mm-66 === So, if there is a proof of statement S and someone >comes along with a proof of the negation of S and both >proofs are valid, then mathematics is doomed. > Doomed? I would think ?ding two such proofs would be the begining of a > most enlightening period for mathematics. Unless, of course, *every* valid > proof of S could somehow be turned into a valid proof of ~S. A small dose > of inconsistency could be a very good thing. > rich > If we take some set theory as the foundation of mathematics (much more than > 99% of current mathematics can be formulated in ZFC for example), then > the existence of correct proofs of some statement S and its negation > immediately implies that ALL statements formulated in the system are > provable. In that sense, there is no such thing as a small dose of > inconsistency. Depends on your point of view. Suppose the shortest formal proofs of any two contradictory statements are very long. There might then still be a lot of useful mathematics left -- provided we use a notion of bounded provability. Whether such a theory can be worked so the close dependency on details of the formalism can be abstracted (as the details of computing machinery can be abstracted from the class of polynomial-time complexity of recursive functions), I don't know. Another thing that could happen is that we ?d out that one of the axioms doesn't really capture what we wanted to capture, and needs to be re?ed. This is what happened a century ago, when (unbounded) comprehension was replaced with separation, and other axioms were introduced to recapture most of what would have been lost. I think I'm not alone in suspecting that the Powerset axiom may be the most vulnerable in this regard. It is in this sense that discovery of an inconsistency would rejuvenate maths. Michel. === >I make three assumptions: > >1) Every member of S can be examined. >2) I can determine if S(i) is less than, equal to, or greater than S(j) >3) S contains every rational number in [0,1) >If you think the proof is wrong please point out >which of my assumptions is false. x is not in S A truly stunning refutation. Russell - 2 many 2 count === >Well, it's not a proof since it contains the errors that others have >pointed out. But I do wish people would stop saying things like you >can't prove false things or you shouldn't attempt to. I'm not very sure >that mathematics is consistent. So, if there is a proof of statement >S and someone comes along with a proof of the negation of S and both >proofs are valid, then mathematics is doomed. > Doomed? I would think ?ding two such proofs would be the begining of a >most > enlightening period for mathematics. Unless, of course, *every* valid >proof of > S could somehow be turned into a valid proof of ~S. A small dose of > inconsistency could be a very good thing. > rich >If we take some set theory as the foundation of mathematics (much more than >99% of current mathematics can be formulated in ZFC for example), then >the existence of correct proofs of some statement S and its negation >immediately implies that ALL statements formulated in the system are >provable. In that sense, there is no such thing as a small dose of >inconsistency. >To reiterate: >Read an introductory mathematical logic text, and you will >see that if a theory is inconsistent, then that theory contains ALL >statements in the language of that theory, i.e., if one contradiction >can be found then ALL statements are provable. So, if Russell's proofs >are correct, then I can prove 1 = 2 and any other statement you could >dream of. Sounds like doom to me. Sure it is bad news for, say, ZFC. But is mathematics in its entirety doomed? There are *no* alternatives to ZFC within which 95% of current mathematics couldn't be re-formulated (without the inconsistency)? I don't know. My guess is that mathematicians, being the very clever folk they are, would ?d some system that avoided the inconsistency. And rather quickly. Like I stated initially, it would be a very enlightening period. Something to look forward to even (unless your work is likely to be part of the 4% that gets lost!). A *little* inconsistency would be a good thing for mathematics, just as it is good for all the other sciences. ;-) rich === > William Elliot >I can't prove that this equation : y^2=x^3 + 7 >has no integer solutions. Can you help me? >Assume some integral solution for y^2 = x^3 + 7 >not 7|x. Otherwise: 7|x, 7|y >0 = y^2 = x^3 + 7 = 7 (mod 7^2) which cannot be >x odd. Otherwise: x even, y odd, let y = 2n+1, x = 2m >4n^2 + 4n + 1 = 8m^3 + 7; 1 = 3 (mod 4) which cannot be >*** Is Z[sqr 7] a UFD, ie a unique factorization domain? *** > Yes, and a Euclidean domain. > LH So (1+sqrt(7))(-1+sqrt(7))=6=2*3. Thus 3 is not prime. What are its prime factors? Jon Miller === > William Elliot >I can't prove that this equation : y^2=x^3 + 7 >has no integer solutions. Can you help me? > Assume some integral solution for y^2 = x^3 + 7 > not 7|x. Otherwise: 7|x, 7|y > 0 = y^2 = x^3 + 7 = 7 (mod 7^2) which cannot be > x odd. Otherwise: x even, y odd, let y = 2n+1, x = 2m > 4n^2 + 4n + 1 = 8m^3 + 7; 1 = 3 (mod 4) which cannot be > *** Is Z[sqr 7] a UFD, ie a unique factorization domain? *** >Yes, and a Euclidean domain. >LH > So (1+sqrt(7))(-1+sqrt(7))=6=2*3. Thus 3 is not prime. What are its prime > factors? > Jon Miller Since 3 is an odd prime and 7 is a quadratic residue mod 3, then 3 splits in Z[sqrt(7)]. Note that X^2 - 7 = (X-1)*(X+1) mod 3. Let all ideals considered here be ideals of Z[sqrt(7)], the ring of integers of Q[sqrt(7)]. Let A = (3), the ideal generated by 3. Let B = (sqrt(7) - 1, 3), the ideal generated by sqrt(7)-1 and 3. Let C = (sqrt(7) + 1, 3), the ideal generated by sqrt(7)+1 and 3. The generators for B come from the ?st factor X-1 mod 3, where I substitute sqrt(7) for X. Likewise for the generators of C. Then, A = B*C. For, (sqrt(7)-1,3)*(sqrt(7)+1,3) = (6, 3*something, 3*something, 9) = (3), since 3 = 9 - 6. Thus, the two prime ideal factors of the ideal (3) are (sqrt(7)-1,3) and (sqrt(7)+1,3). Since Z[sqrt(7)] has class number 1, we can ?d a single generator for these two prime ideals. I would imagine that there are algorithms to do this. I don't know one off-hand. But, ?dling with the ideals can lead to the single generators. For example, (sqrt(7)-1,3) = (sqrt(7)+2,3) = (sqrt(7)+2), since (sqrt(7)+2)*(sqrt(7)-2) = 7-4 = 3. Thus, one prime factor of 3 is sqrt(7) + 2. I also stumbled upon the other prime factor in the above calculation: sqrt(7)-2. But, that second factor could also be obtained from ?ding the single generator of the ideal (sqrt(7)+1,3) in a similar manner. -- Bill Hale === >So (1+sqrt(7))(-1+sqrt(7))=6=2*3. Thus 3 is not prime. What are its prime >factors? >Since 3 is an odd prime and 7 is a quadratic residue mod 3, > then 3 splits in Z[sqrt(7)]. Aaaarrrggghh! I must be under too much stress. Of course, I could have scrounged up a scratch of paper to verify my calculations, but noooo, I didn't have time for that. Just post away without checking your work! Duh! Jon Miller === > *** Is Z[sqr 7] a UFD, ie a unique factorization domain? *** > Yes, and a Euclidean domain. > So (1+sqrt(7))(-1+sqrt(7))=6=2*3. Thus 3 is not prime. What are its prime > factors? sqrt(7)-2 and sqrt(7)+2 -- Don Reble djr@nk.ca === I would like to announce a new online mathematical journal math.e http://www.math.hr/~mathe/ for secondary school and undergraduate students, published by Croatian Mathematical Society. Matrix transformations of the plane; Vigenere cipher; Algebraic method for solving constructive problems; Mathematicians on postage stamps. Any comments are welcomed. Andrej Dujella (editor) === >I have determined that %77 of all ghosts are spotted within a 5 mile radius >of UFO sightings, and that %82 of all UFO sightings are accompanied by a >general increase in the intensity and overall magnitude of ghost related >hauntings. Consider the case where >77% of the potential population of observers live within a 5 mile radius of a putative UFO sighting. > >Therefore, I conclude, via statistics, that ghosts and UFO's are somehow >related, and that ghosts are in all probability using UFO's for >transportation purposes. > You have at best demonstrated a correlation, and not any particular causal link between the two classes of entity. Consider the hypothesis that the propulsion system of UFOs disturbs the spiritual harmony and hence cause an increase in ghostly activity. I'm sure anyone with a little imagination can come up with other hypotheses to explain your putative correlation. -- === >I have determined that %77 of all ghosts are spotted within a 5 mile radius >of UFO sightings, and that %82 of all UFO sightings are accompanied by a >general increase in the intensity and overall magnitude of ghost related >hauntings. Consider the case where >77% of the potential population of observers > live within a 5 mile radius of a putative UFO sighting. >Therefore, I conclude, via statistics, that ghosts and UFO's are somehow >related, and that ghosts are in all probability using UFO's for >transportation purposes. > You have at best demonstrated a correlation, and not any particular > causal link between the two classes of entity. Consider the hypothesis > that the propulsion system of UFOs disturbs the spiritual harmony and > hence cause an increase in ghostly activity. I'm sure anyone with a > little imagination can come up with other hypotheses to explain your > putative correlation. I have pretty much scrapped my original example because it did not really serve it's intended purpose. I guess that I was after something more like the following- A researcher is studying Earth's 5th moon. The researcher derives mountains of data regarding the Earth's 5th moon. A theory is developed, and others begin to study it as well. Pretty soon, it is regarded as a discipline unto itself. There is now a huge following of researchers and theorists. But, there is just one problem. There _is_no_5th_moon. I claim that this is the basis of a huge fraud currently known as psychology. You need not weigh in on this claim of mine, but the example above should pretty much illuminate what I was after in the original post. === >I'm not sure about math not qualifying as science, however. I think that it >may. What is the de?ition of a science ? Ask any American dean: a ?ld is a science if the major funding comes from the National Science Foundation. Or: who's going to vote your way in the university-wide decisions? Math obviously differs from biology or geology, but it's a science by these criteria! dave === >I'm not sure about math not qualifying as science, however. I think that it >may. What is the de?ition of a science ? Ask any American dean: a ?ld is a science if the major funding > comes from the National Science Foundation. Dutch deans evidently have a different rule. My philosophy department receives most of their funding from the NWO. Sadly, not the New World Order, nor the National Wrestling Nonetheless, they st this department under Technical Management . Oh, the agony. I'm practically part of a business school. -- Jesse F. Hughes [Mathematical] society has evolved far enough away from mainstream society that it has become rogue, and now is willing to push its needs against that of the majority. -- James S. Harris === >I'm not sure about math not qualifying as science, however. I think that it >may. What is the de?ition of a science ? Results must be veri?ble, >falsi?ble, and reproducible. I think that math satis?s these three >better than any physical science, and it does so with complete precision >and >an exactness which is unique to math. If you know something that I dont >please post. >Where are the experiments? >They are called conjectures and proofs of special cases of conjectures. > True enough. I suppose experiments can also be used to formulate > conjectures. Conjectures don't seem to play the same role as experiments in empirical science[1]. Conjectures in mathematics are surely analogous to conjectures in science. So, if there are analogues to experiments in mathematics, then they must provide (partial) evidence for or refutation of conjectures. Proofs of special cases can't be the analogues of experiments either. At best, they would have to be the analogues of positive outcomes of particular experiments, since such proofs provide some support (but never any negative support) for the conjecture. If there is anything that corresponds to experiments in mathematics, it would have to be the activity of *searching* for proofs or counterexamples. But in the empirical sciences, an important feature of experimentation is its repeatability. In order to ensure that what you've told me is correct, I may repeat your experiment. In mathematics, one does not really repeat the search, but only evaluates the outcome of the search (proof or counterexample) for correctness[2]. So, even here, the analogy is a bit strained -- at least if one takes repeatability as an essential feature of experimentation in science. As a matter of personal taste, I've never thought that the similarities between mathematics and empirical sciences were suf?ient to support the claim that mathematics is a kind of science. The fundamental feature of empirical science is its use of inductive, not deductive, reasoning. Empirical science is always open to later correction due to new data, whereas mathematics is only later corrected if there was an error on the part of mathematicians. Mathematics deals with (hypothetical) certainty and science with theory and conjecture as their basic bits. At least, if one stresses these aspects of the two activities, then there is reason to reject the claim that mathematics is a science. I recognize that one may think that other aspects of each of these ?lds are more essential, and so one may ?d the claim that mathematics is a science more plausible. Hence my explicit admission that these comments are really a matter of my personal taste. Footnotes: [1] Throughout, I'll take a traditional view of the scienti? method --- something like Hempel. One may complain that these explanations about the role of experiment and theory in empirical science are naive and maybe just plain wrong, but I think they'll serve well enough for now. Besides, naivete was always my forte (ooh, I sound so French). [2] Of course, repeating the search even when one has the proof or counterexample in front of him may be useful, just to give one additional understanding of *why* the proof/counterexample works . But this additional understanding is not necessary to con?m that the proof/counterexample is correct. -- [R]eality has a fascinating ability to check us when we get a little too big for our britches... Make no mistake. There isn't a mathematician alive today that I can't now touch, and not a mathematical career on the planet that I can't now affect. --James Harris, render of worlds === [.snip.] >Is the scienti? method taught in math classes? > Why is this relevant to whether or not mathematics, as practiced, > employ the scienti? method or not? >If it does not employ the scienti? method, then JSH's conjectures might as >well be taught at Harvard or Yale. Would you want that ? No. Math would >disintegrate without the scienti? method. I think you have me confused. I happen to think that mathematics is > science and uses the scienti? method, though the way in which math > is usually presented hides this. I agree. I think that math is a science as well. De?ately a science. > === === === > It's not denial. I'm just very selective about > what I accept as reality. > --- Calvin ( Calvin and Hobbes ) > === === === Arturo Magidin > magidin@math.berkeley.edu > === >I'm not sure about math not qualifying as science, however. I think that it >may. What is the de?ition of a science ? Results must be veri?ble, >falsi?ble, and reproducible. I think that math satis?s these three >better than any physical science, and it does so with complete precision >and >an exactness which is unique to math. If you know something that I dont >please post. >Where are the experiments? >They are called conjectures and proofs of special cases of conjectures. >True enough. I suppose experiments can also be used to formulate conjectures. Exactly. You do special cases, you do a bunch of examples, you test a number of speci? instances, and you make the conjecture. Sometimes, the conjecture is immediately followed by proof. Sometimes there is a large gap between one and the other in time. Often, parts of the conjecture are proven, others not. The main difference between mathematics and other sciences is that the standard of proof required in mathematics is far more stringent than that required in, say, physics. > Is the scienti? method taught in math classes? >Why is this relevant to whether or not mathematics, as practiced, >employ the scienti? method or not? >It isn't, really. I just assumed that if the scienti? method was a vital >part of mathematics it would be a vital part of math courses. Bad >assumption. Mathematics, as practiced, is clearly different from mathematics, as taught. >Why? Because mathematics places such a premium on proof. Just like in physics you don't describe all the experiments that you attempted to design but was unable to, and you only describe the experiments that were actually performed, if you ?d a proof for a theorem you do not usually go through explaining all the conjectures you made which proved to be false, or all the speci? examples you worked out (especially if you have a proof for the general case). Nonetheless, you will sometimes see a textbook (or more frequently, a or discussing possible guesses followed by counterexamples, or guesses followed by proofs that need only to ? certain details on the guesses, and so on. Because mathematics does have an ultimate arbiter (proof), it is not necessary in most instances to discuss the prior steps in the scienti? method as you do in empirical sciences, where no amount of veri?ation is enough to establish truth . There is also a tradition to polish proofs, which usually tends to draw them away from the experimentation or the thought process that led to the proof. Gauss used to say that the thought process that led to a proof was like the scaffolding in construction: once you have the building (the proof/theorem), you take away the scaffolding. (This tendency seems to be eroding some, in that more and more papers are taking the time to describe the intuition behind certain particularly technical proofs). But if you read papers that discuss evidence for conjectures, or that propose conjectures, you will see much of the same sort of things that you see in physics and other empirical sciences. -- === === === It's not denial. I'm just very selective about what I accept as reality. --- Calvin ( Calvin and Hobbes ) === === === Arturo Magidin magidin@math.berkeley.edu === What does it mean geometrically that I(X) + I(Y) is not a radical ideal, where X,Y are subsets of af?e n-space and I(X) for example is the ideal of functions in k[x_1,...,x_n] vanishing on it (k is algebraically closed) === I am somehow confused by the de?ition of limit in my math-book. At ?st an example: Let f(x) = x, x~=1 ( ~= is not equal to ) It's no question that lim[x->1] = 1, right? Now consider: g(x) = {x when x ~= 1, 10 when x = 1} Now, what is lim[x->1], if it exists at all? The de?ition of limit can be found on MathWorld at http://mathworld.wolfram.com/Limit.html (the epsilon-delta-de?ition ), and my mathbook de?es limit the same way, but I just can't really completely grasp the de?ition. Even though it seems to me that the de?ition would make no difference of lim[x->1] f(x) and lim[x->1] g(x). Which leads me to believing that lim[x->1] f(x) and lim[x->1] g(x) both equals 1. I guesse this is wrong, because my mathbook de?es a function f as continuous in x0 if x0 is member of f's domain and limit lim[x->x0] f(x) exists (and therefore is also equal to f(x0) . That parenthesis in the end troubles me, especialy as the de?ition of continuousity in x0 is by MathWorld (http://mathworld.wolfram.com/ContinuousFunction.html) de?ed as: 1. f(x0) is de?ed, so that x0 is in the domain of f 2. lim[x->x0] f(x) exists for x in the domain of f 3. lim[x->x0] f(x) = f(x0) ...and it's the third condition that makes me believe that something misses in my mathbooks de?ition. Anyway, my mathbook and MathWorld's de?itions of continuousity are different - my mathbook seems to consider condition 3. as a consequence of condition 1. and 2. in MathWorlds de?ition (this is what the parenthesis indicate) - while MathWorld seems to see all three conditions as necessary. Can anyone help me out? Carl windows-nt) Cancel-Lock: sha1:eFp8sRvuobj8zWqYL+pkUibWWF0= === Now consider: > g(x) = {x when x ~= 1, 10 when x = 1} > Now, what is lim[x->1], if it exists at all? The limit is still 1. The value at x=1 is ignored when determining the limit. > Which leads me to believing that lim[x->1] f(x) and lim[x->1] g(x) > both equals 1. Right. > I guesse this is wrong, because my mathbook de?es a function f as > continuous in x0 if x0 is member of f's domain and limit lim[x->x0] > f(x) exists (and therefore is also equal to f(x0) . Your text is wrong: for f to be continuous at x, the limit must exist, and it must equal f(x). Existence of the limit does not allow you to conclude that f(x) equals the limit: you yourself have given a counterexample above. > (http://mathworld.wolfram.com/ContinuousFunction.html) de?ed as: > 1. f(x0) is de?ed, so that x0 is in the domain of f > 2. lim[x->x0] f(x) exists for x in the domain of f > 3. lim[x->x0] f(x) = f(x0) ...and it's the third condition that makes me believe that something > misses in my mathbooks de?ition. Quite right. Undergrad textbooks vary tremendously. Some of them are shockingly poor. Len. === On Mon, 05 Jan 2004 23:30:42 +0100, c.j[dot]w >I am somehow confused by the de?ition of limit in my math-book. >At ?st an example: >Let f(x) = x, x~=1 ( ~= is not equal to ) > >It's no question that lim[x->1] = 1, right? >Now consider: >g(x) = {x when x ~= 1, 10 when x = 1} >Now, what is lim[x->1], if it exists at all? >The de?ition of limit can be found on MathWorld at >http://mathworld.wolfram.com/Limit.html (the > epsilon-delta-de?ition ), and my mathbook de?es limit the same way, >but I just can't really completely grasp the de?ition. Even though it >seems to me that the de?ition would make no difference of lim[x->1] >f(x) and lim[x->1] g(x). >Which leads me to believing that lim[x->1] f(x) and lim[x->1] g(x) both >equals 1. >I guesse this is wrong, No, everything is exactly right so far, at least with the usual de?ition of limit . >because my mathbook de?es a function f as > continuous in x0 if x0 is member of f's domain and limit lim[x->x0] >f(x) exists (and therefore is also equal to f(x0) . >That parenthesis in the end troubles me, especialy as the de?ition of >continuousity in x0 is by MathWorld >(http://mathworld.wolfram.com/ContinuousFunction.html) de?ed as: >1. f(x0) is de?ed, so that x0 is in the domain of f >2. lim[x->x0] f(x) exists for x in the domain of f >3. lim[x->x0] f(x) = f(x0) >...and