Are natural numbers isomorphic to complex numbers?
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From: Ostap Bender on 5 Jun 2010 02:39 On Jun 4, 11:19 pm, Akira Bergman <akiraberg...(a)gmail.com> wrote: > "Wasn't it decided by almost complete consensus back in the 19th > century that Euler was right and that it is best to set > 0^0 = 1 > ? > Is that what you advocate too? " > > It is self explanatory from x^y = e^(y*log(x)). It has been on the net > for years and still gets good traffic. It was agreed by sci.math crowd > years ago. I don't know how it has been missed by so many math-heads > for so many years. This is what I mean by the pitfalls of formalism > heavy thinking. > > I don't have to advocate it. It advocates itself. So, you agree with what was decided 180 years ago: that it is best to define 0^0 as 1? Then what is your innovative contribution to this question? And what is this thing: > 0^0 = {0,1} ? What kind of a number is {0,1}? It is not an element of R nor of C, is it? What multiplicaton group is it part of? You wrote: > > > I had been rewarded for following my hunch few times and did not want > > > to abandon it because of formalism. Look at the 0^0 discussion I > > > initiated many years ago as a good example. What do you mean by "rewarded"? What reward did you receive and who (other than yourself) bestowed this reward on you? > "Are you self-employed?" > > I don't have to work for a while. There is nothing like freedom from > doing others work. If you don't have a boss, you don't have to worry about anonymity as much.
From: Akira Bergman on 5 Jun 2010 02:51 "What kind of a number is {0,1}? It is not an element of R nor of C, is it? What multiplicaton group is it part of? " It just means 0^0 is double valued. It is equal to either 0 or 1, depending on how you look at it. I do not know the implications of it. I suspect it will shed more light to the universality of duality. It may even shed light to the Higgs field. "What do you mean by "rewarded"?" Recognized.
From: Ostap Bender on 5 Jun 2010 03:00 On Jun 4, 11:51 pm, Akira Bergman <akiraberg...(a)gmail.com> wrote: > "What kind of a number is {0,1}? It is not an element of R nor of C, > is > it? What multiplicaton group is it part of? " > > It just means 0^0 is double valued. It is equal to either 0 or 1, > depending on how you look at it. I do not know the implications of it. > I suspect it will shed more light to the universality of duality. It > may even shed light to the Higgs field. Are you saying that you have discovered something there that the mathematical community didn't know before you? Which part is you "contribution"? That 0^0 can be defined as 1? That 0^0 can be defined as 0? That 0^0 is not uniquely defined? >> What do you mean by "rewarded"? What reward did you receive and who (other than yourself) bestowed this reward on you? > Recognized. Recognized by whom (other than yourself)?
From: Akira Bergman on 5 Jun 2010 03:06 On Jun 5, 5:00 pm, Ostap Bender <ostap_bender_1...(a)hotmail.com> wrote: > On Jun 4, 11:51 pm, Akira Bergman <akiraberg...(a)gmail.com> wrote: > > > "What kind of a number is {0,1}? It is not an element of R nor of C, > > is > > it? What multiplicaton group is it part of? " > > > It just means 0^0 is double valued. It is equal to either 0 or 1, > > depending on how you look at it. I do not know the implications of it. > > I suspect it will shed more light to the universality of duality. It > > may even shed light to the Higgs field. > > Are you saying that you have discovered something there that the > mathematical community didn't know before you? Which part is you > "contribution"? That 0^0 can be defined as 1? That 0^0 can be defined > as 0? That 0^0 is not uniquely defined? That 0^0 is not uniquely defined. Its unique definition to 1 conflicts with the definition of logarithm. > > >> What do you mean by "rewarded"? What reward did you receive and who > > (other than yourself) bestowed this reward on you? > > > Recognized. > > Recognized by whom (other than yourself)? By sci.math and by the continuing traffic on it on the net.
From: Ostap Bender on 5 Jun 2010 03:54
On Jun 5, 12:06 am, Akira Bergman <akiraberg...(a)gmail.com> wrote: > On Jun 5, 5:00 pm, Ostap Bender <ostap_bender_1...(a)hotmail.com> wrote: > > > On Jun 4, 11:51 pm, Akira Bergman <akiraberg...(a)gmail.com> wrote: > > > > "What kind of a number is {0,1}? It is not an element of R nor of C, > > > is > > > it? What multiplicaton group is it part of? " > > > > It just means 0^0 is double valued. It is equal to either 0 or 1, > > > depending on how you look at it. I do not know the implications of it.. > > > I suspect it will shed more light to the universality of duality. It > > > may even shed light to the Higgs field. > > > Are you saying that you have discovered something there that the > > mathematical community didn't know before you? Which part is you > > "contribution"? That 0^0 can be defined as 1? That 0^0 can be defined > > as 0? That 0^0 is not uniquely defined? > > That 0^0 is not uniquely defined. But this has been argued for many centuries, starting with Cauchy: http://www.math.hmc.edu/funfacts/ffiles/10005.3-5.shtml Zero to the Zero Power It is commonly taught that any number to the zero power is 1, and zero to any power is 0. But if that is the case, what is zero to the zero power? Well, it is undefined (since x^y as a function of 2 variables is not continuous at the origin). http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0to0/ Newsgroups: sci.math From: alopez-o(a)neumann.uwaterloo.ca (Alex Lopez-Ortiz) Subject: sci.math FAQ: What is 0^0? Date: Fri, 17 Nov 1995 17:14:58 GMT According to some Calculus textbooks, 0^0 is an ``indeterminate form''. Some people feel that giving a value to a function with an essential discontinuity at a point, such as x^y at (0,0) , is an inelegant patch and should not be done. Most mathematicians agreed that 0^0 = 1 , but Cauchy [5, page 70] had listed 0^0 together with other expressions like 0/0 and oo - oo in a table of undefined forms. http://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_zero_power Zero to the zero power # Regarding 00 as an empty product of zeros assigns it the value 1. # The combinatorial interpretation of 0^0 is the number of empty tuples of elements from the empty set. There is exactly one empty tuple. On the other hand, 0^0 must be handled as an indeterminate form when it is an algebraic expression obtained in the context of determining limits Some argue that the best value for 0^0 depends on context, and hence that defining it once and for all is problematic.[9] According to Benson (1999), "The choice whether to define 0^0 is based on convenience, not on correctness."[10] > Its unique definition to 1 conflicts > with the definition of logarithm. Isn't that what Wikipedia and other sources, listed above, say? > > > It is self explanatory from x^y = e^(y*log(x)) How do you define log(x) when x=0? How do you define y*log(x) when y = 0 and x=0? > > >> What do you mean by "rewarded"? What reward did you receive and who > > (other than yourself) bestowed this reward on you? > > > > Recognized. > > > Recognized by whom (other than yourself)? > > By sci.math and by the continuing traffic on it on the net. Neither "sci.math" nor "traffic on on the net" are human beings. Are there any competent people (or any intelligent human beings at all) who "recognized" your thoughts as novel and important? To make it easier for you to list intelligent people who "recognized" your "achievement", I found a link to your discussion back in 2006: http://groups.google.com/group/sci.math/browse_frm/thread/e072332adf896034/de4be8e11d740e40?q=Akira+Bergman+0^0+sci.math#de4be8e11d740e40 So, who (other than yourself) "recognized" your "discovery"? Give me their names. I noticed no such people. You seem to be "a legend in your own mind". |