1 + 2 + 3 + 4 + . . . = ?<\/title>\n<meta name=\"description\" content=\"So my kids have been watching "Numberphile" videos on youtube lately. Like this one: https:\/\/www.youtube.com\/watch?v=w-I6XTVZXww and this one:\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.patheos.com\/blogs\/janetheactuary\/2015\/05\/1-2-3-4.html\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"1 + 2 + 3 + 4 + . . . = ?\" \/>\n<meta property=\"og:description\" content=\"So my kids have been watching "Numberphile" videos on youtube lately. 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{"id":2088,"date":"2015-05-30T08:46:04","date_gmt":"2015-05-30T14:46:04","guid":{"rendered":"http:\/\/admin.patheos.com\/blogs\/janetheactuary\/?p=2088"},"modified":"2015-05-30T08:46:04","modified_gmt":"2015-05-30T14:46:04","slug":"1-2-3-4","status":"publish","type":"post","link":"https:\/\/www.patheos.com\/blogs\/janetheactuary\/2015\/05\/1-2-3-4.html","title":{"rendered":"1 + 2 + 3 + 4 + . . . = ?"},"content":{"rendered":"\n<\/head>

So my kids have been watching \u201cNumberphile\u201d videos on youtube lately.<\/p>\n

Like this one:<\/p>\n

https:\/\/www.youtube.com\/watch?v=w-I6XTVZXww<\/a><\/p>\n

and this one:<\/p>\n

https:\/\/www.youtube.com\/watch?v=0Oazb7IWzbA<\/a><\/p>\n

in which we learned that 1 + 2 + 3 + 4 + . . . , that is, the sum of all natural numbers to infinity, equals \u2013 1\/12.<\/p>\n

Yeah, I know, but watch the video \u2014 maybe a couple times. \u00a0There\u2019s a step that looks like a \u201ctrick\u201d partway along the way, like cheating, but the physicist explaining this says that this result, as \u201ctricky\u201d as it seems (and there are other mathematical proofs that don\u2019t employ this \u201ctrick\u201d) is used in such fields as string theory, and, in that sense, has been validated as being \u201ctrue.\u201d<\/p>\n

The second video consists of another mathematician providing more context: \u00a0to the objection that there is no way for an infinite, divergent series to have a sum, let alone one as crazy as \u2013 1\/12, he says that, basically, the rules are simply different once you add in the element of infinity. \u00a0He calls them \u201cregularized sums\u201d, and this page<\/a> (from a google search) explains this a bit more.<\/p>\n

The other interesting thing from the 2nd video is this: \u00a0in response to the objection that 1 + 2 + 3 + . . . \u00a0= \u2013 1\/12 isn\u2019t \u201creal\u201d he provided counter-examples:<\/p>\n

i<\/em>, that is, the imaginary number of \u221a-1, is, by its name, of course, imaginary. \u00a0But mathematicians have developed a whole set of useful calculations around it. \u00a0And likewise, it was one taken as a given that such numbers as the square root of \u221a2 weren\u2019t \u201creal\u201d because they couldn\u2019t be expressed in a finite or fractional way.<\/p>\n

Now, I thought that I would take this idea and use it as a springboard for deep musings on the nature of truth, and the challenges of truths which are counter-intuitive, and the fact that those young people today don\u2019t want to believe in anything that they can\u2019t see right in front of their faces, or at any rate, don\u2019t want to believe in any moral truth that\u2019s not immediately obvious to them. \u00a0But that\u2019s not my skill set. \u00a0(Though I have been meaning to type something up on how young people are not necessarily godless; it\u2019s more that their god is Science, an all-powerful deity that grants their wishes without caring about their behavior.) \u00a0And, at any rate, just because something works well as a metaphor doesn\u2019t mean that it proves anything.<\/p>\n

But it\u2019s still interesting, eh?<\/p>\n<\/body><\/html>\n","protected":false},"excerpt":{"rendered":"