it's the third condition that makes me believe that something >misses in my mathbooks de?ition. You're right, the two de?itions are absolutely not the same. _What_ is the de?ition of limit in the book? You should tell us _exactly_ what that de?ition is, word for word. If the book is using the word limit in the usual way, then the de?ition of continuous in the book is simply _wrong_, because of the parentheses and the word therefore - the correct de?ition is continuous at x0 if x0 is member of f's domain and limit lim[x->x0] f(x) exists and is equal to f(x0) . (But if the book's de?ition of limit is an unusual de?ition then the book's de?ition of continuous is correct. So you need to tell us what that book says limit means.) >Anyway, my mathbook and MathWorld's de?itions of continuousity are >different - my mathbook seems to consider condition 3. as a consequence >of condition 1. and 2. in MathWorlds de?ition (this is what the >parenthesis indicate) - while MathWorld seems to see all three >conditions as necessary. >Can anyone help me out? >Carl ************************ David C. Ullrich === In my reply to Toni I have written which of the two common de?itions of limit my book uses, but just to be sure, I'll give it exactly as it says (as before, as well translated into English as possible): Let f be a function and assume that every x near the point a is a member of f's domain. Then f is said to have the limit A when x approaches a if for every e > 0 there is some d = d(e) > 0 such that {|x - a| < d, x is member of f's domain} => |f(x) - A| < e. I guesse this is the answer to our concerns. The book isn't wrong, as earlier discussed, it just uses another de?ition of the concept limit than MathWorld (and most of the world?) does, and with this other de?ition that other de?ition on continuity is right. Is that so? > On Mon, 05 Jan 2004 23:30:42 +0100, c.j[dot]w >I am somehow confused by the de?ition of limit in my math-book. >At ?st an example: >Let f(x) = x, x~=1 ( ~= is not equal to ) >It's no question that lim[x->1] = 1, right? >Now consider: >g(x) = {x when x ~= 1, 10 when x = 1} >Now, what is lim[x->1], if it exists at all? >The de?ition of limit can be found on MathWorld at >http://mathworld.wolfram.com/Limit.html (the > epsilon-delta-de?ition ), and my mathbook de?es limit the same way, >but I just can't really completely grasp the de?ition. Even though it >seems to me that the de?ition would make no difference of lim[x->1] >f(x) and lim[x->1] g(x). >Which leads me to believing that lim[x->1] f(x) and lim[x->1] g(x) both >equals 1. >I guesse this is wrong, > No, everything is exactly right so far, at least with the usual > de?ition of limit . >because my mathbook de?es a function f as > continuous in x0 if x0 is member of f's domain and limit lim[x->x0] >f(x) exists (and therefore is also equal to f(x0) . >That parenthesis in the end troubles me, especialy as the de?ition of >continuousity in x0 is by MathWorld >(http://mathworld.wolfram.com/ContinuousFunction.html) de?ed as: >1. f(x0) is de?ed, so that x0 is in the domain of f >2. lim[x->x0] f(x) exists for x in the domain of f >3. lim[x->x0] f(x) = f(x0) >...and it's the third condition that makes me believe that something >misses in my mathbooks de?ition. > You're right, the two de?itions are absolutely not the same. > _What_ is the de?ition of limit in the book? You should tell us > _exactly_ what that de?ition is, word for word. > If the book is using the word limit in the usual way, then the > de?ition of continuous in the book is simply _wrong_, > because of the parentheses and the word therefore - > the correct de?ition is continuous at x0 if x0 is member of f's domain and limit lim[x->x0] > f(x) exists and is equal to f(x0) . > (But if the book's de?ition of limit is an unusual de?ition > then the book's de?ition of continuous is correct. So you > need to tell us what that book says limit means.) >Anyway, my mathbook and MathWorld's de?itions of continuousity are >different - my mathbook seems to consider condition 3. as a consequence >of condition 1. and 2. in MathWorlds de?ition (this is what the >parenthesis indicate) - while MathWorld seems to see all three >conditions as necessary. >Can anyone help me out? >Carl > ************************ > > David C. Ullrich === >In my reply to Toni I have written which of the two common de?itions >of limit my book uses, but just to be sure, I'll give it exactly as it >says (as before, as well translated into English as possible): Let f be a function and assume that every x near the point a is a >member of f's domain. Then f is said to have the limit A when x >approaches a if for every e > 0 there is some d = d(e) > 0 such that >{|x - a| < d, x is member of f's domain} => |f(x) - A| < e. >I guesse this is the answer to our concerns. The book isn't wrong, as >earlier discussed, it just uses another de?ition of the concept > limit than MathWorld (and most of the world?) does, and with this >other de?ition that other de?ition on continuity is right. Is that so? Yes, with this de?ition, the de?ition given of continuity agrees with the usual notion of continuity. Unfortunately, it seems like a poor de?ition of limit, since it does not allow for a function to have a limit at a point where it is de?ed UNLESS it happens to be continuous at that point. -- === It's not denial. I'm just very selective about what I accept as reality. --- Calvin ( Calvin and Hobbes ) === Arturo Magidin magidin@math.berkeley.edu === > I am somehow confused by the de?ition of limit in my math-book. > At ?st an example: > Let f(x) = x, x~=1 ( ~= is not equal to ) > It's no question that lim[x->1] = 1, right? You probably mean lim_{x->1}(f(x)) = 1. The limit operation requires you to specify what variable is going to what value (in this case, it's {x->1}), as well as the function you are attempting to take to whatever limiting value (if any) exists for the corresponding value of the variable (in this case, f(x)). > Now consider: > g(x) = {x when x ~= 1, 10 when x = 1} > Now, what is lim[x->1], if it exists at all? > Here, I suspect you mean lim_{x->1}(g(x)). That limit is the same as for the preceding example, that is, lim_{x->1} (g(x)) = 1. The limit of a function is unrelated to its value at the point in question; the de?ition of continuity relates the two values. > The de?ition of limit can be found on MathWorld at > http://mathworld.wolfram.com/Limit.html (the > epsilon-delta-de?ition ), and my mathbook de?es limit the same way, > but I just can't really completely grasp the de?ition. Even though it > seems to me that the de?ition would make no difference of lim[x->1] > f(x) and lim[x->1] g(x). > Which leads me to believing that lim[x->1] f(x) and lim[x->1] g(x) both > equals 1. > That's correct. > I guesse this is wrong, because my mathbook de?es a function f as > continuous in x0 if x0 is member of f's domain and limit lim[x->x0] > f(x) exists (and therefore is also equal to f(x0) . > You have guessed wrong. That's no crime, but rather an opportunity to correct whatever mistakes you have made in your understanding of this topic. You have misread the de?ition. It surely says that the function f(x) is continuous at x=x0 if lim_{x->x0} (f(x)) = f(x0). That statement itself requires that the value x=x0 is in the domain of f (otherwise, the expression f(x0) is unde?ed). The fact that the limit, as x-->x0, of f(x) exists has no bearing on whether f(x0) is equal to that limit. Your function g(x) is not continuous at x=1. If I am mistaken in what your book's de?ition says, and it does indeed state that the function is continuous if the limit exists, then the book is incorrect. If it further states that if the limit exists, then it's equal to f(x0), it's wrong again. > That parenthesis in the end troubles me, especialy as the de?ition of > continuousity in x0 is by MathWorld > (http://mathworld.wolfram.com/ContinuousFunction.html) de?ed as: > 1. f(x0) is de?ed, so that x0 is in the domain of f > 2. lim[x->x0] f(x) exists for x in the domain of f > 3. lim[x->x0] f(x) = f(x0) > ...and it's the third condition that makes me believe that something > misses in my mathbooks de?ition. > Anyway, my mathbook and MathWorld's de?itions of continuousity are > different - my mathbook seems to consider condition 3. as a consequence > of condition 1. and 2. in MathWorlds de?ition (this is what the > parenthesis indicate) - while MathWorld seems to see all three > conditions as necessary. > The de?ition provided in MathWorld is the standard de?ition of continuity. The expression continuosity is nonstandard, so you might as well drop it in favor of the correct terminology. > Can anyone help me out? > As a point of information, please reread your textbook's de?ition of continuity. If it (a) uses the word continuosity , or (b) makes the claim that the equality of f(x0) and lim_{x->x0} (f(x)) follows from the existence of that limit, do us all a favor and let us know what text this is (in particular, give us the title, author, and publisher) so we can have a good laugh. > Carl > Dale === I'm surprised by the number of quick answers to my post - thank you all. I will read through your answers and replys and answer them more precisely tomorrow, but before I go to bed, while there's indications of my mathbook maybe being wrong, I would just like to write down, as clear as possible, exactly what my mathbook says on continuity. As well translated from Swedish as I can: De?ition. A function f is said to be *continuous in x0* if x0 is member of f's domain and if the limit lim[x->x0] f(x) exists (and therefore is automatically equal to the value of f in x0). - If a function is continous in every x that is a member of f's domain, the function is called *continuous*. I would be surprised if the book is really wrong - if you claim it is, can't the above be true for some other de?ition of the limit concept, or something like that? I'll read through your and the other's posts more carefully tomorrow. > I am somehow confused by the de?ition of limit in my math-book. > At ?st an example: > Let f(x) = x, x~=1 ( ~= is not equal to ) > It's no question that lim[x->1] = 1, right? > You probably mean lim_{x->1}(f(x)) = 1. The > limit operation requires you to specify what > variable is going to what value (in this case, > it's {x->1}), as well as the function you are > attempting to take to whatever limiting value > (if any) exists for the corresponding value > of the variable (in this case, f(x)). > > Now consider: > g(x) = {x when x ~= 1, 10 when x = 1} > Now, what is lim[x->1], if it exists at all? > Here, I suspect you mean lim_{x->1}(g(x)). That > limit is the same as for the preceding example, > that is, lim_{x->1} (g(x)) = 1. > The limit of a function is unrelated to its value > at the point in question; the de?ition of > continuity relates the two values. > The de?ition of limit can be found on MathWorld at > http://mathworld.wolfram.com/Limit.html (the > epsilon-delta-de?ition ), and my mathbook de?es limit the same > way, but I just can't really completely grasp the de?ition. Even > though it seems to me that the de?ition would make no difference of > lim[x->1] f(x) and lim[x->1] g(x). > Which leads me to believing that lim[x->1] f(x) and lim[x->1] g(x) > both equals 1. > That's correct. > > I guesse this is wrong, because my mathbook de?es a function f as > continuous in x0 if x0 is member of f's domain and limit lim[x->x0] > f(x) exists (and therefore is also equal to f(x0) . > You have guessed wrong. That's no crime, but rather an opportunity to > correct whatever mistakes you have made in your understanding of this > topic. > You have misread the de?ition. It surely says that the function f(x) > is continuous at x=x0 if lim_{x->x0} (f(x)) = f(x0). That statement > itself requires that the value x=x0 is in the domain of f (otherwise, > the expression f(x0) is unde?ed). The fact that the limit, as x-->x0, > of f(x) exists has no bearing on whether f(x0) is equal to that limit. > Your function g(x) is not continuous at x=1. > If I am mistaken in what your book's de?ition says, and it does indeed > state that the function is continuous if the limit exists, then the book > is incorrect. If it further states that if the limit exists, then it's > equal to f(x0), it's wrong again. > That parenthesis in the end troubles me, especialy as the de?ition > of continuousity in x0 is by MathWorld > (http://mathworld.wolfram.com/ContinuousFunction.html) de?ed as: > 1. f(x0) is de?ed, so that x0 is in the domain of f > 2. lim[x->x0] f(x) exists for x in the domain of f > 3. lim[x->x0] f(x) = f(x0) > ...and it's the third condition that makes me believe that something > misses in my mathbooks de?ition. > Anyway, my mathbook and MathWorld's de?itions of continuousity are > different - my mathbook seems to consider condition 3. as a > consequence of condition 1. and 2. in MathWorlds de?ition (this is > what the parenthesis indicate) - while MathWorld seems to see all > three conditions as necessary. > The de?ition provided in MathWorld is the standard de?ition of > continuity. The expression continuosity is nonstandard, so you might > as well drop it in favor of the correct terminology. > Can anyone help me out? > As a point of information, please reread your textbook's de?ition > of continuity. If it (a) uses the word continuosity , or (b) makes > the claim that the equality of f(x0) and lim_{x->x0} (f(x)) follows > from the existence of that limit, do us all a favor and let us know > what text this is (in particular, give us the title, author, and > publisher) so we can have a good laugh. > Carl > Dale > === Just before I go to sleep, worried to wrongfully indicate my mathbook of being wrong... I was wrong before, my mathbook *does* have another de?ition of the limit concept than MathWorld. See my reply to Toni's post for the details. > I'm surprised by the number of quick answers to my post - thank you all. > I will read through your answers and replys and answer them more > precisely tomorrow, but before I go to bed, while there's indications of > my mathbook maybe being wrong, I would just like to write down, as > clear as possible, exactly what my mathbook says on continuity. > As well translated from Swedish as I can: De?ition. A function f is said to be *continuous in x0* if x0 is > member of f's domain and if the limit > lim[x->x0] f(x) > exists (and therefore is automatically equal to the value of f in x0). > - If a function is continous in every x that is a member of f's domain, > the function is called *continuous*. > I would be surprised if the book is really wrong - if you claim it is, > can't the above be true for some other de?ition of the limit concept, > or something like that? > I'll read through your and the other's posts more carefully tomorrow. > I am somehow confused by the de?ition of limit in my math-book. > At ?st an example: > Let f(x) = x, x~=1 ( ~= is not equal to ) > It's no question that lim[x->1] = 1, right? > You probably mean lim_{x->1}(f(x)) = 1. The > limit operation requires you to specify what > variable is going to what value (in this case, > it's {x->1}), as well as the function you are > attempting to take to whatever limiting value > (if any) exists for the corresponding value > of the variable (in this case, f(x)). > Now consider: > g(x) = {x when x ~= 1, 10 when x = 1} > Now, what is lim[x->1], if it exists at all? > Here, I suspect you mean lim_{x->1}(g(x)). That > limit is the same as for the preceding example, > that is, > lim_{x->1} (g(x)) = 1. > The limit of a function is unrelated to its value > at the point in question; the de?ition of > continuity relates the two values. > The de?ition of limit can be found on MathWorld at > http://mathworld.wolfram.com/Limit.html (the > epsilon-delta-de?ition ), and my mathbook de?es limit the same > way, but I just can't really completely grasp the de?ition. Even > though it seems to me that the de?ition would make no difference of > lim[x->1] f(x) and lim[x->1] g(x). > Which leads me to believing that lim[x->1] f(x) and lim[x->1] g(x) > both equals 1. > That's correct. > I guesse this is wrong, because my mathbook de?es a function f as > continuous in x0 if x0 is member of f's domain and limit lim[x->x0] > f(x) exists (and therefore is also equal to f(x0) . > You have guessed wrong. That's no crime, but rather an opportunity to > correct whatever mistakes you have made in your understanding of this > topic. > You have misread the de?ition. It surely says that the function f(x) > is continuous at x=x0 if lim_{x->x0} (f(x)) = f(x0). That statement > itself requires that the value x=x0 is in the domain of f (otherwise, > the expression f(x0) is unde?ed). The fact that the limit, as x-->x0, > of f(x) exists has no bearing on whether f(x0) is equal to that limit. > Your function g(x) is not continuous at x=1. > If I am mistaken in what your book's de?ition says, and it does indeed > state that the function is continuous if the limit exists, then the book > is incorrect. If it further states that if the limit exists, then it's > equal to f(x0), it's wrong again. > That parenthesis in the end troubles me, especialy as the de?ition > of continuousity in x0 is by MathWorld > (http://mathworld.wolfram.com/ContinuousFunction.html) de?ed as: > 1. f(x0) is de?ed, so that x0 is in the domain of f > 2. lim[x->x0] f(x) exists for x in the domain of f > 3. lim[x->x0] f(x) = f(x0) > ...and it's the third condition that makes me believe that something > misses in my mathbooks de?ition. > Anyway, my mathbook and MathWorld's de?itions of continuousity are > different - my mathbook seems to consider condition 3. as a > consequence of condition 1. and 2. in MathWorlds de?ition (this is > what the parenthesis indicate) - while MathWorld seems to see all > three conditions as necessary. > The de?ition provided in MathWorld is the standard de?ition of > continuity. The expression continuosity is nonstandard, so you might > as well drop it in favor of the correct terminology. > Can anyone help me out? > As a point of information, please reread your textbook's de?ition > of continuity. If it (a) uses the word continuosity , or (b) makes > the claim that the equality of f(x0) and lim_{x->x0} (f(x)) follows > from the existence of that limit, do us all a favor and let us know > what text this is (in particular, give us the title, author, and > publisher) so we can have a good laugh. > Carl > Dale > === > .... > Let f(x) = x, x~=1 ( ~= is not equal to ) > It's no question that lim[x->1] = 1, right? Right. > Now consider: > g(x) = {x when x ~= 1, 10 when x = 1} > Now, what is lim[x->1], if it exists at all? This also is 1. > The de?ition of limit can be found .... (the epsilon-delta- > de?ition ), but I just can't really completely grasp the de?ition. > Even though it seems to me that the de?ition would make no difference > of lim[x->1] f(x) and lim[x->1] g(x). > Which leads me to believing that lim[x->1] f(x) and lim[x->1] g(x) both > equals 1. You're doing well. The epsilon-delta machinery took some of the world's best mathematicians a couple of centuries to sort out. If Newton and Euler didn't manage to get it really clear, you needn't be ashamed if it takes you a few weeks or months. :-) Something that may help you right here is that lim[x->1] f(x) depends on the values of x _near_ 1 but not _at_ 1 itself. > .... my mathbook de?es a function f as continuous in x0 if .... Continuity is another matter. The property of being continuous at 1 involves the values of x _both_ near 1 _and_ at 1 itself. Do you see why that makes a difference? Ken Pledger. === On Mon, 05 Jan 2004 23:30:42 +0100, c.j[dot]w >I am somehow confused by the de?ition of limit in my math-book. >At ?st an example: >Let f(x) = x, x~=1 ( ~= is not equal to ) >It's no question that lim[x->1] = 1, right? >Now consider: >g(x) = {x when x ~= 1, 10 when x = 1} >Now, what is lim[x->1], if it exists at all? This depends on your de?ition of limit. Some texts use: For any epsilon > 0 such that |f(x) - f(x0)| < epsilon we can ?d delta > 0 such that 0 < |x - x0| < delta. Some use: For any epsilon > 0 such that |f(x) - f(x0)| < epsilon we can ?d delta > 0 such that |x - x0| < delta. According to the ?st one the limit of g exists, according to the second one not. One of problems of picking up elementary analysis texts is that even the seasoned mathematicians in this group rarely seem to come together on which de?itions should be used. === > On Mon, 05 Jan 2004 23:30:42 +0100, c.j[dot]w >I am somehow confused by the de?ition of limit in my math-book. >At ?st an example: >Let f(x) = x, x~=1 ( ~= is not equal to ) >It's no question that lim[x->1] = 1, right? >Now consider: >g(x) = {x when x ~= 1, 10 when x = 1} >Now, what is lim[x->1], if it exists at all? > This depends on your de?ition of limit. Some texts use: For any epsilon > 0 such that |f(x) - f(x0)| < epsilon we can ?d > delta > 0 such that 0 < |x - x0| < delta. > Some use: For any epsilon > 0 such that |f(x) - f(x0)| < epsilon we can ?d > delta > 0 such that |x - x0| < delta. I had not looked careful enough, my book's de?ition *does* differ from MathWorld's. My books de?ition is the second one, while MathWorlds is the ?st one, that, is, my book uses |x - x0| < delta and MathWorld uses 0 < |x - x0| < delta . Might this explain the whole thing? > According to the ?st one the limit of g exists, according to the > second one not. > One of problems of picking up elementary analysis texts is that even > the seasoned mathematicians in this group rarely seem to come together > on which de?itions should be used. === * c. j. w. > I guesse this is wrong, because my mathbook de?es a function f as > continuous in x0 if x0 is member of f's domain and limit lim[x->x0] > f(x) exists (and therefore is also equal to f(x0) . This would have mislead me too. > That parenthesis in the end troubles me, especialy as the de?ition > of continuousity in x0 is by MathWorld > (http://mathworld.wolfram.com/ContinuousFunction.html) de?ed as: > 1. f(x0) is de?ed, so that x0 is in the domain of f > 2. lim[x->x0] f(x) exists for x in the domain of f > 3. lim[x->x0] f(x) = f(x0) This is correct. The limit lim[x->x0] f(x) does not need to be identical to f(x0) as both books indicate. -- Jon Haugsand === c.j[dot]w < c.j[dot]w @telia.com> scribbled the following on sci.math: > I am somehow confused by the de?ition of limit in my math-book. > At ?st an example: > Let f(x) = x, x~=1 ( ~= is not equal to ) > It's no question that lim[x->1] = 1, right? > Now consider: > g(x) = {x when x ~= 1, 10 when x = 1} > Now, what is lim[x->1], if it exists at all? I guess it doesn't exist. lim[x->1-]g(x) and lim[x->1+]g(x) are both 1, but g(1) is 10. Because g(1) exists but is different from the side limits, lim[x->1]g(x) can't exist. That's my opinion at least. Feel free to differ. -- /-- Joona Palaste (palaste@cc.helsinki.? ------------- Finland -------- -- http://www.helsinki.?