you guys are missing the point…to borrow a popular concept, imagine there’s a bug trying to reach an apple and that it travels 9/10 of the remaining distance each minute. If it starts at 1 unit away from the apple, the total distance it travels in infinite amount of time will be 0.99999… But using common sense, we all know that since the bug only travels 9/10 of the remaining way, it never reaches the apple. so we can be sure that the total distance it travels is less than 1. Thus, 0.99999…cannot equal 1.</p>
<p>But, mathematicians have DEFINED 0.999…to be equal to 1, since it is more convenient this way. Same with infinite series. All kinds of paradoxes can rise in the frame of infinity, but mathematicians has simplified this problem. After all, mathematics is human invention.</p>
<p>So yes, for all mathematical intentions and purposes, 0.999… equals 1. and you can prove this mathematically using various methods…but those are only mathematical proof…the real nature of the two numbers are, I would argue, not identical.</p>
<p>I believe you can argue both ways…it’s only a matter of what perspective you take.</p>
<p>a12, if we are allowed to define 0.9999… to be whatever we want, as you did, I can prove that 0.9999 > 1! But there is only one definition for 0.999… that makes any sense, and it involves a ‘conceptual’ infinity, and NOT SOME SORT OF PROCESS WITH FINITE STEPS. There is no debate. Someone should close this thread.</p>
<p>Gandhiji, the problem is that the rigorous proof of 1/3 = 0.333… is identical to the proof of 1 = 0.999…
So if we are using the assumption that 1/3 = 0.333…, we already know the answer to the question.</p>