Path: csiph.com!v102.xanadu-bbs.net!xanadu-bbs.net!feeder.erje.net!eu.feeder.erje.net!feeds.phibee-telecom.net!newsfeed.xs4all.nl!newsfeed3a.news.xs4all.nl!xs4all!post.news.xs4all.nl!not-for-mail Return-Path: X-Original-To: python-list@python.org Delivered-To: python-list@mail.python.org X-Spam-Status: OK 0.086 X-Spam-Evidence: '*H*': 0.83; '*S*': 0.00; 'argument': 0.05; 'mathematics': 0.05; 'messing': 0.09; 'subject:set': 0.09; 'cc:addr:python-list': 0.11; 'believes': 0.16; 'cc:name:python list': 0.16; 'dig': 0.16; 'finite': 0.16; 'happily': 0.16; 'question)': 0.16; 'really?': 0.16; 'roy': 0.16; 'to:addr:pearwood.info': 0.16; "to:name:steven d'aprano": 0.16; 'wrote:': 0.18; 'normally': 0.19; '(the': 0.22; 'example': 0.22; 'cc:addr:python.org': 0.22; '(or': 0.24; 'cc:2**0': 0.24; 'possibly': 0.26; 'header:In-Reply-To:1': 0.27; 'tried': 0.27; 'point': 0.28; 'absolute': 0.30; 'message-id:@mail.gmail.com': 0.30; "i'm": 0.30; 'that.': 0.31; 'url:wiki': 0.31; 'usually': 0.31; 'convince': 0.31; "d'aprano": 0.31; 'steven': 0.31; 'subject:numbers': 0.31; 'subject:size': 0.31; 'url:wikipedia': 0.31; 'probably': 0.32; 'another': 0.32; 'quite': 0.32; 'up.': 0.33; 'not.': 0.33; 'sense': 0.34; 'subject:the': 0.34; 'subject: (': 0.35; 'subject:with': 0.35; 'but': 0.35; 'received:google.com': 0.35; 'add': 0.35; 'there': 0.35; "didn't": 0.36; 'url:org': 0.36; 'application': 0.37; 'clear': 0.37; 'being': 0.38; 'stopped': 0.38; 'fact': 0.38; 'rather': 0.38; 'that,': 0.38; 'little': 0.38; 'obtain': 0.39; 'sure': 0.39; 'how': 0.40; 'even': 0.60; 'most': 0.60; 'tell': 0.60; 'march': 0.61; 'numbers': 0.61; 'entire': 0.61; 'kind': 0.63; 'sum': 0.64; 'total': 0.65; 'series': 0.66; 'between': 0.67; 'mar': 0.68; 'natural': 0.68; 'answer.': 0.68; 'attention.': 0.68; 'invalid': 0.68; 'smith': 0.68; 'results': 0.69; 'therefore': 0.72; 'apart': 0.72; 'attention': 0.75; 'divergent': 0.84; 'existence.': 0.84; 'oscar': 0.84; 'absolutely': 0.87 DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=20120113; h=mime-version:in-reply-to:references:from:date:message-id:subject:to :cc:content-type; bh=mc/N+4+4mJxr9OvIH708WnFRHgUOBrhRGKewmHioJU0=; b=j1NFT2rs1r7ezwDVB70Fno+1Fap3TnTqQ5S5da6gJTvnRzJ4ucUnTj7WCD50N1nxkN ctEan6QTcdkN2gbUcIDhz7Z4VFUPhQyezkhEgDdlGQa/c8WXS+tMbdL3KNWH/cj5WdIb m3qhvR1dmkeCBtqPliNOBelfQRJ1ce/8FWP6FP6DpP6AwA5hb9a2DNts31ucoazVe5Um tbPBFgFjCod5olwRDX8PU4lYUf9tVo9IVLD7lPMdRQkRaO34K6Gz1kSRxxeM19f0pK4P TjK+PzFXp2BnqnBlnS6vgEaMQ1QaHdd7EFH2FDTTdoCFoun/wOSp6p6GAOOd1wYkLOTp mI7g== X-Received: by 10.58.228.35 with SMTP id sf3mr117499vec.7.1394022117284; Wed, 05 Mar 2014 04:21:57 -0800 (PST) MIME-Version: 1.0 In-Reply-To: <5316d7a0$0$2923$c3e8da3$76491128@news.astraweb.com> References: <8e4c1ab1-e65d-483f-ad9d-6933ae2052c3@googlegroups.com> <53153e66$0$24931$e4fe514c@dreader36.news.xs4all.nl> <531688b1$0$25064$e4fe514c@dreader37.news.xs4all.nl> <53169cd9$0$2923$c3e8da3$76491128@news.astraweb.com> <5316d7a0$0$2923$c3e8da3$76491128@news.astraweb.com> From: Oscar Benjamin Date: Wed, 5 Mar 2014 12:21:37 +0000 Subject: Re: Working with the set of real numbers (was: Finding size of Variable) To: "Steven D'Aprano" Content-Type: text/plain; charset=ISO-8859-1 Cc: Python List X-BeenThere: python-list@python.org X-Mailman-Version: 2.1.15 Precedence: list List-Id: General discussion list for the Python programming language List-Unsubscribe: , List-Archive: List-Post: List-Help: List-Subscribe: , Newsgroups: comp.lang.python Message-ID: Lines: 52 NNTP-Posting-Host: 2001:888:2000:d::a6 X-Trace: 1394022463 news.xs4all.nl 2878 [2001:888:2000:d::a6]:48445 X-Complaints-To: abuse@xs4all.nl Xref: csiph.com comp.lang.python:67839 On 5 March 2014 07:52, Steven D'Aprano wrote: > On Tue, 04 Mar 2014 23:25:37 -0500, Roy Smith wrote: > >> I stopped paying attention to mathematicians when they tried to convince >> me that the sum of all natural numbers is -1/12. > > I'm pretty sure they did not. Possibly a physicist may have tried to tell > you that, but most mathematicians consider physicists to be lousy > mathematicians, and the mere fact that they're results seem to actually > work in practice is an embarrassment for the entire universe. A > mathematician would probably have said that the sum of all natural > numbers is divergent and therefore there is no finite answer. Why the dig at physicists? I think most physicists would be able to tell you that the sum of all natural numbers is not -1/12. In fact most people with very little background in mathematics can tell you that. The argument that the sum of all natural numbers comes to -1/12 is just some kind of hoax. I don't think *anyone* seriously believes it. > Well, that is, apart from mathematicians like Euler and Ramanujan. When > people like them tell you something, you better pay attention. Really? Euler didn't even know about absolutely convergent series (the point in question) and would quite happily combine infinite series to obtain a formula. > Normally mathematicians will tell you that divergent series don't have a > total. That's because often the total you get can vary depending on how > you add them up. The classic example is summing the infinite series: > > 1 - 1 + 1 - 1 + 1 - ... There is a distinction between absolute convergence and convergence. Rearranging the order of the terms in the above infinite sum is invalid because the series is not absolutely convergent. For this particular series there is no sense in which its sum converges on an answer but there are other series that cannot be rearranged while still being convergent: http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#Alternating_harmonic_series Personally I think it's reasonable to just say that the sum of the natural numbers is infinite rather than messing around with terms like undefined, divergent, or existence. There is a clear difference between a series (or any limit) that fails to converge asymptotically and another that just goes to +-infinity. The difference is usually also relevant to any practical application of this kind of maths. Oscar