~palaste --------------------- rules! --------/ Insanity is to be shared. - Tailgunner === > c.j[dot]w < c.j[dot]w @telia.com> scribbled the following on sci.math: > I am somehow confused by the de?ition of limit in my math-book. > At ?st an example: > Let f(x) = x, x~=1 ( ~= is not equal to ) > It's no question that lim[x->1] = 1, right? > Now consider: > g(x) = {x when x ~= 1, 10 when x = 1} Now, what is lim[x->1], if it > exists at all? > I guess it doesn't exist. lim[x->1-]g(x) and lim[x->1+]g(x) are both 1, > but g(1) is 10. Because g(1) exists but is different from the side limits, > lim[x->1]g(x) can't exist. > That's my opinion at least. Feel free to differ. My understanding of the limit concept is that a function need not be de?ed at a point for a limit to exist at that point. For example, the OP's function g has lim_{x->1}g(x) = 1. Proving the limit exists at x = 1... Let e > 0 be given. We need to ?d d > 0, such that |x-1| < e if |x-1| < d. Clearly we can choose d = e and the limit is proven to exist at x = 1. Now to prove continuity of the function at x = 1, we would have to show that lim_{x->1} g(x) = g(1). We have g(1) = 10, and lim{x->1} g(x) = 1, so g is not continuous at x = 1. === >c.j[dot]w < c.j[dot]w @telia.com> scribbled the following on sci.math: >I am somehow confused by the de?ition of limit in my math-book. >At ?st an example: >Let f(x) = x, x~=1 ( ~= is not equal to ) >It's no question that lim[x->1] = 1, right? >Now consider: >g(x) = {x when x ~= 1, 10 when x = 1} Now, what is lim[x->1], if it >exists at all? >I guess it doesn't exist. lim[x->1-]g(x) and lim[x->1+]g(x) are both 1, >but g(1) is 10. Because g(1) exists but is different from the side limits, >lim[x->1]g(x) can't exist. >That's my opinion at least. Feel free to differ. > My understanding of the limit concept is that a function need not > be de?ed at a point for a limit to exist at that point. No, there's no doubt about that. > For > example, the OP's function g has lim_{x->1}g(x) = 1. > Proving the limit exists at x = 1... > Let e > 0 be given. We need to ?d d > 0, such that > |x-1| < e if |x-1| < d. Clearly we can choose d = e and > the limit is proven to exist at x = 1. > Now to prove continuity of the function at x = 1, we would have to > show that lim_{x->1} g(x) = g(1). We have g(1) = 10, and > lim{x->1} g(x) = 1, so g is not continuous at x = 1. What you say is that lim[x->x0] g(x) = 1 *although* f(1) = 10. Sounds reasonable to me and seems to ? MathWorlds de?ition of continuousity. *But*, let me cite my mathbook again: [...] continuous in x0 if x0 is member of f's domain and limit lim[x->x0] f(x) exists (and therefore is also equal to f(x0) that is, the existence of lim[x->x0] g(x) implies lim[x->x0] g(x) = g(x0). At least that's what my mathbook is indicating. But this is not true, since if g(x) = (as before) = {x when x~= 1, 10 when x=1} then lim[x->1] g(x) *does* exist *but* is different from g(1). _But_ my mathbook has the same de?ition of limits as MathWorld, and if your proof above is right, my mathbook[*] contradicts itself! Am I right? Could that be possible? [*]What is referred to as my mathbook , by the way, is the most used mathbook on analysis of single variable functions used at university level at technical educations in Sweden. I would think it would be reliable. === >c.j[dot]w < c.j[dot]w @telia.com> scribbled the following on sci.math: >I am somehow confused by the de?ition of limit in my math-book. >At ?st an example: >Let f(x) = x, x~=1 ( ~= is not equal to ) >It's no question that lim[x->1] = 1, right? >Now consider: >g(x) = {x when x ~= 1, 10 when x = 1} Now, what is lim[x->1], if it >exists at all? >I guess it doesn't exist. lim[x->1-]g(x) and lim[x->1+]g(x) are both 1, >but g(1) is 10. Because g(1) exists but is different from the side limits, >lim[x->1]g(x) can't exist. >That's my opinion at least. Feel free to differ. >My understanding of the limit concept is that a function need not >be de?ed at a point for a limit to exist at that point. No, there's no doubt about that. >For >example, the OP's function g has lim_{x->1}g(x) = 1. >Proving the limit exists at x = 1... >Let e > 0 be given. We need to ?d d > 0, such that >|x-1| < e if |x-1| < d. Clearly we can choose d = e and >the limit is proven to exist at x = 1. >Now to prove continuity of the function at x = 1, we would have to >show that lim_{x->1} g(x) = g(1). We have g(1) = 10, and >lim{x->1} g(x) = 1, so g is not continuous at x = 1. What you say is that lim[x->x0] g(x) = 1 *although* f(1) = 10. Sounds > reasonable to me and seems to ? MathWorlds de?ition of continuousity. *But*, let me cite my mathbook again: > [...] continuous in x0 if x0 is member of f's domain and limit > lim[x->x0] f(x) exists (and therefore is also equal to f(x0) This is the de?ition of continuity, not of limit. The limit of f as x -> x0 is L if for any e > 0 there exists a number f where for all x0-f < y < x0 + f, |f(y)-L| < e Essentially meaning that the limit of a function as x -> x0 is L if I can get arbitrarily close to L by taking f(y) where y is suf?iently close to x0. The de?ition of continuity has to do with the left hand limit equalling the right hand limit equalling f(x). that is, the existence of lim[x->x0] g(x) implies lim[x->x0] g(x) = > g(x0). At least that's what my mathbook is indicating. But this is not > true, since if > g(x) = (as before) = {x when x~= 1, 10 when x=1} > then lim[x->1] g(x) *does* exist *but* is different from g(1). > _But_ my mathbook has the same de?ition of limits as MathWorld, and if > your proof above is right, my mathbook[*] contradicts itself! Am I > right? Could that be possible? > [*]What is referred to as my mathbook , by the way, is the most used > mathbook on analysis of single variable functions used at university > level at technical educations in Sweden. I would think it would be reliable. > === [snipped] > Now to prove continuity of the function at x = 1, we would have to show > that lim_{x->1} g(x) = g(1). We have g(1) = 10, and lim{x->1} g(x) = 1, > so g is not continuous at x = 1. > What you say is that lim[x->x0] g(x) = 1 *although* f(1) = 10. Sounds > reasonable to me and seems to ? MathWorlds de?ition of continuousity. > *But*, let me cite my mathbook again: [...] continuous in x0 if x0 is > member of f's domain and limit lim[x->x0] f(x) exists (and therefore is > also equal to f(x0) This de?ition sounds wrong to me. It's entirely possible that a function has a limit at a point but is not de?ed at that point or has a different value to the limit at that point. Your de?ition above would be correct if you omit the word therefore ...I would also have thought they would use the term continuous at x0 instead of continuous in x0. > that is, the existence of lim[x->x0] g(x) implies lim[x->x0] g(x) = g(x0). If you use your de?ition above... > At least that's what my mathbook is indicating. But this is not true, > since if > g(x) = (as before) = {x when x~= 1, 10 when x=1} then lim[x->1] g(x) > *does* exist *but* is different from g(1). > _But_ my mathbook has the same de?ition of limits as MathWorld, and if > your proof above is right, my mathbook[*] contradicts itself! Am I > right? Could that be possible? Entirely possible!! Has it been translated? > [*]What is referred to as my mathbook , by the way, is the most used > mathbook on analysis of single variable functions used at university level > at technical educations in Sweden. I would think it would be reliable. === The more I think about it the more Decker's example bugs me, despite my thinking that I had a handle on it that actually suited my purposes! Maybe I'm just wrong here. I think I'll think about it some more and see if I can ?d an answer other than just being wrong. But being wrong wouldn't be so bad, after all, if I have it wrong then the sooner I ?d that out, the better I think. That doesn't change the fact that there are quite a few annoying posters here, but they can keep posting all they want, and I doubt I'll stop posting, even if I'm wrong with this argument here. I ?d myself at peace with posting. I like doing it, so there's no reason to stop just for being wrong. Then again, I haven't decided I am just yet. It'll take some more thinking. James Harris === [snip] > I ?d myself at peace with posting. I like doing it, so there's no > reason to stop just for being wrong. Keep in mind that no one has objected to your posting wrong theories or arguments. The objections have been to your vitriolic and militant defense of your errors when the exact nature of those errors have been spoon-fed to you. Your character ?which prompt you to attack your critics with accusations of being liars, co-conspirators or simply heads , represent a bigger problem. -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === > Maybe I'm just wrong here. Now I've seen everything. === >The more I think about it the more Decker's example bugs me, despite >my thinking that I had a handle on it that actually suited my >purposes! >Maybe I'm just wrong here. I knew it! The other day when you told Nora you didn't want to talk to her anymore I _said_ this meant you were starting to realize she was right. Just now when came up with the clever reply You stupid head!!! What the is wrong with you Ullrich? No matter how many ing times I tell you to off, you keep replying to me!!! What the is your problem you head? You Ullrich are a stupid piece of dumb who refuses to get the message when someone does NOT want to talk to you, you stupid ing ty asshole. You are an ASSHOLE Ullrich!!! Now why don't you take your dumb ass stupid self somewhere to GET A ING CLUE and QUIT ING REPLYING TO ME AS IF I EVER WANT TO TALK TO YOU!!!!!!!!!!!!! OFF!!!! Can't you get it through your stupid head? OFF!!!!!!!!!!!!!!!!! to something I said I said it was good to see you were starting to realize you were wrong - I swear I hadn't seen this post when I >I think I'll think about it some more and see if I can ?d an answer >other than just being wrong. >But being wrong wouldn't be so bad, after all, if I have it wrong then >the sooner I ?d that out, the better I think. That's exactly correct. You could ?d these things out _much_ faster, you know. This one has taken you it seems like more than a year. When I say something wrong I ?d out about it much quicker. (Has to do with the difference in our reactions when >That doesn't change the fact that there are quite a few annoying >posters here, but they can keep posting all they want, and I doubt >I'll stop posting, even if I'm wrong with this argument here. Golly, I hope _my_ posts don't annoy you. >I ?d myself at peace with posting. I like doing it, so there's no >reason to stop just for being wrong. Remind me to write this one down - it goes next to the time you said you didn't want to hear about it if you were wrong, for use when you complain that none of _us_ care about the Truth... >Then again, I haven't decided I >am just yet. > >It'll take some more thinking. >James Harris ************************ David C. Ullrich === >Maybe I'm just wrong here. I guess there's a ?st time for everything. Doug === > The more I think about it the more Decker's example bugs me, despite > my thinking that I had a handle on it that actually suited my > purposes! Maybe I'm just wrong here. I think I'll think about it some more and see if I can ?d an answer > other than just being wrong. But being wrong wouldn't be so bad, after all, if I have it wrong then > the sooner I ?d that out, the better I think. That doesn't change the fact that there are quite a few annoying > posters here, but they can keep posting all they want, and I doubt > I'll stop posting, even if I'm wrong with this argument here. I ?d myself at peace with posting. I like doing it, so there's no > reason to stop just for being wrong. Then again, I haven't decided I > am just yet. It'll take some more thinking. > James Harris You know, if you have a sincere interest in mathematics, why not help answer questions that others post on the group? David Moran === >The more I think about it the more Decker's example bugs me, despite >my thinking that I had a handle on it that actually suited my >purposes! >Maybe I'm just wrong here. >I think I'll think about it some more and see if I can ?d an answer >other than just being wrong. >But being wrong wouldn't be so bad, after all, if I have it wrong then >the sooner I ?d that out, the better I think. >That doesn't change the fact that there are quite a few annoying >posters here, but they can keep posting all they want, and I doubt >I'll stop posting, even if I'm wrong with this argument here. >I ?d myself at peace with posting. I like doing it, so there's no >reason to stop just for being wrong. Then again, I haven't decided I >am just yet. >It'll take some more thinking. > >James Harris > You know, if you have a sincere interest in mathematics, why not help answer > questions that others post on the group? I'm *using* this newsgroup. I've been pointing that out for years now, but I still keep having to repeat it. I use the newsgroup for my own purposes, and what I do, how I post, is a part of those purposes. And since I'm so successful with the current plan, I don't see any reason to change. James Harris === > The more I think about it the more Decker's example bugs me, despite > my thinking that I had a handle on it that actually suited my > purposes! > Maybe I'm just wrong here. > I think I'll think about it some more and see if I can ?d an answer > other than just being wrong. > But being wrong wouldn't be so bad, after all, if I have it wrong then > the sooner I ?d that out, the better I think. > That doesn't change the fact that there are quite a few annoying > posters here, but they can keep posting all they want, and I doubt > I'll stop posting, even if I'm wrong with this argument here. > I ?d myself at peace with posting. I like doing it, so there's no > reason to stop just for being wrong. Then again, I haven't decided I > am just yet. > It'll take some more thinking. > James Harris >You know, if you have a sincere interest in mathematics, why not help answer >questions that others post on the group? I'm *using* this newsgroup. I've been pointing that out for years > now, but I still keep having to repeat it. I use the newsgroup for my own purposes, and what I do, how I post, is > a part of those purposes. And since I'm so successful with the current plan, I don't see any > reason to change. > James Harris I just thought that someone with your obvious interest in mathematics could continue your research efforts and also help people with their own questions. I am not a mathematician, but I use mathematics EXTENSIVELY at work (I work for a meteorology website, http://www.oklahomastormteam.8m.com/)and despite that, I enjoy solving mathematical problems. That's why I come on and help others on Usenet. David Moran === >Dear all, > >I am facing with this dif?ult problem. Please help me! > >In this problem, I need to construct some matrices which satisfy the >following matrix equation: >(( A * A )V1 + ( B * B) V2) V = (D * D) >where * denotes Kronecker product , A, B, V1, V2, V are unknown >matrices >that needs solving; they are all square. Some structure needs to be >imposed: >I have certain pattern for A and B; and V1, V2, and V are required to be >diagonal... Matrix D is given... >The task is to ?d the best approximation of the above-mentioned A, B, >V1, >V2 and V... I have been thinking about this for long time... can anybody >give me some hints? >I guess it is dif?ult to ?d closed form analytical solution.. > How dii?ult is it - it seems to me that the problem has an awful lot > of structure. > >. How to >design iterative algorithm to let computer search for the answer? It is >really hard... please help me! > You are trying to solve the nonlinear system of equations > (( A * A )V1 + ( B * B) V2) V - (D * D) = 0 > If the problem is not too large use Newton's method. If the problem > is too big for that look for a canned program that will solve > nonlinear systems without needing to generate and then solve the > jacobian. >Dear Joe, >a) (( A * A )V1 + ( B * B) V2) V - (D * D) = 0 >These * are Kronecker products not the normal multiplier, how to design >Newton's iteration algorithm for this equation? >b) The worst thing is that this problem is not continuous; the elements in V >can be arbitary; but the elements in A, B, V1, V2 need to be integer... >hence I lost completely how to do iterative algorithm for these kind of >mix-integer-continuous problem... >Could you please help me more? >-Walalla You have not told us how big this problem is. You should have mentioned that some of the matrices have to be integer - it changes the nature of the problem a lot. It seems to me that: 1. if the matices A B and D each have m rows and m columns then you have a total of 5 x m x m unknowns and only m x m equations. You will have to look to your other conditions to narrow the solution down. 2. There may be no unique solution. If you have a solution then you can take any number and multiply V1 and V2 by that number and divide V by the same number and you have another solution. You could also interchange A and B and V1 and V2 and get a second answer. Your other conditions may be able to narrow it down to a unique solution. 3. (( A * A )V1 + ( B * B) V2) will have all integer elements. 4. therefore V must have the property so that inverse of V times D*D has all integer elements - I suggest you start by looking for a V with that property and the further property that the elements of ( D*D times inverse V) do not have any common factors. Once you have that V you know that the V in a solution must either be that V or that V divided by an integer; 5. Look ?st at those combinations of elements where the i j th element of A is squared and multiplied by an element of V1 then the i j th element of B is squared and multiplied by an element of V2 and the sum of the two has to be equal to an element D*D times inverse of V . To take the easiest example A(1,1) x A(1,1) x V1(1) + B(1,1) x B(1,1) x V2(1) = D(1,1) x D(1,1) / V(1) and must be an integer. Those conditions will signi?antly restrict the possible solutions. The values in D will put upper limits on the values in A B V1 and V2. You may be able to ?d a solution simply by exhaustively searching the set of possible solutions - using my point number 5 to narrow the search as much as possible as the ?st test to be applied. I think you need to sit back and look at the structure of your problem before running off looking for a numerical algorithm. I hope this helps. === >Dear all, >I am facing with this dif?ult problem. Please help me! >In this problem, I need to construct some matrices which satisfy the >following matrix equation: >(( A * A )V1 + ( B * B) V2) V = (D * D) > >where * denotes Kronecker product , A, B, V1, V2, V are unknown matrices >that needs solving; they are all square. Some structure needs to be imposed: >I have certain pattern for A and B; and V1, V2, and V are required to be >diagonal... Matrix D is given... >The task is to ?d the best approximation of the above-mentioned A, B, V1, >V2 and V... I have been thinking about this for long time... can anybody >give me some hints? >I guess it is dif?ult to ?d closed form analytical solution.. How dii?ult is it - it seems to me that the problem has an awful lot > of structure. >. How to >design iterative algorithm to let computer search for the answer? It is >really hard... please help me! You are trying to solve the nonlinear system of equations (( A * A )V1 + ( B * B) V2) V - (D * D) = 0 If the problem is not too large use Newton's method. If the problem > is too big for that look for a canned program that will solve > nonlinear systems without needing to generate and then solve the > jacobian. Dear Joe, a) (( A * A )V1 + ( B * B) V2) V - (D * D) = 0 These * are Kronecker products not the normal multiplier, how to design Newton's iteration algorithm for this equation? b) The worst thing is that this problem is not continuous; the elements in V can be arbitary; but the elements in A, B, V1, V2 need to be integer... hence I lost completely how to do iterative algorithm for these kind of mix-integer-continuous problem... Could you please help me more? -Walalla === > [lots of stuff] > I think you've got it. Good show. > Lee Rudolph This is a revision of the previous similar post which has been deleted. There are two approaches to the construction of a Riemann surface from a given relation f(w,z)=0: the original which goes from relation to system of branches to Riemann surface, and the modern which bypasses the introduction of branches. But one may think that branches are worth studying for there own sake, and then in context just think of Riemann surfaces as a byproduct. This is about branches. There are already many websites devoted to the more abstract geometrical theory. Branches cannot be studied from the given equation on its own. Special entities have to be introduced, ?st an ordinary point z0 in the z-plane to serve as an initial value for z. This done the solutions of the equation f(w,z0)=0, say w1, w2, w3... will be initial values for branches. Now, permutations on branches produced by analytic continuation round closed circuits are already determined without the branches as yet being fully speci?d. A circuit from z0 passing among but not through singularities and returning to z0 sends each initial value into an initial value which may be the same or different. The permutation depends only on the homotopy class of the circuit. These classes depend in turn on the branch points which are determined by the given equation. At this point the fact that branches are not yet fully de?ed comes into play. To proceed, a second special entity, namely a system of cuts has to be introduced. Then the branches depend more immediately on these cuts which de?e a domain for them than on the branch points. Discussion of the consequences of this requires a few more paragraphs and will be for another time. So one arrives at a theory but one which depends on special entities which are arbitrary. At this point it is useful to take a short digression into the philosophy of language. A statement may be made in different languages but must be made in some language. So it can be claimed that a statement is language-independent only in the sense that it is translatable. This is the best that can be done. By analogy what is needed in the present theory is to show that from a result for one set of special entities a result for any other can be derived. This is easy and well known for an alternative initial point. The two points are joined by an arc and the arc added to the circuit. It is not quite as simple for an alternative set of cuts but it can be done. With this, branches and permutations on branches are treated in a way which is as general as it can be. If it is desired to go on to construct a Riemann surface coincidences of branch boundaries indicate how sheet boundaries have to be identi?d. boundary= ------------A63D68CC994D48313070B3EF === ------------------------------------------------------------- -------- Sorry about the delay (holidays). Here is the adjacency list in alphabetical order of 2-letter state abbreviation. I don't have access to Mathematica right now, or i