But even here, I think there is a non-trivial axiom lurking in the background. I don't know if it has been given a name.
I am not even sure that I can describe it correctly. It says that when we use an expression like "0. Perhaps it is a statement that the inductive definition of the natural numbers completely defines them. I welcome informed correction or discussion. Each of the steps could have been different. Many people would not agree with the principle of Archimedes; they would insist that infinitesimals "really exist.
Exclusion of infintesimals is just part of defining the "standard real numbers" as a subset of "all the real numbers. There is a theory of "surreal numbers," initiated by J.
Conway, which goes even farther than nonstandard analysis. In non-standard analysis, there are infinitesimals, but they are somewhat anonymous. A non-standard argument may say, "let epsilon be an infinitesimal; then In the surreal numbers, there is no reason why you could not give a name to the difference between 0.
I have a vague idea that models of nonstandard analysis can be constructed within Conway's surreal number Field.
In any case, there is no unique limit to the sequence 0. Well, if steps A and C of my pitiful attempt at a proof can be set aside, what about step B? I don't know. I don't really doubt that we all know how many beans make five, and we are all agreed on the answer. Your assumption is incorrect. There is no need to assume this. We need to be careful with unwarranted assumptions. Here's how: you show it later, the same way.
It is simple algebra, following all the rules. Stuff deleted. The formula is for proving that a non-terminating, repeating number is. The number of digits the decimal needs to move before the next set of. You have everything you need, right here! For the next series of equations the number of digits the decimal needs. Just as I said. Why not? You have just shown by "proofs" done exactly the same way how other decimal equivalences to fractions are calculated. Why are you afraid to face the validity of your own proof?
You have admirably proven what you are trying to disprove! John Pimentel. You showed it yourself. The decimal system does not claim unique representation all of the time. But we deal with the problems of non-unique representation all the time in algebra.
That is why we "simplify. I hope you will rethink your position and be guided by the algebraic proofs you have so competently performed. You are almost to the understanding you need. Regards, Raymond E. Well, it fits the pattern, which you conveniently editted from Earle's quote of your post:. I'm confused by your reply. It looks to me as if. I tink I done found two of them thar critters. I done give ya two of them above. I think you're the one who "proved" that.
Do you agree that. Maybe a definition of a brain that understands math in this universe. This is true aside from the speling erers So what? Sorry to disagree with you, Albert, but only on this point. I like giving him enough rope to make a noose and stick his head through it figuratively speaking, of course. I found a number well, a few between. Then I challenged him to find a similar number between.
This is fun. Until it gets boring. I have yet to see why this proof is incorrect John. Maybe you can enlighten me. Show me this logic please.
No, you have made a blatanly false assumption. You certainly haven't proven anything. And just how do you assume that 0. Maybe you are in another universe, in which case a lot would be explained.
Although it is a strange universe for 9 to equal It has to do with the way the number system is built. I would suggest a good course in analysis and proof at your local university to have it fully explained.
You really need this explained? Ok, I guess it is obvious from what you have written above that you do. If we assume that. I am still confused how you got your assumption that 0. Ok, how about 0.
Next challenge please. You see John, 0. If you can't see this then I'll put them under a common denominator for you. Actually there are an infinitely many numbers in between. Ok, go ahead. Oh, and I would like a common denominator too please. Nope, I understand them perfectly well. No fudging either. If you would like I will search for a textbook that shows this proven so you can read it for yourself?
Although any analysis textbook will give you all the tools to do this yourself. Just in case you really need a very formal proof I will give you one.
Ok, now that we have that under our belt, let us move on to our proof. This is obviously true though, but I will gladly prove it by induction if you feel it is necessary? There, overkill, but bullet proof. Have a nice day, Dave Weisbeck z The terms of inifinite sums may not be swapped, so 9.
It approaches 1 for an infinite number of terms, but there is a difference between "being 1" and "approaching 1 asymptotically".
People who are not convinced of this should take a course on differentials. Of course, for people in this thread who are, implicitly or explicitly mostly implicitly, it appears to me, and also mostly incompletely , defining.
Of course, my comments assume that we are not using conflicting definitions of "limit" or "approaches infinity" or "sum" or "goes from 1 to n", etc. But with the possible exception of the definition of "limit", those terms aren't where the difficulties in this thread lie. Alan Beban be Read carefully, and I think you'll find that convergent series can have their terms interchanged. These are not convergent series. Therefore rearranging terms cannot be directly justified if at all.
One must have a mathematically precise definition for "infinity" before math can be done with it. Tell me what it is. If you can't find one, they're two names for the same number. Please reread the definition of limit. If the limit exists, it is a number, not a process.
Again, if you define. If you define it some other way, we're not using the same language. Which is fine. Just explain how your definition differs from the standard one. See Principles of Mathematical Analysis, by Dr. In fact, your whole approach is rather strange. I believe you would accept that. Adding and subtracting term by term is allowed.
Perhaps you are confusing series summation with limits. Upcoming Events. Featured on Meta. Now live: A fully responsive profile. The unofficial elections nomination post. Linked Related Hot Network Questions. Mathematics Stack Exchange works best with JavaScript enabled. Accept all cookies Customize settings.
0コメント