would translate it to integer form. If you have Mathematica, use the line of code below the list to convert it. You can ?d all the work i have done on this problem at http://www.eichblatt.us/USMapProblem/ USmapList[] := ( (* Build the graph from the state adjacency list *) stateAdjacency = { {AL, GA, MS, TN, FL}, {AK}, {AR, TX, LA, OK, MO, TN, MS},{AZ, CA, NM, UT, NV}, {CA, NV, OR, AZ}, {CO, UT, WY, NM, NE, KS,OK}, {CT, RI, MA, NY}, {DE, MD, PA, NJ}, {FL, GA, AL}, {GA, FL, SC, NC, AL, TN}, {HI}, {IA, IL, WI, MN, SD, NE, MO}, {ID, WA, OR, NV, UT, WY, MT}, {IL, IA, WI, IN, KY, MO}, {IN, OH, MI, IL, KY},{KS, OK, CO, MO, NE}, {KY, IL, MO, TN, VA, WV, OH, IN},{LA, TX, MS, AR}, {MA, VT, NH, NY, RI, CT}, {MD, DE, PA, VA, WV}, {ME, NH}, {MI, IN, OH, WI}, {MN, WI, IA, SD, ND}, {MO, IA, NE, KS, OK, AR, TN, KY, IL},{MS, AL, LA, AR, TN}, {MT, ID, WY, SD, ND},{NC, SC, VA, TN, GA}, {ND, SD, MN, MT}, {NE, KS, CO, WY, SD, IA, MO}, {NH, ME, VT, MA}, {NJ, NY, PA, DE},{NM, TX, AZ, CO, OK}, {NV, CA, AZ, UT, ID, OR}, {NY, NJ, VT, PA, MA, CT}, {OH, IN, WV, PA, KY, MI}, {OK, TX, CO, KS, NM, AR, MO}, {OR, CA, NV, ID, WA}, {PA, WV, DE, MD, NJ, NY, OH}, {RI, MA, CT}, {SC, GA, NC}, {SD, ND, MT, WY, NE, IA, MN}, {TN, KY, MO, AR, MS, AL, GA, NC, VA}, {TX, OK, LA, NM, AR}, {UT, CO, WY, ID, NV, AZ}, {VA, WV, MD, NC, TN, KY},{VT, NH, MA, NY}, {WA, OR, ID}, {WI, IL, MI, IA, MN}, {WV, VA, OH, PA, MD, KY}, {WY, MT, ID, UT, CO, NE, SD}}; I will send the list in integer form when i return home. >I thought of a simple problem and i suspect that it has already been >worked on, but I haven't been able to ?d any discussion of it in intro >Graph Theory books. Does anyone know where to look for a treatment of >this problem? >The speci? example I thought of is: >Supposing that every person in the United States can name what state >they live in and all of its adjacent states. What is the smallest group >of people required to name all 50 states, and what states are they from? >I can show by simple arguments (involving only the degree of the >vertices) that there must be at least 9 states. And i have found several >solutions with 13 states through random trial and error. >If you will send me (or post) a list of the states together with the > adjacent states, I will see what > I can do with the question. Preferred form: a list of lists of the form > [i, j1, j2,...,jk] >where i is a number between 1 and 50 representing the i-th state and j1, j2, > ...,jk are the numbers of the adjacent states. (I have worked on similar > problems.) >--Edwin Clark >PS If you want to check out the literature yourself do a Google or > MathSciNet search on dominating set or domination number(s) > === > Here is a shocking admission: I'm curiously growing weary of this > fascinating exchange. Major snippage below. > > [...] > You used this claim to support the claim > that Arturo is incompetent. This is just utter bull. >you have some reading comprehension problems. my initial follow up in >no way claims that arture is incompetent. > Explain this post then. > > ,---- > | > | [snip] > | > | > I maintain that Magidin is indeed either incompetent as a > | > mathematician, or more likely a liar, or both. > | > | well, he is an ADJUNCT assistant professor. ADJUNCTS are the pariahs > | in academic caste system. > | > | so whomever you are, may well be right on this one... > `---- > How is jstevh@yahoo.com right, given that Arturo is an adjunct? >well, some who happen to be adjuncts happen to be incompetent - duh. > You demonstrate your usual blunt reasoning, maky. > For the record, your post was an insult to me, and you know it. That's > how I read it, that's how everyone reasonable read it. >No, but that's just because I ?d your opinions to be beneath >insult, and your current claims a show of cowardice. You can't even >stick by your sly attempts at an insult. >shoudn't you be insulted by your employers instead? >No. Your statement is plain. It is an attack on me. well, it was not inteded as such. if apologies you seek, i apologise for insulting you. now onto the obvious topic of my post. don't you agree with my assessment of the adjunct rank in academia? >please explain. may well be right can only refer to the fact that I was > being called (through copying an old post of James > Harris) incompetent as a mathematician, or more likely a > liar, or both. You are agreeing with the possibility, which > means you are stating your opinion that it may > very well be true that I am an incompetent and/or a liar. what i said, does it have mean that? > Just the latest in your campaign to attack me because I would not > support you in a.a. >come again? For someone as ignorant as you are, you don't fake ignorance > too well. >http://groups.google.com/groups?selm= d40a9e18.0212311124.449af944%40posting.google.comhttp://groups.google.com/groups?selm=188f56bf .0302161606.63662fd8%40posting.google.com oh. i do not recall asking for your support on that particular remark. but if you disagree with the assessment, i have no objection. >-- > === == === = > It's not denial. I'm just very selective about > what I accept as reality. > --- Calvin ( Calvin and Hobbes ) > === == === = >Arturo Magidin >magidin@math.berkeley.edu (in case mathforum's post did not show up on your reader) === [.snip.] >(in case mathforum's post did not show up on your reader) Don't much mind breaking this to you, but the reason I did not reply to you before is simple: I have absolutely no interest in engaging in an exchange with you. You have absolutely no interest in discussing anything, all you want is someone to agree with you. Even how you phrase questions shows that you have already made up your mind (and as usual, in an absurd way and frozen it in an absurd, ignorant, and stupid position). So, I am not interested in discussing adjuncts with you, and I am not interested in your pathetic attempts to back off from your insult, or your apologies , with their quotation marks, that are nothing but further insults to my intelligence. -- === === === It's not denial. I'm just very selective about what I accept as reality. --- Calvin ( Calvin and Hobbes ) === === === Arturo Magidin magidin@math.berkeley.edu === > > Here is a shocking admission: I'm curiously growing weary of this > fascinating exchange. Major snippage below. > [...] > Explain this post then. > ,---- > | > | [snip] > | > > | > I maintain that Magidin is indeed either incompetent as a > | > mathematician, or more likely a liar, or both. > > | > | well, he is an ADJUNCT assistant professor. ADJUNCTS are the pariahs > | in academic caste system. > > | > | so whomever you are, may well be right on this one... > `---- > How is jstevh@yahoo.com right, given that Arturo is an adjunct? >well, some who happen to be adjuncts happen to be incompetent - duh. > How does that increase the likelihood that jstevh@yahoo.com is right? >what makes you assume that's i claim i maintain? didn't you read my >previous reply? > Well what the did [you] may well be right on this one... mean, > if it didn't mean that your comments increased the likelihood that the > original [im]poster was right? if i had written you could be correct... > that you are not alleging that adjuncts are likely to be liars, so you must > be insinuating that adjuncts are incompetent. >geez, i would hate to assume that you are having problems with >existential quanti?rs. is that it? > Well, I think I know a thing or two about logic, but please enlighten > me about what error I made regarding existentials. >well, ?st tell me how much logic you know. can you handle venn >diagrammes? > I have more than suf?ient background in logic, I am sure. Honest. evidence? > Did you mean only this? >what else could i have meant? >do you have further objections? > (1) Some adjuncts are incompetent. > (2) Arturo is an adjunct. > ------- > Therefore, it is possible that Arturo is incompetent. > I just want to be clear: The argument above is the argument you know > claim to have advocated? what do you think i said? no baseless assumptions. > *This* is an argument that you think is a good argument? did i ever qualify my statements as being a good logical argument? what i recall is pointing out your innef?ient reading comprehension. > This from the man that wants to teach me about existentials? > You're adorable. > Know what? I agree. I still don't. I have no idea *what* your post > meant, >why the objections then? a bit emotional? what? > > unless my bad argument above captures it. >the argument where you are trying to patch poor reading >comprehension? > > No, the argument with premises numbered (1) and (2) and with > unlabeled conclusion. The one you seem to think is just peachy. what makes you think i qualify your argument as peachy? or for that matter, what makes you assume i adopted your argument as my own? >you also made a tonne of assumptions about it. but worry not, i'll >gladly dissect it for you, should you need further assistance >undertanding it... > Assist on. >now, where are your answers to the questions i asked? > Which ones? I won't respond to your fantasies regarding those slave > laborers known as adjuncts, as it's not particularly my interest. >then, what are you doing in this discussion? > Mostly, mocking a complete moron. Shame you hadn't noticed. moron? where did that come from? isn't it true that you engaged in this discussion on some sort of assumption? === >[snip] > A much worse example: Say as usual (i) when a function is de?ed > by a formula we take the domain to be the set of all reals for which > the formula makes sense, >I'll also assume, in that case, that you're intending the range to be >a subset of R. But even then there can be differences of opinion about >those reals for which the formula makes sense . As a simple example: >If we were asked for the implied domain of > f(x) = Sqrt(x^2 (x-1)), >I would hope that everyone would say that it is {0} U [1, +oo). But >suppose that we were asked for the implied domain of > g(x) = |x| Sqrt(x-1). >I suspect that some would say that 0 is in the implied domain of g (and >also that f and g are the same function), while others would say that 0 >is not in the implied domain of g (and thus that f and g are different >functions). > If we were talking about actual math we'd need to be a lot more > careful with all of this, or better yet simply throw out the notion > of implied domain . I think we're talking about the way terminology > is used in a typical calculus class - in a context like that there are > no complex numbers, so the domain does not include 0. > I hope so, anyway. > I think it's interesting to point out that, when de?ing continuity, some authors, like Rudin in his book on Analysis, prefer to consider the domain X of the function as a kind of universal set, without paying any attention to another space where X might be embedded. So, supposing X and Y are metric spaces, or even general topological spaces, such authors say f:X -> Y is continuous at an element a of X if, for every neighborhood V of (f(a), there is a nighborhood U of a (depending on V), such that f(x) is in V for every x in U. Rudin gives this de?ition, adapting it to metric spaces by means of the epsilon-delta de?ition. But some other authors, like Bartle, prefer to consider a larger set in which the domain of the function is embedded. So, using neighborhoods and considering functions between real vector spaces, Bartle says f:D -> R^m, D a subset of R^n, is continuous at an element a of D if, for every neighborhood V of f(a), there's a neighborhood U of a, (depending on V),such that, if x is in U cap D, then f(x) is in V. Authors who adopt the ?st kind of de?tion prove that f is continuous on X iff the inverse image under f of every open set of Y is an open set of X. But Bartle, in the case of real vector spaces, prefer to prove that f is continuous on D iff the inverse image under f of every open set of R^m is given by the intersection of D and an open set of R^n. I think these 2 forms of de?ing continuity turns out to convey the same idea, and a remarkable fact is that, in both cases, the concepts of continuity and discontinuity only make sense in the domain of the function. But, at least when it comes to real vector spaces, there seems to be a certain tendency to consider the universal space R^n (if this a good adjective). So, there seems to be a certain natural tendency to consider limit points of the domain, even if the don't belong to it. Possibly, this comes from the fact that, when computing limits, limit points are considered even if the are not in the domain of the function. I've heard some people say that f(x) = 1/x has an essential discontinuity (opposed to removable discontinuity) at x=0 because it doesn`t have a limit there. This is a completely wrong statement, because is simply not de?ed at x=0. Artur X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: george_cox@btinternet.com X-Punge: Micro$oft === In <8tEJb.2572$uF6.1092753@news1.news.adelphia.net>, on 01/03/2004 at 07:11 PM, Leonard M. Wapner said: >But does it make sense to call the function continuous? Yes. >(It's the gap in the domain that bothers me.) That might be relevant in a Calculus textbook. It's certainly not relevant in Topology, and I'm not aware of any Analysis text that uses a de?ition of continuous inconsistent with that used in Topology. -- Shmuel (Seymour J.) Metz, SysProg and JOAT not reply to spamtrap@library.lspace.org === I was kind of freaking out earlier as I ?ally considered the situation with a factorization like that presented by Rick Decker, a professor at Hamilton College: (5a_1(x) + 7)(5a_2(x) + 7) = 7(25x^2 + 30x + 2) where his a's are roots of a^2 - (x - 1)a + 7(x^2 + x). I didn't start feeling better until I cursed out David Ullrich and Nora Baron , as well as realized that hey, if I'm wrong, I'm just wrong, and it's not that big of a deal. What I just want is to know what's the *truth*. Well I ?ally remembered that one of the a's has internal structure, as at x=1 *one* of the a's equals -1, and in fact, what happens when x=1 is a radical bit of surgery to the equation, which in zeroing out that middle coef?ient, changes the constant terms by directly deleting out. Seeing how it changes them is just a matter of considering what results, if you have that coef?ient gone for all x, as then you have a^2 - 7(x^2 + x) and checking at x=0, gives constant terms of 0. The important thing to notice here is that in a^2 - (x - 1)a + 7(x^2 + x) that middle coef?ient is multipled times an ?a', and when it goes, it takes away that ?a', as well as its contribution to the constant term. That is in fact the *only* way to change the constant term with a dependency on x, so it represents a special case. It turns out that factorizations which allow that are what I call imperfect factorizations, while my own is what I call a perfect factorization. Remember that with my factorization I have (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) where b_3(x) = a_3(x) - 3 and the a's are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) and when x=0, a_1(0) = a_2(0) = b_3(0) = 0. And notice that it doesn't have that problem while 7 is not a unit. James Harris === > I was kind of freaking out earlier as I ?ally considered the > situation with a factorization like that presented by Rick Decker, a > professor at Hamilton College: > (5a_1(x) + 7)(5a_2(x) + 7) = 7(25x^2 + 30x + 2) > where his a's are roots of > a^2 - (x - 1)a + 7(x^2 + x). > Well I ?ally remembered that one of the a's has internal structure, > as at x=1 *one* of the a's equals -1, and in fact, what happens when > x=1 is a radical bit of surgery to the equation, which in zeroing out > that middle coef?ient, changes the constant terms by directly > deleting out. Not at all. At x = 1, we have a^2 + (7)(2) and the roots of that are sqrt(-14) and -sqrt(-14), neither of which is equal to -1. > Seeing how it changes them is just a matter of considering what > results, if you have that coef?ient gone for all x, as then you have > a^2 - 7(x^2 + x) > Of course you don't. When x = 1 you have a^2 + (7)(2) The coef?ient of the a term is hardly gone for all x -- only for x = 1. > and checking at x=0, gives constant terms of 0. > Are you saying that the constant term of my a(x) polynomial (namely, the term not involving a, i.e., 7(x^2 + x)) only vanishes when x = 0 or x = -1? If so, I'm in complete agreement. Otherwise, I'm puzzled about how to justify looking at the a(x) polynomial simultaneously at x = 1 and at x = 0. a(0) and a(1) are different polynomials, namely a(0) = a^2 - a a(1) = a^2 + 14 It turns out that factorizations which allow that are what I call > imperfect factorizations, while my own is what I call a perfect > factorization. > What is the referent of that in the sentence above? Do you mean that one or more coef?ients of the a(x) polynomial vanish? But that's precisely what you WANT to have happen, so you can get a reducible polynomial and perhaps the splits of 49 into 7, 7, 1 (in your case) or of 7 into 7, 1 (in my example). > Remember that with my factorization I have > (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) > where b_3(x) = a_3(x) - 3 and the a's are roots of > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) > and when x=0, a_1(0) = a_2(0) = b_3(0) = 0. > And notice that it doesn't have that problem while 7 is not a unit. > So are you saying that in your a polynomial, the coef?ients never vanish for any values of x? That's certainly not true for -49(2401x^3 - 147x^2 - 3x), though I grant it is true for 3(-1 + 49x), as long as you restrict your attention to rational integers x. I must be missing something here. Are you claiming that your a(x) polynomial is irreducible unless x = 0? That's exactly what you DON'T want to happen, since then you'll never get the splitting you want. Rick === > I didn't start feeling better until I cursed out David Ullrich and > Nora Baron , as well as realized that hey, if I'm wrong, I'm just > wrong, and it's not that big of a deal. What I just want is to know > what's the *truth*. The last statement is a lie, but I will still help you: Your proof of Fermat's Last Theorem is nonsense. Your opinions about the core error in mathematics are nonsense. Your discoveries about prime counting functions are two hundred years old and quite trivial. Is that enough truth for you? === I am replying to this message because it's the ?st one that popped onto my screen, but this would apply to many, many other messages. I don't know much about James Harris' messages because I haven't read any of them beyond skimming; perhaps, though, it would be more savory of our mathematics newsgroup to restrain our emotional outbursts against this human being. I enjoy seeing the comradery amongst our fellow math enthusiasts, but when I run across a James Harris thread, I become very disappointed in the way that people are ?sed in' to a group bashing session. If I may offer a possible solution for those who become enraged at James' postings: one need not continue to read a message that is causing anger - this certainly provokes an outburst in reply. As a tip - not directed towards anyone in particular - it re?poorly on an individual when they indulge their anger and post a nasty reply. >I didn't start feeling better until I cursed out David Ullrich and > Nora Baron , as well as realized that hey, if I'm wrong, I'm just >wrong, and it's not that big of a deal. What I just want is to know >what's the *truth*. The last statement is a lie, but I will still help you: Your proof of Fermat's Last Theorem is nonsense. > Your opinions about the core error in mathematics are nonsense. > Your discoveries about prime counting functions are two hundred years > old and quite trivial. Is that enough truth for you? === >We almost all are aware that, for n = integer >= 2, we can write a >non-integer real with base-n digits (0 through {n-1}), some digits >following after a decimal-point if necessary. >But what about in base-1? But, I don't understand. Surely there are only countably many ?ite or in?ite sequences of > 1's with a decimal point somewhere. So we can't do very much for > representations of the reals here, can we? > Base 1 can represent all the rational numbers. We even de?e irrational numbers if we allow multiple radix points. I describe the factorial base in another thread. 321 base ! = 3*3! + 2*2! + 1*1! = 23 base 10 .123 base ! = 1/2! + 2/3! + 3/4! = 0.958333... base 10 We can convert any ?ite string of 1's (or 0's we want fo follow convention) into an integer. By converting the fractional part of a base 1 number into base ! we can represent any rational number. For example, 23 0's followed by a radix point followed by 23 0's could be the base 1 representation of 321.123 (base !) = 23.0.958333... (base 10) We can de?e a sequence of rationals by using multiple radix points. Consider this sequence: 1 base ! = 1 base 10 11 base ! = 1*2! + 1*1! = 3 base 10 111 base ! = 1*3! + 1*2! + 1*1! = 9 base 10 The following base ! sequence de?es e-2: (.1, .11, .111, ...) We could represent e-2 in base 1 with: .0.000.000000000... Russell - The universe is one dimensional === > But what about in base-1? > Integers are easy (though base-one representations are not exactly > analogous to higher bases, since we do not write base-1 integers using > only zeros). > Example: 7 (base 10) = > 1111111 (base 1) > But what about non-integers? > Have you any clever schemes for writing, say, 1/2 or pi in base-1?? > 2/2 = 0.111111... > 1/2 = 0.1010... > 1/3 = 0.100100... > 2/3 = 0.110110... > 1/4 = 0.10001000... > 2/4 = 0.10101010... > 3/4 = 0.11101110... > ... I thought 0 wasn't allowed. -- Jesse F. Hughes I'm better than you, and you know it. -- James Harris === Defending myself: By the way, I am very aware that base-1 is not a base in the same sense that we typically refer to our commonly-used number-system as base-10 and to binary as base-2 . (Although some repliers seem to believe I am unaware of my less-than-literal use of the word base .) My particular de?ition of the term base-1 is not my own, yet I cannot recall where else I have seen it used in the sense I use it in my original post, but I have seen the term used this way in several different (and reputable) places, I am sure. thanks, Leroy Quet > I am posting this as more a fun challenge rather than a serious > question. > {So, that is why I have cross-posted this to rec.puzzles AND > sci.math.} > We almost all are aware that, for n = integer >= 2, we can write a > non-integer real with base-n digits (0 through {n-1}), some digits > following after a decimal-point if necessary. > But what about in base-1? > Integers are easy (though base-one representations are not exactly > analogous to higher bases, since we do not write base-1 integers using > only zeros). > > Example: 7 (base 10) = > 1111111 (base 1) > But what about non-integers? > Have you any clever schemes for writing, say, 1/2 or pi in base-1?? > [The best I can come up with right now is to write the continued > fraction of the real, with each term consisting of a base-1 positive > integer. But this is really a list of base-1 integers. Still, > anything better??] > thanks, > Leroy Quet === > In Base-0 the integers exist, but you can't tell two integers apart. > > ObPuzzle: Do non-integers exist? Yes. Either of these newsgroups constitutes a constructive proof. -- Aatu Koskensilta (aatu.koskensilta@xortec.? Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === > Can we use a fraction as a radix, such as r = 3/2? and the Mensanator replied: > I don't know. How many digits are in Radix 3/2? One and a half? I'd say two-- zero and one. I'm making this up as I go, but for radix r, digits could range from 0 to ceiling(r)-1. So for example in base 3/2 11 represents 3/2 + 1 = 5/2 101 represents 9/4 + 1 = 13/4 .1 represents 2/3 I think we can represent any number. For example, how do we represent 2? 2 = 3/2 + 1/2 = 3/2 + 4/9 + 1/18 = 3/2 + 4/9 + 1/18 = ... = 3/2 + 4/9 + 256/6561 + 2048/177147 + ... Working out more digits, you can get two can be represented by 1.00100 00010 01001 01000 ... I also think we probably have more than one representation for most numbers. For example, 1 represents 1, but we also have 1 = 2/3 + 1/3 = 2/3 + 8/27 + 1/27 = 2/3 + 8/27 + 512/19683 + ... which leads to one can be represented by .10100 00010 01001 01000 ... In fact, given any representation of a number with a 1 in some position, we could subtract the 1 from that position, and add the string 1010000010... at the next position to get another (probably) representation of the same number. When adding two numbers, carries are funky. If you add 1 and 1 at some position, you get a result the has to be equivalent to a two (at that position). But two can't be represented by a ?ite string. So the carry propogates one digit to the left *and* in?itely to the right. Bob H === Today is the third anniversary of the very ?st message I posted to sci.math. I discovered the Usenet while doing an Internet search for information on the Collatz Conjecture. I immediately became addicted to this wonderful forum. At home and at work I am considered the math whiz , but I quickly discovered that I actually know almost nothing compared to real mathematicians (which is a sad commentary on the state of math knowledge of the general public). But that hasn't stopped me from putting in my 2 cents whenever a subject came up that I was interested in. Sure, I occasionally make mistakes which inevitably provoke embarassing replies. But I look at it as a learning experience. If you never speak up, you never discover the ?n your knowledge. And it is gratifying to be thanked by those who appreciate my butting into their threads when I just happen to have the answer to their problems. But what was I really after? The answer to my Big Problem that I've been trying to ?d for going on 4 years. I tried asking for help once, but did not get a single reply. Did I turn bitter (like James Harris) because no one would help me? No, I ?ured that possibly no one read it or it wasn't clear what I was asking or maybe it was just too hard a problem for the casual sci.math reader. So I just continued with the usual smart-ass remarks, puzzle solving and ponti?ating on trivially easy problems. And I think I have come away from sci.math with more than I arrived with. Three years ago I thought congruent was something that only applied to triangles. Without sci.math, I might never have learned about Python or discovered Mathworld. For a special case of the Big Problem, I was able to come up with a closed form solution which I was quite proud of. I actually seemed to be learning something from sci.math. And then it happened. I was posting a response to the issue of twin primes in Collatz sequences when something I said got me to go back and review my Big Problem from a new perspective. Before long I was on Mathworld looking up linear congruence. And there it was: the formula I've been searching for! All I need is the PowerMod function from Mathematica. Ha! Mathematica indeed. The answer may just as well been on the far side of the Moon for all the good that does me. But wait, PowerMod sounds like one of those functions that I always skip past thinking I'll never need to use this. Maybe I've got something equivalent to it already. Sure enough, there in the GMPY module of Python is gmpy.divm(a, b, M): x such that b*x == a modulo M Could the answer to the Big Problem really be that simple? I plugged in some numbers and out popped the correct answer, which I already knew since simple Hailstone Functions can be found by brute force iteration. What makes it a Big Problem is when the formula is something like (2199023255552*n - 1180195622215159)/617673396283947 and the brute force method involves checking 617 quadrillion numbers which would take 19 years of computer time. And that's just for the second generation. To do the stats I want, I need at least a half dozen or so generations. For this Big Problem, the answer 159543494874899925553253413448 was calculated in an instant by the gmpy.divm function. solve those also. The 20th generation of the above has 295 digits. So, although I will take full credit for working this out entirely on my own, I could never have done it without sci.math. All the ?ars, JSH bull and thankless help that is part of the gestalt of sci.math has all been worth it. === There can be no force exerted without duration: Even the slightest touch requires a starting point in time, followed by the thrust, and an ending point; during which the entire impulse occurs: Therefore all forces must be classi?d as impulses [ft]. A forced change in position [s] too is not instantaneous: The displacement [s] requires some time to occur, and is called a rate of displacement [s/t]: For any given object; body and/or mass of material matter, the ratio of the impulse [ft], exerted on and/or by it, divided by the rate of displacement [s/t] that it causes is a constant [ft/(s/t = ft^2/s), and is a measure of the body's mass, and/or its inertia: http://newsone.net/ -- Free reading and anonymous posting to 60,000+ groups NewsOne.Net prohibits users from posting spam. If this or other posts === There can be no If this or other posts abuse@newsone.net Done. === > How many digits of pi have you memorized? I did 100. The world record >I currently can give 17 digits of pi from memory. I was able to >(March 14th), but I slowly forgot the digits as I didn't practice >reciting them. >Anthony > pi day??? Pi Day is most widely recognized as March 14th. since March is the 3rd month of the year, it is written in the USA as 3/14, representing the beginning digits of pi, 3.14. It isn't an of?ial holiday (it isn't of?ially recognized by the government), but it is still quite a recognized math holiday . A lot of people often celebrate it for the number pi itself, for its importance today in society, etc. There are many different activities people do, including reciting digits of pi, calculating digits, eating pie, creating/reciting mnemonics for the digits of pi, etc... It's also celebrated often in math classes as well. Here is a webpage with many different pi-related links, inlcuding a fair bit concerning Pi-Day: http://www.wellesley.mec.edu/wms/library/pages/curric_pages/ pi/pi_day.htm Anthony === > A point I made is that if you have a constant times a polynomial, like > 49(x+1)(x+2)(x+3) = (7x + 7)(7x + 14)(x+3) > you can set x=0, to see how factors of 49 distributed, which is > trivial here as you can *see* how it distributed easily enough. > So the short answer is that my position is that the distributive > property *always* holds, while these posters are trying to argue that > there are ways to break the distributive property: a(b+c) = ab + ac. > When I've given examples like > > 49(x+1)(x+2)(x+3) = (7x + 7)(7x + 14)(x+3) > in the past, they have claimed that it matters that those are > polynomials versus other expressions I use that are not polynomials, > but the reality is that they're STILL attacking the distributive > property, which is a bizarre math position to take. > Actually, no one is attacking the distributive property. Your argument goes as follows. You start with (a1(x) + p)(a2(x) + q) = pP(x) with a1(0)=a2(0)=0 and p and q coprime. You then claim (i) p clearly divides (a1(x) + p)(a2(x) + q). At x=0 p must divide (a1(x) + p), therefore p must divide (a1(x) + p) for all x (ii) if p divides (a1(x) + p) then by the reverse distributive law p must divide a1(x) And then note that (i) and (ii) lead to the conclusion a1(x) is divisible by p for all x. Your critics reply: No, (i) is wrong so you are wrong. You reply: If you think that I am wrong then you must think that (ii) is wrong. You are attacking the distributive property. But no one is saying that (ii) is wrong, What people are saying is that (i) is wrong. This is where polynomials come in. If a1(x) and a2(x) are polynomials then (i) is correct. The examples you give to support your reasoning all use polynomials for a1(x) and a2(x). However, if a1(x) and a2(x) are not polynomials, then (i) is not correct (explicit examples have been given). - William Hughes === >An interesting deduction chain but I do have to ask the obvious >question as to whether induction works on uncountable ordinals. > In general, no. Induction is based on the well ordering principle > (every subset of natural numbers has a lowest term within it), which > does not apply for real subsets. It would also impose an ordering on > the reals, which is not kosher. There is lots more you can say about > it. >Huh? An ordinal is by de?ition well-ordered. > But simple induction does not work beyond the ?st in?ite ordinal > because there will be more than one member without predecessor. What is simple induction ? I think of simple induction as requiring the following: Ay [(Ax P(y)] Note that the second is different from: Ax [P(x) -> P(x+1)] I think of mine as simpler, because it doesn't require a separate proof of P(0). Thomas === > >.999... is an in?ite series. >The limit of .999... is 1. >.999... is not 1, because it is an in?ite series and 1 is not an >in?ite series. > 1 , as a decimal is actually 1.0000.... > But the value of .999... *is* the limit of the partial sums. > .9999.... is a different series of digits from 1.0000.... > But they are the same number. > >.9999... and 1.0000... are two different in?ite series, with the >same limit (the possibility of which should be of no surprise.) >Charlie Volkstorf > Says he who swallows camels but srains at gnats. You can swallow whatever you please, man (except the truth, apparently.) C-B === The square of .999... is not equal to 1, therefore .999... is not equal to 1. 9^2 = 81 .9^2 = .81 99^2 = 9801 .99^2 = .9801 999^2 = 998001 .999^2 = .998001 9999^2 = 99980001 .9999^2 = .99980001 99999^2 = 9999800001 .99999^2 = .9999800001 999999^2 = 999998000001 .999999^2 = .999998000001 9999999^2 = 99999980000001 .9999999^2 = .99999980000001 99999999^2 = 9999999800000001 .99999999^2 = .9999999800000001 999999999^2 = 999999998000000001 .999999999^2 = .999999998000000001 9999999999^2 = 99999999980000000001 .9999999999^2 = .99999999980000000001 99999999999^2 = 9999999999800000000001 .99999999999^2 = .9999999999800000000001 999999999999^2 = 999999999998000000000001 .999999999999^2 = .999999999998000000000001 9999999999999^2 = 99999999999980000000000001 .9999999999999^2 = .99999999999980000000000001 99999999999999^2 = 9999999999999800000000000001 .99999999999999^2 = .9999999999999800000000000001 999999999999999^2 = 999999999999998000000000000001 .999999999999999^2 = .999999999999998000000000000001 9999999999999999^2 = 99999999999999980000000000000001 .9999999999999999^2 = .99999999999999980000000000000001 99999999999999999^2 = 9999999999999999800000000000000001 .99999999999999999^2 = .9999999999999999800000000000000001 999999999999999999^2 = 999999999999999998000000000000000001 .999999999999999999^2 = .999999999999999998000000000000000001 9999999999999999999^2 = 99999999999999999980000000000000000001 .9999999999999999999^2 = .99999999999999999980000000000000000001 99999999999999999999^2 = 9999999999999999999800000000000000000001 .99999999999999999999^2 = .9999999999999999999800000000000000000001 etc., etc., etc. The limited minds of the limited mathematicians saith not. Garry Denke, Geologist Denoco Inc. of Texas === >The limited minds of the limited mathematicians saith not. Namecalling will get you nowhere. Doug === >The limited minds of the limited mathematicians saith not. > Namecalling will get you nowhere. > Doug There's nothing wrong with having a limited mind. That's why there is education. Garry Denke, Geologist Denoco Inc. of Texas === On 5 Jan 2004 13:16:32 -0800, garrydenke@dontmesswithtexas.com (Garry >The limited minds of the limited mathematicians saith not. > Namecalling will get you nowhere. > Doug >There's nothing wrong with having a limited mind. >That's why there is education. >Garry Denke, Geologist >Denoco Inc. of Texas Right. You know some of the people with limited minds you're trying to educate are professional mathematicians, like with PhD's and jobs and publications and everything. Is it your superior education that explains why you realize that 0.999... <> 1 even though the mathematicians are too dense to realize this? And a related question: If I wanted to understand Baroque music, I should go to school and study astrophysics, right? ************************ David C. Ullrich === > Is it your superior education that explains why you realize that 0.999... > <> 1 even though the mathematicians are too dense to realize this? it's only simple multiplication 9^2 = 81 .9^2 = .81 99^2 = 9801 .99^2 = .9801 999^2 = 998001 .999^2 = .998001 9999^2 = 99980001 .9999^2 = .99980001 99999^2 = 9999800001 .99999^2 = .9999800001 999999^2 = 999998000001 .999999^2 = .999998000001 9999999^2 = 99999980000001 .9999999^2 = .99999980000001 99999999^2 = 9999999800000001 .99999999^2 = .9999999800000001 999999999^2 = 999999998000000001 .999999999^2 = .999999998000000001 9999999999^2 = 99999999980000000001 .9999999999^2 = .99999999980000000001 99999999999^2 = 9999999999800000000001 .99999999999^2 = .9999999999800000000001 999999999999^2 = 999999999998000000000001 .999999999999^2 = .999999999998000000000001 9999999999999^2 = 99999999999980000000000001 .9999999999999^2 = .99999999999980000000000001 99999999999999^2 = 9999999999999800000000000001 .99999999999999^2 = .9999999999999800000000000001 999999999999999^2 = 999999999999998000000000000001 .999999999999999^2 = .999999999999998000000000000001 9999999999999999^2 = 99999999999999980000000000000001 .9999999999999999^2 = .99999999999999980000000000000001 99999999999999999^2 = 9999999999999999800000000000000001 .99999999999999999^2 = .9999999999999999800000000000000001 999999999999999999^2 = 999999999999999998000000000000000001 .999999999999999999^2 = .999999999999999998000000000000000001 9999999999999999999^2 = 99999999999999999980000000000000000001 .9999999999999999999^2 = .99999999999999999980000000000000000001 99999999999999999999^2 = 9999999999999999999800000000000000000001 .99999999999999999999^2 = .9999999999999999999800000000000000000001 etc., etc., etc., you do the math > [snip unrelated question] > David C. Ullrich garry denke, geologist denoco inc. of texas === >you do the math We have. Get your head out of the multiplication books and start doing some actual *math*, you pseudo-math fan. Doug === >And a related question: If I wanted to understand Baroque >music, I should go to school and study astrophysics, right? I'm afraid that, with the cost of higer education today, you'll quickly go baroque with that plan of attack. Doug === > And a related question: If I wanted to understand Baroque > music, I should go to school and study astrophysics, right? Rather, that's what you should study if you want to understand the Music of the Spheres. DWC === > And a related question: If I wanted to understand Baroque > music, I should go to school and study astrophysics, right? > >Rather, that's what you should study if you want to understand the Music >of the Spheres. Typical sort of thing you limited-mind people would say. Lee Rudolph (they're *ellipsoids*, dammit) === > And a related question: If I wanted to understand Baroque > music, I should go to school and study astrophysics, right? >Rather, that's what you should study if you want to understand the Music >of the Spheres. Very good! (I spent a minute trying to think of something with _no_ connection to Baroque music - rejected all sorts of possibilities when I saw tenous connections. I ?ured no matter what I used someone here would come up with something - nice work.) >DWC ************************ David C. Ullrich === >The limited minds of the limited mathematicians saith not. > Namecalling will get you nowhere. >There's nothing wrong with having a limited mind. Namecalling will get you nowhere. Doug === >The limited minds of the limited mathematicians saith not. > Namecalling will get you nowhere. >There's nothing wrong with having a limited mind. > Namecalling will get you nowhere. and here is i calling you norris@rintintin 9^2 = 81 .9^2 = .81 99^2 = 9801 .99^2 = .9801 999^2 = 998001 .999^2 = .998001 9999^2 = 99980001 .9999^2 = .99980001 99999^2 = 9999800001 .99999^2 = .9999800001 999999^2 = 999998000001 .999999^2 = .999998000001 9999999^2 = 99999980000001 .9999999^2 = .99999980000001 99999999^2 = 9999999800000001 .99999999^2 = .9999999800000001 999999999^2 = 999999998000000001 .999999999^2 = .999999998000000001 9999999999^2 = 99999999980000000001 .9999999999^2 = .99999999980000000001 99999999999^2 = 9999999999800000000001 .99999999999^2 = .9999999999800000000001 999999999999^2 = 999999999998000000000001 .999999999999^2 = .999999999998000000000001 9999999999999^2 = 99999999999980000000000001 .9999999999999^2 = .99999999999980000000000001 99999999999999^2 = 9999999999999800000000000001 .99999999999999^2 = .9999999999999800000000000001 999999999999999^2 = 999999999999998000000000000001 .999999999999999^2 = .999999999999998000000000000001 9999999999999999^2 = 99999999999999980000000000000001 .9999999999999999^2 = .99999999999999980000000000000001 99999999999999999^2 = 9999999999999999800000000000000001 .99999999999999999^2 = .9999999999999999800000000000000001 999999999999999999^2 = 999999999999999998000000000000000001 .999999999999999999^2 = .999999999999999998000000000000000001 9999999999999999999^2 = 99999999999999999980000000000000000001 .9999999999999999999^2 = .99999999999999999980000000000000000001 99999999999999999999^2 = 9999999999999999999800000000000000000001 .99999999999999999999^2 = .9999999999999999999800000000000000000001 go ?ure, you do the math garrydenke@dontmesswithtexas === >The limited minds of the limited mathematicians saith not. >Namecalling will get you nowhere. But he's getting there _fast_! === In sci.logic, Garry Denke on 5 Jan 2004 00:10:21 -0800 : > The square of .999... is not equal to 1, > therefore .999... is not equal to 1. 9^2 = 81 > .9^2 = .81 99^2 = 9801 > .99^2 = .9801 999^2 = 998001 > .999^2 = .998001 [rest snipped for brevity] Interesting logic, but doesn't quite work... :-) The square of .999... is .999... , as can readily be proven. If one solves the equation x^2 = x, one gets two values: 1 and 0. -- #191, ewill3@earthlink.net It's still legal to go .sigless. === > In sci.logic, Garry Denke > on 5 Jan 2004 00:10:21 -0800 > : >The square of .999... is not equal to 1, >therefore .999... is not equal to 1. > 9^2 = 81 >.9^2 = .81 > 99^2 = 9801 >.99^2 = .9801 > 999^2 = 998001 >.999^2 = .998001 > [rest snipped for brevity] > Interesting logic, thanks, its called unlimited logic > but doesn't quite work... not for those with limited logic, no > :-) (;o > The square of .999... is .999... , for those of limited ability > as can readily be proven. to others of limited ability, yes > If one solves the equation x^2 = x, one gets two values: 1 and 0. let's see, quote the ghost If 1 solves the equation x^2 = x, 1 = x x^2 = x 1^2 = 1 1 gets 2 values: 1 and 0. nope, 1 is the only answer, sorry ghost garry denke, geologist denoco inc. of texas === >The square of .999... is not equal to 1, >therefore .999... is not equal to 1. >.9^2 = .81 >.99^2 = .9801 [snip] >.99999999999999999999^2 = .9999999999999999999800000000000000000001 >etc., etc., etc. >The limited minds of the limited mathematicians saith not. Saith not what? Note that your argument has EXACTLY the same structure as: .9 does not equal 1 .99 does not equal 1 ... .99999999999999999999 does not equal 1 etc., etc., etc. You then infer that since none of the individual terms in the sequence equals 1, the *limit* of the entire sequence can not equal 1 either. This is mistaken, since the limit of a sequence of real numbers can be another real number that is NOT one of the terms of the sequence. And note that (.999...)^2 = lim(n->oo) (1 - 1/n)^2 = lim(n->oo) (1 - 2/n - 1/(n^2)) = 1 - 0 - 0 = 1. >Garry Denke, Geologist >Denoco Inc. of Texas -- --------------------------- | B B aa rrr b | | BBB a a r bbb | | B B a a r b b | | BBB aa a r bbb | ----------------------------- === >The square of .999... is not equal to 1, >therefore .999... is not equal to 1. > >.9^2 = .81 >.99^2 = .9801 > [snip] >.99999999999999999999^2 = .9999999999999999999800000000000000000001 >etc., etc., etc. >The limited minds of the limited mathematicians saith not. > Saith not what? The square of .999... is not equal to 1, therefore .999... is not equal to 1. [snip your inference] > And note that (.999...)^2 > = lim(n->oo) (1 - 1/n)^2 > = lim(n->oo) (1 - 2/n - 1/(n^2)) > = 1 - 0 - 0 > = 1. The limited mind of the limited mathematician saith 1. Garry Denke, Geologist Denoco Inc. of Texas === > The square of .999... is not equal to 1, What is the square of .9999.... then? === >The square of .999... is not equal to 1, > >therefore .999... is not equal to 1. >.9^2 = .81 > >.99^2 = .9801 > [snip] >.99999999999999999999^2 = .9999999999999999999800000000000000000001 >etc., etc., etc. >The limited minds of the limited mathematicians saith not. > Saith not what? > The square of .999... is not equal to 1, > therefore .999... is not equal to 1. Can you name a number between (0.999...)^2 and 1? -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Can you name a number between (0.999...)^2 and 1? > This must be the thread that never dies. Even though I already gave my opinion that 0.999... = 1.0 is true by de?ition, I will add fuel to the ?e by showing there are numbers between 0.999... (base 10) and 1. Consider these Cauchy sequences: (9/10, 99/100, 999/1000, ...) = 0.999... base 10 (10/11, 120/121, 1330/1331, ...) = 0.AAA... base 11 The terms in the second sequence are closer to 1 than the corresponding terms in the ?st sequence. One could argue that 0.AAA... base 11 is between 0.999... base 10 and 1. Russell - 2 many 2 count windows-nt) Cancel-Lock: sha1:IGO83rfdzboGt1jdbf2Mv/? === One could argue that 0.AAA... base 11 is between 0.999... base 10 > and 1. Of course it is--because all three numbers equal 1. --Len. === >One could argue that 0.AAA... base 11 is between 0.999... base 10 >and 1. Of course it is--because all three numbers equal 1. > So my statement is not false. Russell - 2 many 2 count windows-nt) Cancel-Lock: sha1:Z48iDc6M+CIM5xJEQblTfqoKXxg= === > One could argue that 0.AAA... base 11 is between 0.999... base 10 > and 1. > > Of course it is--because all three numbers equal 1. So my statement is not false. Where I come from, Of course it is means, Your statement is true. Unless mathematics has changed since my grad-school days, true is not false. So I guess you're right. :-) Len. === > Can you name a number between (0.999...)^2 and 1? > This must be the thread that never dies. > Even though I already gave my opinion > that 0.999... = 1.0 is true by de?ition, > I will add fuel to the ?e by showing there > are numbers between 0.999... (base 10) and 1. > Consider these Cauchy sequences: > (9/10, 99/100, 999/1000, ...) = 0.999... base 10 > (10/11, 120/121, 1330/1331, ...) = 0.AAA... base 11 > The terms in the second sequence are closer > to 1 than the corresponding terms in the > ?st sequence. > One could argue that 0.AAA... base 11 is between > 0.999... base 10 and 1. That makes exactly as much sense as claiming that 6/4 is bigger than 3/2. The sum of the 0.AAA... series is exactly 1. Otherwise stated, each of your Cauchy sequences above belongs to the same equivalence class. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === >The square of .999... is not equal to 1, >therefore .999... is not equal to 1. >.9^2 = .81 >.99^2 = .9801 > [snip] >.99999999999999999999^2 = .9999999999999999999800000000000000000001 >etc., etc., etc. >The limited minds of the limited mathematicians saith not. > Saith not what? >The square of .999... is not equal to 1, >therefore .999... is not equal to 1. > Can you name a number between (0.999...)^2 and 1? Yes. Can you? Garry Denke, Geologist Denoco Inc. of Texas === >The square of .999... is not equal to 1, >therefore .999... is not equal to 1. > Can you name a number between (0.999...)^2 and 1? > Yes. Can you? No, and neither can you. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === >The square of .999... is not equal to 1, >therefore .999... is not equal to 1. > Can you name a number between (0.999...)^2 and 1? >Yes. Can you? > No, and neither can you. you, no i, yes garry denke, geologist denoco inc. of texas === >The square of .999... is not equal to 1, >therefore .999... is not equal to 1. > > Can you name a number between (0.999...)^2 and 1? >Yes. Can you? > No, and neither can you. > you, no > i, yes You haven't, and you can't. If you think otherwise, it merely demonstrates that you have failed to grasp the concept of a real number. Both 0.999... and its square are exactly equal to 1. And no, there are no 8's in the decimal expansion of (0.999...)^2. Either of them. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === > The limited mind of the limited mathematician saith 1. And what saith the limited mind of the limited usenet crank? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === > >The limited mind of the limited mathematician saith 1. > And what saith the limited mind of the limited usenet crank? Garry Denke, Geologist Denoco Inc. of Texas === Yes, the limit is zero. The phrase should be: the expression tends to a limit as n tends to in?ity. However, the point is that such limits must remain unrealised ?at' in?ity. Tony Thomas >Mathematicians avoid this awkwardnwess by the weasel words: > the limit of 1/10^n tends to zero as n tends to in?ity. Nit: the limit *IS* zero. 1/10^n *TENDS TO* zero. The terms are going > someplace ; the limit is not. Len. > === > Yes, the limit is zero. The phrase should be: > the expression tends to a limit as n tends to in?ity. However, the point is that such limits must remain unrealised ?at' > in?ity. Perhaps I don't understand what you mean by must remain unrealised . But I suppose that I disagree with you. Of course, one may _choose_ to speak of only what happens as n increases without bound, thereby avoiding any reference to anything happening ?at' in?ity. There's nothing wrong with doing that. But if one works in, say, the extended reals, then 1/10^x is continuous (from the left) at x = +oo, having the value 0 there. I suppose that is what you would call realisation of the limit at in?ity. David Cantrell > Mathematicians avoid this awkwardnwess by the weasel words: > the limit of 1/10^n tends to zero as n tends to in?ity. > >Nit: the limit *IS* zero. 1/10^n *TENDS TO* zero. The terms are going >someplace ; the limit is not. === >What do you mean refer to ? Its limit is 1 but the series itself is >not equal to 1. Nor does it ever reach 1. .999999... is a name; it's a symbol. Just as 1 is a symbol. The > symbol refers to a thing: the limit of a series of partial sums. As > it happens, the following are two different names, but names for the > same thing: 0.999999.... > 1.000000.... > We use quotation marks when we speak of the name, and drop the marks > when we refer to the thing that the name names. So we have: 0.999999... is not equal to 1.000000... > but > 0.999999... = 1.000000... > That is, these are two names for the same thing. We even have a rule > for which number a decimal names: it names the limit of a series of > partial sums. > You are mistaken if you think that 1 behaves in some special way > that 0.999.... does not: they both work the same way: they are both > names for a number. > Thomas I would argue that .999... is the term resulting from substituting 9 for N in the term .NNN... which represents the in?ite series .N, .NN, .NNN, ... which has a limit of N/9. But then, I just like to argue. Charlie Volkstorf === > I would argue that .999... is the term resulting from substituting 9 > for N in the term .NNN... which represents the in?ite series .N, > .NN, .NNN, ... which has a limit of N/9. You are telling us what .999.... is, typographically. What is it numerically? === >Abe Lincoln: If you call a tail a leg, how many legs does a dog >have? >Spectator: 5. >Abe Lincoln: No, 4. Just because you call a tail a leg doesn't make >it a leg. > But in mathematics if we say > De?ition: a _tail_ is a leg > then a tail _is_ a leg. Any mathematician who rede?es an existing term ought to lose his job (IMHO.) >Get over it. Talk about something signi?ant (e.g. Program Synthesis >or Quine Atoms.) >Charlie Volkstorf > ************************ > David C. Ullrich === >Abe Lincoln: If you call a tail a leg, how many legs does a dog > >have? >Spectator: 5. >Abe Lincoln: No, 4. Just because you call a tail a leg doesn't make >it a leg. > But in mathematics if we say > De?ition: a _tail_ is a leg > then a tail _is_ a leg. >Any mathematician who rede?es an existing term ought to lose his job (IMHO.) Well, that's a bit more evidence regarding how much you know about how math works - mathematicians rede?e existing terms all the time. >Uh, no. By de?ition 0.999... is the _sum_ _of_ a certain in?ite >series. That in?ite series has sum 1. So 0.999... = 1. That's not a _re_de?ition, by the way, it's simply the de?ition. Anyone pretending to mathematical competence who insists that 0.999... <> 1 has no idea what he's talking about (in the literal sense - he doesn't know the _meaning_ of the symbols he's using.) >Get over it. Talk about something signi?ant (e.g. Program Synthesis >or Quine Atoms.) >Charlie Volkstorf > ************************ > David C. Ullrich ************************ David C. Ullrich === >Abe Lincoln: If you call a tail a leg, how many legs does a dog >have? >Spectator: 5. >Abe Lincoln: No, 4. Just because you call a tail a leg doesn't make >it a leg. > > But in mathematics if we say > De?ition: a _tail_ is a leg > then a tail _is_ a leg. >Any mathematician who rede?es an existing term ought to lose his job (IMHO.) > Well, that's a bit more evidence regarding how much you know about how > math works - mathematicians rede?e existing terms all the time. So what does integer mean these days? Do we have to reprint all our old math books now? Maybe if you programmed computers for awhile you'd learn to pay more attention to details. Or have you? >Uh, no. By de?ition 0.999... is the _sum_ _of_ a certain in?ite >series. That in?ite series has sum 1. So 0.999... = 1. > That's not a _re_de?ition, by the way, it's simply the de?ition. > Anyone pretending to mathematical competence who insists > that 0.999... <> 1 has no idea what he's talking about (in > the literal sense - he doesn't know the _meaning_ of the > symbols he's using.) You're just being sloppy with your statements. If you treat 0.999... as a literal equal to 1.000... then you have two different literals being equal, which is not a good idea. But if you are clear that you mean the limits of two series, then there is no problem. >Get over it. Talk about something signi?ant (e.g. Program Synthesis >or Quine Atoms.) >Charlie Volkstorf > ************************ > David C. Ullrich === > >Abe Lincoln: If you call a tail a leg, how many legs does a dog >have? >Spectator: 5. >Abe Lincoln: No, 4. Just because you call a tail a leg doesn't make >it a leg. > But in mathematics if we say > De?ition: a _tail_ is a leg > then a tail _is_ a leg. >Any mathematician who rede?es an existing term ought to lose his job (IMHO.) > Well, that's a bit more evidence regarding how much you know about how > math works - mathematicians rede?e existing terms all the time. >So what does integer mean these days? Do we have to reprint all our >old math books now? Huh? Who said that integer means something different than it did 50 years ago? >Maybe if you programmed computers for awhile you'd learn to pay more >attention to details. Or have you? Uh, >Uh, no. By de?ition 0.999... is the _sum_ _of_ a certain in?ite >series. That in?ite series has sum 1. So 0.999... = 1. > That's not a _re_de?ition, by the way, it's simply the de?ition. > Anyone pretending to mathematical competence who insists > that 0.999... <> 1 has no idea what he's talking about (in > the literal sense - he doesn't know the _meaning_ of the > symbols he's using.) >You're just being sloppy with your statements. No, the person being sloppy, ignoring standard de?itions, is _you_. Everything I've said about what notation means is absolutely standard (I can't prove that because it's not a mathematical fact, but it's true) and everything I've said _using_ those notations is something I can _prove_ from the de?itions. > If you treat 0.999... >as a literal equal to 1.000... then you have two different literals >being equal, which is not a good idea. you shouldn't make cracks about paying attention to details if you're going to turn around and babble like this. Neither 0.999... nor 1.000... is a literal. 0.999... and 1.000... are literals, and nobody's claimed that they're equal. > But if you are clear that you >mean the limits of two series, then there is no problem. The only person to whom this is _not_ clear is you, which is why you're the only one speaking nonsense in this discussion. It's clear to anyone who understands the standard meaning of the notation that the symbol 0.999... denotes the sum of a certain series. (A series has a sum, not a limit - _sequences_ have limits. Pay attention to details , indeed.) We're making some progress here - recently you've said to me and to others that _if_ the notation means what it in fact means then there's no problem saying that 0.999... = 1. Now all we have to do is learn what the notation actually standard meaning of a bit of notation by thinking about it. Notation means what it means, by de?ition. In particular 0.999... _does_ mean the sum of a certain series, ie the limit of the sequence of partial sums of that series.) >Get over it. Talk about something signi?ant (e.g. Program Synthesis >or Quine Atoms.) > >Charlie Volkstorf > ************************ > David C. Ullrich ************************ David C. Ullrich === > > One of the common ways of de?ing the real numbers is to consider the > equivalence classes of Cauchy sequences of rationals. According to this > de?ition, the sequences > > 9/10, 99/100, 999/1000, ... > and > 1, 1, 1, 1, ... > are members of the same equivalence class, and therefore are > representatives of the same real number. >Abe Lincoln: If you call a tail a leg, how many legs does a dog >have? >Spectator: 5. >Abe Lincoln: No, 4. Just because you call a tail a leg doesn't make >it a leg. > Not the least bit relevant. >de?e = call >9/10, 99/100, 999/1000, ... = tail >1, 1, 1, 1, ... = leg > There are two common ways of de?ing the > real numbers. One is using equivalence classes of Cauchy sequences of > rationals, as I indicated. The other is using Dedekind cuts. By either > method, it can be rigorously shown that 0.999... and 1 represent the same > real number. >That is what is not relevant. (It is a red herring.) > Surely the de?ition of what the real numbers are is of some relevance > in deciding whether two real numbers are equal or not. You are misusing the word de?ition . Real numbers existed long before Cauchy and Dedekind. If we were talking about a particular term that someone coined to de?e a new construction (e.g. triangle ), then we could refer to its de?ition to make sure we were using the term correctly. But the term real number does not refer to something originated by either of these two de?itions. Real number is a term coined to represent the intuitive concept of (e.g.) measurement of a physical quantity to unlimited precision. It has its meaning independent of Cauchy sequences and Dedekind cuts. The above uses of Cauchy sequences and Dedekind cuts are attempts to construct mathematical objects that correspond to the real numbers. They de?e the construction of these objects and the correspondance with the real numbers. They do not de?e the real numbers themselves. > I have explained to you the following facts: 1. The real numbers are equivalence classes of Cauchy sequences > of rationals. 2. The real number 0.999... is the equivalence class containing > the Cauchy sequence 9/10, 99/100, 999/1000, .... 3. The real number 1.000... is the equivalence class containing > the Cauchy sequence 1, 1, 1, 1, .... 4. The equivalence classes mentioned in statements 2 and 3 are > in fact the same equivalence class. >.999... is an in?ite series, not an integer, with a limit of 1, an > Nobody said anything about integers. The statement that .999... = 1 refers to the integer 1. > The fact that you have not seen this de?ition before How can you substantiate the above assertion? > does not change > the fact that it is indeed a standard way of de?ing the real numbers. No, it is a way of modeling the real numbers. Real number was de?ed long before that. Charlie Volkstorf === > > There are two common ways of de?ing the > real numbers. One is using equivalence classes of Cauchy sequences of > rationals, as I indicated. The other is using Dedekind cuts. By either > method, it can be rigorously shown that 0.999... and 1 represent the same > real number. >That is what is not relevant. (It is a red herring.) > Surely the de?ition of what the real numbers are is of some relevance > in deciding whether two real numbers are equal or not. > You are misusing the word de?ition . Real numbers existed long > before Cauchy and Dedekind. You don't understand how the term de?ition is used in mathematics. The real numbers were only vaguely understood before Cauchy and Dedekind. There were no de?itions to make the concepts precise. > If we were talking about a particular term that someone coined to > de?e a new construction (e.g. triangle ), then we could refer to > its de?ition to make sure we were using the term correctly. But the > term real number does not refer to something originated by either of > these two de?itions. Real number is a term coined to represent > the intuitive concept of (e.g.) measurement of a physical quantity to > unlimited precision. It has its meaning independent of Cauchy > sequences and Dedekind cuts. If you don't have a de?ition, then how are you going to tell whether you are using the term correctly or not? Consider the title of Dedekind's paper: Was sind und was sollen die Zahlen? . Dedekind was concerned with the question of how the real numbers *ought* to be de?ed, precisely because they hadn't been de?ed yet. We now know that there are other number systems that could also be used to represent the intuitive concept of measurement of a physical quantity to unlimited precision. , but these other number systems (such as the hyperreals or the surreals) do not have the property of being isomorphic to the real numbers. > The above uses of Cauchy sequences and Dedekind cuts are attempts to > construct mathematical objects that correspond to the real numbers. > They de?e the construction of these objects and the correspondance > with the real numbers. They do not de?e the real numbers > themselves. Wrong. By de?ition, the real numbers are the objects constructed by these methods. > I have explained to you the following facts: > 1. The real numbers are equivalence classes of Cauchy sequences > of rationals. > 2. The real number 0.999... is the equivalence class containing > the Cauchy sequence 9/10, 99/100, 999/1000, .... > 3. The real number 1.000... is the equivalence class containing > the Cauchy sequence 1, 1, 1, 1, .... > 4. The equivalence classes mentioned in statements 2 and 3 are > in fact the same equivalence class. >.999... is an in?ite series, not an integer, with a limit of 1, an > Nobody said anything about integers. > The statement that .999... = 1 refers to the integer 1. Only in the sense that the integers are a subring of the reals. The question of whether .999... = 1 makes no sense unless both sides are understood to be real numbers. > The fact that you have not seen this de?ition before > How can you substantiate the above assertion? It's obvious that you have completely misunderstood the de?ition, if you have seen it. > does not change > the fact that it is indeed a standard way of de?ing the real numbers. > No, it is a way of modeling the real numbers. Real number was > de?ed long before that. By whom? All this is by the way, since you have already conceded the argument in another branch of this thread by admitting that there is no problem in de?ing 0.999... to mean the limit of the sequence 0.9, 0.99, 0.999, .... -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === >.999... is an in?ite series, not an integer, with a limit of 1, an > Nobody said anything about integers. >The statement that .999... = 1 refers to the integer 1. > Only in the sense that the integers are a subring of the reals. Concession accepted. > The fact that you have not seen this de?ition before >How can you substantiate the above assertion? > It's obvious that you have completely misunderstood the de?ition, if > you have seen it. So you don't know that I haven't seen it after all. Now how have I demonstrated a misunderstanding? > does not change > the fact that it is indeed a standard way of de?ing the real numbers. >No, it is a way of modeling the real numbers. Real number was >de?ed long before that. > By whom? By its use in texts since antiquity. > All this is by the way, since you have already conceded the argument in > another branch of this thread by admitting that there is no problem in > de?ing 0.999... to mean the limit of the sequence 0.9, 0.99, 0.999, > .... You can de?e a symbol (e.g. 0.999... ) to mean anything you want, as long as it has no meaning already. You can't de?e an intuitive concept (e.g. real number or integer ) because it already has an intuitive meaning. You can only derive properties of it. That is the distinction that you are missing. C-B === >.999... is an in?ite series, not an integer, with a limit of 1, an > Nobody said anything about integers. > The statement that .999... = 1 refers to the integer 1. >Only in the sense that the integers are a subring of the reals. > Concession accepted. > The fact that you have not seen this de?ition before > How can you substantiate the above assertion? > >It's obvious that you have completely misunderstood the de?ition, if >you have seen it. > So you don't know that I haven't seen it after all. Now how have I > demonstrated a misunderstanding? > does not change > the fact that it is indeed a standard way of de?ing the real numbers. > No, it is a way of modeling the real numbers. Real number was > de?ed long before that. >By whom? > By its use in texts since antiquity. > That isn't the same as having a rigorous mathematical de?ition that gets rid of all these issues you are having with different representations of the same real number. So, have you seen the de?ition of the real numbers as cauchy sequences? Or dedekind cuts? You are side stepping the issue a little >All this is by the way, since you have already conceded the argument in >another branch of this thread by admitting that there is no problem in >de?ing 0.999... to mean the limit of the sequence 0.9, 0.99, 0.999, >.... > You can de?e a symbol (e.g. 0.999... ) to mean anything you want, > as long as it has no meaning already. You can't de?e an intuitive > concept (e.g. real number or integer ) because it already has an > intuitive meaning. You can only derive properties of it. > That is the distinction that you are missing. > C-B === >.999... is an in?ite series, not an integer, with a limit of 1, an > Nobody said anything about integers. >The statement that .999... = 1 refers to the integer 1. > Only in the sense that the integers are a subring of the reals. > Concession accepted. An integer may be an equivalence class of ordered pairs of natural numbers, where (a,b) ~ (c,d) if a+d = b+c. Or an integer may be an equivalence class of ordered pairs of Cauchy sequences of rationals, with the obvious equivalence relation, in which one of the members of the equivalence class has the form of a constant sequence with all terms equal to n/1 for some integer n. I wanted to make it clear that I was talking about the second kind of integer, not the ?st. Both 0.999... and 1 = 1.000... are of this type. > The fact that you have not seen this de?ition before >How can you substantiate the above assertion? > It's obvious that you have completely misunderstood the de?ition, if > you have seen it. > So you don't know that I haven't seen it after all. Now how have I > demonstrated a misunderstanding? You demonstrated that with your tail = leg nonsense, in which you clearly did not realize that when a real number is de?ed to be an equivalence class of Cauchy sequences of rationals, that means precisely that a real number is an equivalence class of Cauchy sequences of rationals. > does not change > the fact that it is indeed a standard way of de?ing the real numbers. >No, it is a way of modeling the real numbers. Real number was >de?ed long before that. > By whom? > By its use in texts since antiquity. As I have said, the ancients had only an intuitive idea of what a real number was. > All this is by the way, since you have already conceded the argument in > another branch of this thread by admitting that there is no problem in > de?ing 0.999... to mean the limit of the sequence 0.9, 0.99, 0.999, > .... > You can de?e a symbol (e.g. 0.999... ) to mean anything you want, > as long as it has no meaning already. You can't de?e an intuitive > concept (e.g. real number or integer ) because it already has an > intuitive meaning. You can only derive properties of it. > That is the distinction that you are missing. There is no such distinction in mathematics. You can't derive properties of a mathematical object unless it either (1) has a de?ition, or (2) has properties that are derived from the basic axioms. The real numbers are an example of case (1). Sets and set membership are examples of case (2). Before the reals had a de?ition, people were merely assuming without proof that the reals had certain properties. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === > 1/10 + 1/10^2 + ...+1/10^n = 1 - 1/10^n > > This is true for all (?ite) values of n. > (1 - 1/10^n) > 1 > This is also true for all ?ite values of n (and trans?ite values too) > Only when n achieves absolute in?ity does 1/10^n become zero. Wow. TWO mistakes in three lines. (The sum is 1/9, not 1-1/10^n. There are no values of n with 1-1/10^n > 1.) That's an 0.666... batting average :-) Or 0.333..., depending on what you count as a success. And of course, 0.666... + 0.333... === == = 0.999... --Ron Br === > 1/10 + 1/10^2 + ...+1/10^n = 1 - 1/10^n > This is true for all (?ite) values of n. > (1 - 1/10^n) > 1 > This is also true for all ?ite values of n (and trans?ite values too) > Only when n achieves absolute in?ity does 1/10^n become zero. I ?d that (1 - 1/10^n) > 1 is false for all real n. Did you mean (1 - 1/10^n) < 1 or did you mean (1 - 1/10^n) > 0 ? For natural number values of n, the values of 1/10^n form a sequence of rationals decreasing towards 0 as n increases without bound. But n is never in?ite since natural numbers are all ?ite, so 1/10^n never becomes zero. The sequence is a Cauchy sequence and therefore has a limiting value, but that limit need not be one of the numbers in the sequence itself, just as least upper bounds and greatest lower bounds, where they exist, of sets of real numbers need not be members of the sets of which they are bounds. === >.999... is an in?ite series. >The limit of .999... is 1. >.999... is not 1, because it is an in?ite series and 1 is not an >in?ite series. >Get over it. Talk about something signi?ant (e.g. Program Synthesis >or Quine Atoms.) > >Charlie Volkstorf > In that case, 0.333... is not 1/3 and 0.666... is not 2/3, and it must > be wrong to write 0.333... = 1/3 or 0.666... = 2/3, or any of the other > non-terminating repeating decimals as fractions!. >No. The limit of .3, .33, .333, ... is in fact 1/3 just as the limit >of .6, .66, .666, ... is 2/3, of course. But when you leave out the >limit of (either in an English description or the de?ition of your >mathematical notation), then you are making a mistake and you >introduce the problems we are now seeing. > Duh, that's what the ellipsis in .999... are for. Nobody claims that > .9 =1, or .99 =1, etc. But .99..., which is to say, the limit of a > series, is 1. Same with .3..., .6..., and so on. Ever heard of > context sensitivity? I am referring to the statement that ?There is no real number between 0.999... and 1, and, therefore, they must be one and the same number!' treating .999... as a number in the same vein as 1, leading to two different (literal) numbers being equal. This problem also pops us in when we diagonalize and produce .999... and mathematicians say Oh no - that's 1 and 1 IS in our list! and needlessly jump through hoops to ? it. Charlie Volkstorf > ?cid === > I am referring to the statement that ?There is no real number between > 0.999... and 1, and, therefore, they must be one and the same > number!' treating .999... as a number in the same vein as 1, leading > to two different (literal) numbers being equal. You are mistaking the symbol for a number with the number. 0.999999.... is a different symbol from 1.000.... So is I , a different symbol for the same number. All three are symbols for exactly the same number. Thomas === Duh, that's what the ellipsis in .999... are for. Nobody claims that >.9 =1, or .99 =1, etc. But .99..., which is to say, the limit of a >series, is 1. Same with .3..., .6..., and so on. Ever heard of >context sensitivity? > I am referring to the statement that ?There is no real number between > 0.999... and 1, and, therefore, they must be one and the same > number!' treating .999... as a number in the same vein as 1, leading > to two different (literal) numbers being equal. This problem also > pops us in when we diagonalize and produce .999... and mathematicians > say Oh no - that's 1 and 1 IS in our list! and needlessly jump > through hoops to ? it. > You miss my point. .999... is a peice of notation that refers to the limit of the obvious in?ite series. (The series, of course, being Sum_j=1 to in?ity of 9/(10^j). This limit is clearly 1. (A little epsilon-N makes this trivial). So, .999... refers to a number which is the limit of a certain series, and this number is 1. Hence, .999... = 1. ?cid === * Acid Pooh > You miss my point. .999... is a peice of notation that refers to > the limit of the obvious in?ite series. (The series, of course, > being Sum_j=1 to in?ity of 9/(10^j). This limit is clearly 1. (A > little epsilon-N makes this trivial). So, .999... refers to a > number which is the limit of a certain series, and this number is 1. > Hence, .999... = 1. FWIW, .999 also refers to an in?ite series, namely 9/10 + 9/100 + 9/1000 + 0/10000 + 0/100000 + .... Incidently, it can be reduced to the ?ite series 9/10 + 9/100 + 9/1000 -- Jon Haugsand === >The limit of .3, .33, .333, ... is in fact 1/3 just as the limit >of .6, .66, .666, ... is 2/3, of course. But when you leave out the >limit of (either in an English description or the de?ition of your >mathematical notation), then you are making a mistake and you >introduce the problems we are now seeing. >Charlie Volkstorf > There is no problem in logic or mathematics _de?ing_ such symbols as > 0.999... or 0.333... as the limit of partial sums rather than as the > sequence of partial sums. Read what I said (above): when you leave out ?the limit of' in the de?ition you introduce problems . You are not leaving out the limit of and thus my statement does not apply to what you are saying. > And there is no obligation on the rest of the world to reject such a > useful and generally accepted de?ition want, as long as it is de?ed only once. Again my statement does not apply to what you are saying. > simply because some dingbat gets anal retentive about it. What's worse, being anal or being illiterate? > Live with it. Charlie Volkstorf === > Read what I said (above): when you leave out ?the limit of' in the > de?ition you introduce problems . You are not leaving out the > limit of and thus my statement does not apply to what you are saying. The symbol .999.... MEANS the limit of the such-and-such a series of partial sums . So when we say .9999.... = 1.00000... there is no leaving out of a limit. Moreover, when you say 1 , there is the same limit. > want, as long as it is de?ed only once. Again my statement does not > apply to what you are saying. But it *has* been de?ed (once), decimals *are* symbols for the limit of the series such-and-such . Thomas === >The limit of .3, .33, .333, ... is in fact 1/3 just as the limit >of .6, .66, .666, ... is 2/3, of course. But when you leave out the >limit of (either in an English description or the de?ition of your >mathematical notation), then you are making a mistake and you >introduce the problems we are now seeing. >Charlie Volkstorf > There is no problem in logic or mathematics _de?ing_ such symbols as > 0.999... or 0.333... as the limit of partial sums rather than as the > sequence of partial sums. > Read what I said (above): when you leave out ?the limit of' in the > de?ition you introduce problems . You are not leaving out the > limit of and thus my statement does not apply to what you are saying. Then you agree that 0.999... = 1. Because: (1) 0.999... means the limit of the sequence 0.9, 0.99, 0.999, ..., (2) The limit of that sequence is 1, and (3) Two things equal to the same thing are equal to each other. Simple, no? -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === > Then you agree that 0.999... = 1. > Because: (1) 0.999... means the limit of the sequence 0.9, 0.99, 0.999, ..., > (2) The limit of that sequence is 1, and > (3) Two things equal to the same thing are equal to each other. > Simple, no? If 0.999... means the limit of .9, .99, .999... (and is consistently used to mean that) then certainly 0.999...=1 C-B === >.999... is an in?ite series. >The limit of .999... is 1. >.999... is not 1, because it is an in?ite series and 1 is not an >in?ite series. >Get over it. Talk about something signi?ant (e.g. Program Synthesis >or Quine Atoms.) >Charlie Volkstorf > In that case, 0.333... is not 1/3 and 0.666... is not 2/3, and it must > be wrong to write 0.333... = 1/3 or 0.666... = 2/3, or any of the other > non-terminating repeating decimals as fractions!. >No. The limit of .3, .33, .333, ... is in fact 1/3 just as the limit >of .6, .66, .666, ... is 2/3, of course. But when you leave out the >limit of (either in an English description or the de?ition of your >mathematical notation), then you are making a mistake and you >introduce the problems we are now seeing. > You've contradicted yourself there. How ?zat? > Answer yes or no > 0.333... = 1/3 If you mean the in?itely long string 0.333... then no. If you mean the limit of the in?ite series .3, .33, .333, ... then yes. Charlie Volkstorf > Although you have an interesting point is 0.99... an integer? no. > Herc windows-nt) Cancel-Lock: sha1:K19z/4ttNyaXSuTtHlswcPFU1QY= === Mathematicians avoid this awkwardnwess by the weasel words: > the limit of 1/10^n tends to zero as n tends to in?ity. Nit: the limit *IS* zero. 1/10^n *TENDS TO* zero. The terms are going someplace ; the limit is not. Len. === > In our ordinary real number system, we say that the number K with >decimal expansion .99999... is the samas 1. An informal argument for >this is sketched below: 10K = 9.999... >- K = .9999... >____________ > 9K = 9 > K = 1 >But maybe this argument is misleading. What if there is some number, >call it 1 - 1/omega, that is greater than any ?ite string .9...9 of >nines, yet less than 1? If K were actually equal to 1 - 1/omega, the >informal argument used in the last paragraph would not work, ... Let t be the difference between 1 and 0.9999... ( t stands for ?tesimal.) K = 1 - t = 0.9999... 10K = 10 - 10t = 9.9999... K + 9 = 10 - t = 9.9999... Uh oh... 10 - 10t = 10 - t 10t = t 9t = 0 t = 0 -- David Canzi === > In our ordinary real number system, we say that the number K with > decimal expansion .99999... is the samas 1. An informal argument for > this is sketched below: 10K = 9.999... > - K = .9999... > ____________ > 9K = 9 > K = 1 > But maybe this argument is misleading. What if there is some number, > call it 1 - 1/omega, that is greater than any ?ite string .9...9 of > nines, yet less than 1? If K were actually equal to 1 - 1/omega, the > informal argument used in the last paragraph would not work, for this > argument overlooks the fact that the difference between 10K and 10 is > ten times as great as the difference between K and 1. There is a > residual in?itesimal quantity below that does not get canceled out: 10K = 10 - 10/omega > - K = 1 - 1/omega > _________________ > 9K = 9 - 9/omega > K = 1 - 1/omega > but how would this string 1-1/aleph0 be represented? in every real base b+1, .bbbbb... repeating is never bigger than .999... in base 10 .aaaaa... in base b = K, follow the same process.. x = a/(b-1). the maximum value of a is b-1, and thus x cannot be greater than 1. this even has the logical extension: limit b to in?ity, where each integer becomes in?itessimal, there's still no breaking of it. multiplication by (b-1) in base b moves the decimal place one to the right, which has _no_ effect on an in?ite list of nines, and therefore 10K still equals 10 - 1/aleph0 === Paul I have been pushing this barrow for some time but the fruit is regarded as rotten or forbidden by the cogniscenti. I do appreciate that the ?doctrine' of real numbers has been forged by some of the world's ?est intellects and is not to be taken lightly, however, your argumant for a bit left over in convergent in?ite series may be ? Consider: Sn = (9/10 + 9/100 + ...+9/10^n) = 1 - 9/10^n which is true for all values of n. The mysterious zen bull's tail never disappears altogether but seems to converge to your omega rather than to zero. But the high priests say that as n tends to in?ity the tail tends to zero. The trick is that this expression and others like it can be said to never exceed its limit, which seems to be true. When we say that 2x is the differential coef?ient of x^2 we are are including such a limit in the expression, but this works OK. The point is that the use of such limits is guaranteed to be harmless, provided that we do not admit any pesky in?itessimals. Some time ago (1950s), mathemeticians invented things like supernatiural numbers and integrated them into analysis, so the whole question is water under the bridge. The annoying residual is all that Cantorian set theoretic doctrine that doesn't seem to be consistent with the newly respectable in?ite and in?itessimal constructions. (but I don't know because I am loath to divert my meagre intellectual resources to climbing these arcane heights.) My preference is for a ?number' system which includes both in?itessimal and in?ite ?numbers' which can be operated on by familiar arithmetical rules. This requires a shift in perspective which This can be illustrated by the following (crude) formulation. Ur = b^rw Where Ur is a cardinal, b is a ?ite base, r is a ?ite integral parameter and w is an in?ite number. When: r = 0, Ur = 1 r = 1, Ur = 1* (the ?st in?ite cardinal relative to (b,w) r = -1, Ur = *1 (the ?st in?itesimal 1/1*) The main conclusion, which Cantor would have seen right away is that this leads to an in?ite hierarchy of in?ities and thrusts beyond all hope the primitive idea of absolute in?ity. Being a religious nut, he couldn't stand the face of God when he saw it. In the doctrine of real numbers, the set of the irrational numbers is de?ed negatively as all other numbers on the real line than the rationals. A detailed classi?ation of these is impossible but their general type is de?ed by the in?ite convergent sequences used in Cantor's diagonal argument. This leaves open the possibility of claiming that the real line is absolutely continuous. This seems fair enough if we allow that Ao stands for absolute in?ity. But as the diagonal argument shows, this Ao is far too small a base to ensure absolute continuity. Here the mysti?ation about the diagonal begins. Suf?e to say, the argument is a peitio pricipii and therfore ? The genii has been out of the bottle for some time but a lot of mathematicians don't seem to know or can't face the prospect of extending the kingdom of real numbers to include all those demons from hell. Tony Thomas > In our ordinary real number system, we say that the number K with > decimal expansion .99999... is the samas 1. An informal argument for > this is sketched below: 10K = 9.999... > - K = .9999... > ____________ > 9K = 9 > K = 1 But maybe this argument is misleading. What if there is some number, > call it 1 - 1/omega, that is greater than any ?ite string .9...9 of > nines, yet less than 1? If K were actually equal to 1 - 1/omega, the > informal argument used in the last paragraph would not work, for this > argument overlooks the fact that the difference between 10K and 10 is > ten times as great as the difference between K and 1. There is a > residual in?itesimal quantity below that does not get canceled out: 10K = 10 - 10/omega > - K = 1 - 1/omega > _________________ > 9K = 9 - 9/omega > K = 1 - 1/omega Intuitively, nothing could be more natural than to go ahead and talk > about 1/omega, 1/Aleph-1, and so on. Just as we move from the natural > numbers to the fractions and then on to the reals, should we not be > able to move from the whole ordinal numbers to some richer number > ?ld? > Curiously, Cantor himself was very much opposed to this step. When a > fellow mathematician attempted to use Cantors trans?ite numbers to > develop a theory of in?itely small quantities, Cantor accused him of > trying to ?infect mathematics with the Cholera-Bacillus of > in?itesimals'. Cantor even constructed a proof that no number can be > in?itesimal. This proof, however, is just as circular and worthless > as ?itist attempts to prove that no number can be in?ite. In both > cases, the desired conclusion is smuggled in as part of the de?ition > of ?number'. > Why was Cantor so vehemently opposed to in?itesimals? In his > valuable essay, ?The Metaphysics of the Calculus', Abraham Robinson > suggests that Cantor already had enough problems trying to defend > trans?ite numbers. It seems likely that, consciously or otherwise, > Cantor deemed it politically wise to go along with othodox > mathematicians on the question of in?itesimals. Cantors stance > might be compared to that of a pro-marijuana Congressional candidate > who advocates harsh penalties for the sale or use of heroin. Yet, as > we shall see, there is almost as much justi?ation for in?itesimals > as there is for Cantors trans?ite ordinals. > Formally speaking, it is as consistent to say that there is a number > between all of .9, .99, .999, ... and 1 as it is to say that there is > a number greater than all of 1, 2, 3, ... . And just as we go on to > ?d more and more ordinals piled atop one another, we can go on to > ?d more and more in?itesimals squeezed beneath each other. > [...] > But so great is the average persons fear of in?ity that to this day > calculus all over the world is being taught as a study of *limit > processes* instead of what it really is: *in?itesimal analysis*. > As someone who has spent a good portion of his adult life teaching > calculus courses for a living, I can tell you how weary one gets of > trying to explain the complex and ?dling theory of limits to wave > after wave of uncomprehending freshman. ?How pleasant it would be to let pass away some of the verbiage I > learnt at school--learnt because teachers must live, I suppose. The > apeing and prolonged caw called grammar, the cackling of the human hen > over the egg of language--I should like to unlearn grammar.' But there is hope for a brighter future. Robinsons investigations of > the hyperreal numbers have put in?itesimals on a logically > unimpeachable basis, and here and there calculus texts based on > in?itesimals have appeared [*]. [*: - Keisler, H. J. (1976). /Elementary Calculus/. Boston: Prindle, > Weber & Schmidt. > - Henle & Kleinberg (1978). /In?itesimal Calculus/. Cambridge, > Mass.: MIT Press.] [Rer, Rudy (1995). /In?ity and the Mind/. Princeton, NJ: > Princeton University Press. (pp. 79/80 + 87)] > PH === > 1 - 1/omega There is another system, besides Robinson's hyperreals, namely Conway's surreals, in which such in?itesimal calculations are common. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === > But there is hope for a brighter future. Robinsons investigations of the > hyperreal numbers have put in?itesimals on a logically unimpeachable > basis, and here and there calculus texts based on in?itesimals have > appeared [*]. What is the current status of this? I read somewhere, but don't recall where that it was proven that Robinson's approach and the standard approach are exactly equivalent in what they can do, so that while Robinson's approach is more intuitive, it cannot lead to any more or less than the standard approach. Thus, while the non-standard approach would be better if everyone was starting from scratch, there is so much already in standard terms, that learning the standard approach is necessary, and once you've done that, there is no need to learn the non-standard approach, since you can do everything using the standard approach. Basically, a mathematical society gets to choose one way or the other, and then they are st with it. We hit on the limit approach rather than the rigorization of infenitesimals, and now are st. So, is that correct? -- --Tim Smith === >But there is hope for a brighter future. Robinsons investigations of the >hyperreal numbers have put in?itesimals on a logically unimpeachable >basis, and here and there calculus texts based on in?itesimals have >appeared [*]. > What is the current status of this? I read somewhere, but don't recall > where that it was proven that Robinson's approach and the standard approach > are exactly equivalent in what they can do, correct > so that while Robinson's > approach is more intuitive, not clear > it cannot lead to any more or less than the > standard approach. There are no new theorems in standard mathematics that can be proved by non-standard methods.[*] In some cases non-standard proofs may be shorter or easier in some other sense. In other cases longer or harder. > Thus, while the non-standard approach would be better if > everyone was starting from scratch, a wild assumption > there is so much already in standard > terms, that learning the standard approach is necessary, and once you've > done that, there is no need to learn the non-standard approach, since you > can do everything using the standard approach. > Basically, a mathematical society gets to choose one way or the other, and > then they are st with it. We hit on the limit approach rather than the > rigorization of in?itesimals, and now are st. a rather extreme way to say it > So, is that correct? Except for specialists, one need only learn one system. And that might as well be the commonly-used system, so one can talk to other people. Specialists may learn, and use, two or more systems. . . . [*] This equivalence does depend on the Axiom of Choice. Proof of the existence of non-standard models relies on the Compactness Theorem of model theory. Non-standard analysis can prove the Boolean Algebra Prime Ideal Theorem without any further use of AC. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === > so that while > Robinson's approach is more intuitive No. it isn't. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === > Basically, a mathematical society gets to choose one way or the other, and > then they are st with it. We hit on the limit approach rather than the > rigorization of infenitesimals, and now are st. > So, is that correct? No, because the original idea is the in?itesimal - Through P let there be drawn to this [ spherical ] surface two lines HK, IL, intercepting very small arcs HI, KL ; ... - Prop. LXX Theor. XXX of Newtons' Principia. This is his proof that there is no gravitational attraction within a hollow uniform sphere. I think nonstandard analysis is a more direct justi?ation of this type of reasoning, as opposed to the limit justi?ation which casts it in somewhat different terms. In physics, one never really makes the distinction between delta x and dx , and I think if one understands delta x as ?st order in x the equation can be made rigorous, as witness the nonstandard construction of in?itesimals in terms of sequences. Lew Mammel, Jr. === > 881c8779.0401041419.1097372c@posting.google.com>... > [...] > But there is hope for a brighter future. Robinsons investigations of > the hyperreal numbers have put in?itesimals on a logically > unimpeachable basis, and here and there calculus texts based on > in?itesimals have appeared [*]. > [*: - Keisler, H. J. (1976). /Elementary Calculus/. Boston: Prindle, > Weber & Schmidt. > - Henle & Kleinberg (1978). /In?itesimal Calculus/. Cambridge, > Mass.: MIT Press.] > [Rer, Rudy (1995). /In?ity and the Mind/. Princeton, NJ: > Princeton University Press. (pp. 79/80 + 87)] I have just discovered that the entire text of Keislers book, to which Rer refers, can be downloaded for free at the following link: http://www.math.wisc.edu/~keisler/calc.html PH === In sci.logic, Paul Holbach on 4 Jan 2004 14:19:31 -0800 <881c8779.0401041419.1097372c@posting.google.com>: > In our ordinary real number system, we say that the number K with > decimal expansion .99999... is the samas 1. An informal argument for > this is sketched below: 10K = 9.999... > - K = .9999... > ____________ > 9K = 9 > K = 1 > But maybe this argument is misleading. That argument suffers from a major problem, as one can readily see by considering the ?ite case. Kn = .999...99 10*Kn = 9.999...90 10*Kn - Kn = 8.999...91 > What if there is some number, > call it 1 - 1/omega, that is greater than any ?ite string .9...9 of > nines, yet less than 1? If K were actually equal to 1 - 1/omega, the > informal argument used in the last paragraph would not work, for this > argument overlooks the fact that the difference between 10K and 10 is > ten times as great as the difference between K and 1. There is a > residual in?itesimal quantity below that does not get canceled out: 10K = 10 - 10/omega > - K = 1 - 1/omega > _________________ > 9K = 9 - 9/omega > K = 1 - 1/omega > Intuitively, nothing could be more natural than to go ahead and talk > about 1/omega, 1/Aleph-1, and so on. Just as we move from the natural > numbers to the fractions and then on to the reals, should we not be > able to move from the whole ordinal numbers to some richer number > ?ld? K = .9999... = 1 - 1/infty K/10 = .09999... = 0.1 - 1/(10*infty) 0.9 + K/10 = 0.9999... = 1 - 1/(10*infty) Might work but then one would have to, as you state below, be very careful as to where the in?tesimals and trans?ites go. Some possible questions, for example, from college freshman (well, OK, one ex-college freshman :-) ). [1] What is 1/infty + 1/infty? [a] 2/infty [b] 1/infty [c] 0 [d] indeterminate [e] the expression cannot be simpli?d, but must be left in this form [2] What is 1/infty - 1/infty? [a] 0 [b] 1/infty [c] -1/infty [d] indeterminate [e] the expression cannot be simpli?d [3] What is (1/infty)/(1/infty)? [a] 1 [b] infty [c] any number you want [d] unde?ed [e] the expression cannot be simpli?d [4] What is 1/infty^2? [a] 1/infty [b] 0 [c] any number you want [d] unde?ed [e] the expression cannot be simpli?d [5] What is 1/sqrt(infty)? [a] 1/infty [b] 0 [c] any number you want [d] unde?ed [e] the expression cannot be simpli?d [6] What is 0.999... + 1/infty? [a] 1 [b] 0.999.... [c] any number you want [d] unde?ed [e] the expression cannot be simpli?d [7] What is the relationship between 1/infty and 0? [a] 1/infty > 0 [b] 1/infty = 0 [c] indeterminate [8] What does 1/3 equal? [a] 0.333.... [b] 0.333.... + 1/infty [c] 0.333.... + 1/(3*infty) [d] indeterminate [e] the expression cannot be expanded into an in?ite decimal [9] What does 3*(1/3) equal? [a] 1 [b] 0.999... [c] 1 - 1/infty [d] 1 - 3/infty [e] indeterminate [f] the expression cannot be simpli?d [10] What does sqrt(1 - 1/infty) equal? [a] 1 - 1/(2*infty) + 3/(8*infty^2) - 5/(16*infty^3) + ... [b] 1 - 1/(2*infty) [c] 1 - 1/infty [d] 1 [e] indeterminate [f] the expression cannot be simpli?d [11] What does lim(h->0+) (1+h)^(1/h) equal? [a] Euler's number, ?e' [b] (1+1/infty)^infty [c] the expression cannot be simpli?d [12] What does 1/(1 - 1/infty) equal? [a] 1 + 1/infty + 1/infty^2 + 1/infty^3 + ... [b] 1 + 1/infty [c] 1 [d] indeterminate [e] the expression cannot be simpli?d [13] What is lim(x->0-) (x^2/x)? [a] 0 [b] -1/infty [c] 1/infty [d] infty/infty^2 [e] indeterminate [f] the expression cannot be simpli?d [14] What is sum(i=1,+infty) (1/infty)? [a] 0 [b] 1 [c] e [d] indeterminate [e] the expression cannot be simpli?d [15] Does the trichotomy principle always hold? [a] yes [b] yes, but it may be tricky to determine when infty is in there [c] only when infty is not involved [d] no [e] unknown All of these will have to be dealt with in some fashion. Standard mathematics uses the sequence 1c2a3d4b5b6a7b8a9a10d11a12c13a14d15a, which implicitly sets 1/infty=0 pretty much everywhere. > Curiously, Cantor himself was very much opposed to this step. When a > fellow mathematician attempted to use Cantors trans?ite numbers to > develop a theory of in?itely small quantities, Cantor accused him of > trying to ?infect mathematics with the Cholera-Bacillus of > in?itesimals'. Cantor even constructed a proof that no number can be > in?itesimal. This proof, however, is just as circular and worthless > as ?itist attempts to prove that no number can be in?ite. In both > cases, the desired conclusion is smuggled in as part of the de?ition > of ?number'. It depends on how one de?es number . If one uses Cauchy sequences one has problems, for example, as Cauchy sequences lead to limits. 0.999... is such a sequence. f_1 = 0.9 f_2 = 0.99 f_3 = 0.999 etc. The classical de?ition of limit for this case is: f = lim(s->+infty) f_s if, for any epsilon > 0 I pick, one can show an N such that for all s > N, abs(f_s - f) < epsilon. If f = 1 - 1/infty, I pick epsilon = 1/2*infty, and then wonder what N satis?s this de?ition, bearing in mind 0 < 1/infty < 1/10^N for any integer N (although the nonpositive integers aren't all that interesting :-) ) You may also recall Douglas Adams' number 2^(infty - 1). > Why was Cantor so vehemently opposed to in?itesimals? In his > valuable essay, ?The Metaphysics of the Calculus', Abraham Robinson > suggests that Cantor already had enough problems trying to defend > trans?ite numbers. It seems likely that, consciously or otherwise, > Cantor deemed it politically wise to go along with othodox > mathematicians on the question of in?itesimals. Cantors stance > might be compared to that of a pro-marijuana Congressional candidate > who advocates harsh penalties for the sale or use of heroin. Yet, as > we shall see, there is almost as much justi?ation for in?itesimals > as there is for Cantors trans?ite ordinals. > Formally speaking, it is as consistent to say that there is a number > between all of .9, .99, .999, ... and 1 as it is to say that there is > a number greater than all of 1, 2, 3, ... . And just as we go on to > ?d more and more ordinals piled atop one another, we can go on to > ?d more and more in?itesimals squeezed beneath each other. > [...] > But so great is the average persons fear of in?ity that to this day > calculus all over the world is being taught as a study of *limit > processes* instead of what it really is: *in?itesimal analysis*. > As someone who has spent a good portion of his adult life teaching > calculus courses for a living, I can tell you how weary one gets of > trying to explain the complex and ?dling theory of limits to wave > after wave of uncomprehending freshman. > ?How pleasant it would be to let pass away some of the verbiage I > learnt at school--learnt because teachers must live, I suppose. The > apeing and prolonged caw called grammar, the cackling of the human hen > over the egg of language--I should like to unlearn grammar.' > But there is hope for a brighter future. Robinsons investigations of > the hyperreal numbers have put in?itesimals on a logically > unimpeachable basis, and here and there calculus texts based on > in?itesimals have appeared [*]. Let us hope Robinson has some answers for my questions. I've not read his works, though. The Leibnitz notation [dy/dx] is currently a limit, even if it does look like the division of two in?itesimals. If y(x) = x^2, then dy/dx = lim(h->0) (y(x+h) - y(x))/h = lim(h->0) (x^2 + 2hx + h^2 - x^2) / h = lim(h->0) (2hx + h^2)/h = lim(h->0) (2x + h) = 2x, for example. Integration is similar. This notation can be abused nastily, and is. > [*: - Keisler, H. J. (1976). /Elementary Calculus/. Boston: Prindle, > Weber & Schmidt. > - Henle & Kleinberg (1978). /In?itesimal Calculus/. Cambridge, > Mass.: MIT Press.] > [Rer, Rudy (1995). /In?ity and the Mind/. Princeton, NJ: > Princeton University Press. (pp. 79/80 + 87)] > PH -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Or, 1/3 = .333333... 3 * 1/3 = 1 = .99999... > In our ordinary real number system, we say that the number K with > decimal expansion .99999... is the samas 1. An informal argument for > this is sketched below: 10K = 9.999... > - K = .9999... > ____________ > 9K = 9 > K = 1 > But maybe this argument is misleading. What if there is some number, > call it 1 - 1/omega, that is greater than any ?ite string .9...9 of > nines, yet less than 1? If K were actually equal to 1 - 1/omega, the > informal argument used in the last paragraph would not work, for this > argument overlooks the fact that the difference between 10K and 10 is > ten times as great as the difference between K and 1. There is a > residual in?itesimal quantity below that does not get canceled out: 10K = 10 - 10/omega > - K = 1 - 1/omega > _________________ > 9K = 9 - 9/omega > K = 1 - 1/omega > Intuitively, nothing could be more natural than to go ahead and talk > about 1/omega, 1/Aleph-1, and so on. Just as we move from the natural > numbers to the fractions and then on to the reals, should we not be > able to move from the whole ordinal numbers to some richer number > ?ld? > Curiously, Cantor himself was very much opposed to this step. When a > fellow mathematician attempted to use Cantors trans?ite numbers to > develop a theory of in?itely small quantities, Cantor accused him of > trying to ?infect mathematics with the Cholera-Bacillus of > in?itesimals'. Cantor even constructed a proof that no number can be > in?itesimal. This proof, however, is just as circular and worthless > as ?itist attempts to prove that no number can be in?ite. In both > cases, the desired conclusion is smuggled in as part of the de?ition > of ?number'. > Why was Cantor so vehemently opposed to in?itesimals? In his > valuable essay, ?The Metaphysics of the Calculus', Abraham Robinson > suggests that Cantor already had enough problems trying to defend > trans?ite numbers. It seems likely that, consciously or otherwise, > Cantor deemed it politically wise to go along with othodox > mathematicians on the question of in?itesimals. Cantors stance > might be compared to that of a pro-marijuana Congressional candidate > who advocates harsh penalties for the sale or use of heroin. Yet, as > we shall see, there is almost as much justi?ation for in?itesimals > as there is for Cantors trans?ite ordinals. > Formally speaking, it is as consistent to say that there is a number > between all of .9, .99, .999, ... and 1 as it is to say that there is > a number greater than all of 1, 2, 3, ... . And just as we go on to > ?d more and more ordinals piled atop one another, we can go on to > ?d more and more in?itesimals squeezed beneath each other. > [...] > But so great is the average persons fear of in?ity that to this day > calculus all over the world is being taught as a study of *limit > processes* instead of what it really is: *in?itesimal analysis*. > As someone who has spent a good portion of his adult life teaching > calculus courses for a living, I can tell you how weary one gets of > trying to explain the complex and ?dling theory of limits to wave > after wave of uncomprehending freshman. > ?How pleasant it would be to let pass away some of the verbiage I > learnt at school--learnt because teachers must live, I suppose. The > apeing and prolonged caw called grammar, the cackling of the human hen > over the egg of language--I should like to unlearn grammar.' > But there is hope for a brighter future. Robinsons investigations of > the hyperreal numbers have put in?itesimals on a logically > unimpeachable basis, and here and there calculus texts based on > in?itesimals have appeared [*]. > [*: - Keisler, H. J. (1976). /Elementary Calculus/. Boston: Prindle, > Weber & Schmidt. > - Henle & Kleinberg (1978). /In?itesimal Calculus/. Cambridge, > Mass.: MIT Press.] > [Rer, Rudy (1995). /In?ity and the Mind/. Princeton, NJ: > Princeton University Press. (pp. 79/80 + 87)] > PH === In our ordinary real number system, we say that the number K with decimal expansion .99999... is the samas 1. An informal argument for this is sketched below: 10K = 9.999... - K = .9999... ____________ 9K = 9 K = 1 But maybe this argument is misleading. What if there is some number, call it 1 - 1/omega, that is greater than any ?ite string .9...9 of nines, yet less than 1? If K were actually equal to 1 - 1/omega, the informal argument used in the last paragraph would not work, for this argument overlooks the fact that the difference between 10K and 10 is ten times as great as the difference between K and 1. There is a residual in?itesimal quantity below that does not get canceled out: 10K = 10 - 10/omega - K = 1 - 1/omega _________________ 9K = 9 - 9/omega K = 1 - 1/omega Intuitively, nothing could be more natural than to go ahead and talk about 1/omega, 1/Aleph-1, and so on. Just as we move from the natural numbers to the fractions and then on to the reals, should we not be able to move from the whole ordinal numbers to some richer number ?ld? Curiously, Cantor himself was very much opposed to this step. When a fellow mathematician attempted to use Cantors trans?ite numbers to develop a theory of in?itely small quantities, Cantor accused him of trying to ?infect mathematics with the Cholera-Bacillus of in?itesimals'. Cantor even constructed a proof that no number can be in?itesimal. This proof, however, is just as circular and worthless as ?itist attempts to prove that no number can be in?ite. In both cases, the desired conclusion is smuggled in as part of the de?ition of ?number'. Why was Cantor so vehemently opposed to in?itesimals? In his valuable essay, ?The Metaphysics of the Calculus', Abraham Robinson suggests that Cantor already had enough problems trying to defend trans?ite numbers. It seems likely that, consciously or otherwise, Cantor deemed it politically wise to go along with othodox mathematicians on the question of in?itesimals. Cantors stance might be compared to that of a pro-marijuana Congressional candidate who advocates harsh penalties for the sale or use of heroin. Yet, as we shall see, there is almost as much justi?ation for in?itesimals as there is for Cantors trans?ite ordinals. Formally speaking, it is as consistent to say that there is a number between all of .9, .99, .999, ... and 1 as it is to say that there is a number greater than all of 1, 2, 3, ... . And just as we go on to ?d more and more ordinals piled atop one another, we can go on to ?d more and more in?itesimals squeezed beneath each other. [...] But so great is the average persons fear of in?ity that to this day calculus all over the world is being taught as a study of *limit processes* instead of what it really is: *in?itesimal analysis*. As someone who has spent a good portion of his adult life teaching calculus courses for a living, I can tell you how weary one gets of trying to explain the complex and ?dling theory of limits to wave after wave of uncomprehending freshman. ?How pleasant it would be to let pass away some of the verbiage I learnt at school--learnt because teachers must live, I suppose. The apeing and prolonged caw called grammar, the cackling of the human hen over the egg of language--I should like to unlearn grammar.' But there is hope for a brighter future. Robinsons investigations of the hyperreal numbers have put in?itesimals on a logically unimpeachable basis, and here and there calculus texts based on in?itesimals have appeared [*]. [*: - Keisler, H. J. (1976). /Elementary Calculus/. Boston: Prindle, Weber & Schmidt. - Henle & Kleinberg (1978). /In?itesimal Calculus/. Cambridge, Mass.: MIT Press.] [Rer, Rudy (1995). /In?ity and the Mind/. Princeton, NJ: Princeton University Press. (pp. 79/80 + 87)] PH === > >Cutting Edge New Physics Ideas >4. Put a chunk of dark energy near a chunk of dark matter and you >basically have a weightless warp drive. This means you feel weightless >and the Universe passes by you seemingly faster than the speed of light. > You can time travel to your past and to your future and beyond under >certain conditions. >Oh yeah, that makes a lot of sense. LMAO. > [EL] > Jack Sarfatti is not insane at all. > He is the ultimate of the 20th Century knowledgeable Physicist and > what makes you laugh your ass out is not Jack's insanity, but the > insanity of mainstream physics, which Jack have mastered to a level of > divinity. So far, everything I've seen him write is mindless drivel. I think you're easily impressed. -E as > accepted widely by the physics community including the darkness of > energy and matter along with holes that have a black colour and bangs > that are big when size did not even make any sense. Not to mention > time that became a street in which Minkowski played hide and seek with > Einstein going to and fro while the twin was getting younger. > Wait until Jack tells you about the quarks with its pink colour and > vanilla ? not only going up and down but dancing jerk and > psychedelic while occasionally doing the samba and the rumba. It is > all in the bubble chamber with empirical evidence and pictures of > sophisticated kids interpreting the maps made by chicken nails feeding > in a barn on special plastic that records the trails scratching the > surfaces. > We are living in a fabulous zoo so come and watch the humans. > Jack is an idol that represents this Century. > He is a physics showman. > Refuting him is futile and I suggest that you go back to your bakery > and bake some more bread before your kids get hungry. Leave the > dreamers dream but give them no bread and they shall be excluded > evolutionary wise. > EL === Portfolio of PAF as of 5JAN03: BCE 550 22.85 $12,567.50 BMY 50 29.26 $1,463.00 MRK 100 48. $4,800.00 Q 27,200 4.52 $122,944.00 SBC 15,000 27.07 $406,050.00 realestate land 3APR03 of 3 lots $19,000. science-art of pictures,porcelain etc starting JAN03 for $12,160. realestate land 30JUL03 another lot $11,500. Today I sold 15,000 shares of Qwest at 4.44 to obtain a pro? and with its proceeds bought 2,250 more shares of SBC at 26.99 and 100 Merck at 47.90. What I am doing is re-arranging the portfolio so that it follows the OS of StockMarket faithfully and no secondary strategies or competing strategies. This year I hope to follow the OS with total committment to the Crossover technique. Archimedes Plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: george_cox@btinternet.com X-Punge: Micro$oft === In , on 01/04/2004 at 03:32 PM, Estela Beslerzewski said: >In other word's, you can't work with real numbers, if your goal is to >?d something that is infact in?ite. What do you mean by in?ite? You can work with the common two point compacti?ation of or or the less common one point compacti?ation, but you have to give up some rules of arithmetic to do so. >By in?ite, there is no end. That depends on what you mean by in?ite and what you mean by end. -- Shmuel (Seymour J.) Metz, SysProg and JOAT not reply to spamtrap@library.lspace.org === >By in?ite, there is no end. > That depends on what you mean by in?ite and what you mean by end. Quite right. I've heard lots of statements by people that Cantor somehow disproved the usual medieval (or Aristotelian) notion of the in?ite; hardly true. I'm not sure whether Cantor made such a claim or not. What Cantor (and others today) mean by the in?ite is not what the ancient or medieval meaning was. The fact that trans?ite numbers can in fact be limited --circumscribed, de?ed, in a word, limited, is the very proof that they are not in?ite in the old sense. In fact, in the ancient terminology, a trans?ite number is one more kind of ?ite (that is, not in?ite) number. Which is not to say that we moderns don't have an advantage. In the old terminology, Cantor proved there are more (?ite) numbers than just the integers. Alternatively, that the old category of ?ite had to come apart into two: what we now call ?ite, and trans?ite. The ancients and medievals had no conception of trans?ite numbers, but they had a quite well-considered conception of what they meant by in?ite, and aleph-0 ain't it. So the presentation in which the ancients medievals had confused notions of the in?ite, which Cantor and others straightened out, is really rather backwards. If anything, later developments in set theory vindicated the convictions of Aristotle and others that a completed in?ite cannot exist as at once. Once we understand that in?ite means without any limit , that is. Nothing is going to cause mathematicians to stop saying in?ite when they should say trans?ite . But you'll see lots of medievalists who know math who say trans?ite all the time, because they understand more about the old concept of in?ite than most mathematicians who borrowed the word for something quite different. Thomas X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: george_cox@btinternet.com X-Punge: Micro$oft === In , on 01/04/2004 at 04:15 AM, Estela Beslerzewski said: >Wouldn't determining a sum for a in?ite series, be a paradox >itself? No, because the term is an abbreviation for the limit of the sequence of partial sums. -- Shmuel (Seymour J.) Metz, SysProg and JOAT not reply to spamtrap@library.lspace.org === > Wouldn't determining a sum for a in?ite series, be a paradox itself? > Or > A sum of in?ite series, would then classify an ?in?ite series', in > fact a > ?ite series, since a sum can be made. >Not if the real numbers behave as advertized. The Cauchy construction of >the real numbers guarantees that for every Cauchy sequence there is a >real number. If the partial sums of an in?ite sum form a Cauchy >sequence, q.v. , there MUST be a unique real number represented by that >sequence. > In other word's, you can't work with real numbers, if your goal is to ?d > something that is infact in?ite. > By in?ite, there is no end. Quite the reverse. If you want to work with the reals, you have to have a basic set that is already uncountably in?ite., and is the set of real numbers is based on an in?ite set of in?ite sets (Dedekind cuts or equivalence classes of Cauchy sequences). === I want to try out Mathematica, but until I know if it's what I really need, I don't want to spring for $1000+ for the latest release. I went to Ebay.com, expecting to ?d some copies of Mathematica 3 or 4 for sale, cheap (since they are up to version 5, now right), and was surprised not to ?d any for sale. Does anyone know why older versions would not be for sale? Would it have something to do with the registration process that I know you have to use? real cheap? (Like $25 or something in that neighborhood?) Steve O.