OKay, I think I get it. Thanks :) On 14/10/10 00:56, David Sletten wrote: > Here's a slightly more informal argument. Suppose you challenge me that > 1 is not equal to 0.9999... What you are saying is that 1 - 0.9999... is > not equal to 0, i.e., the difference is more than 0. But for any > positive value arbitrarily close to 0 I can show that 0.999... is closer > to 1 than that. If you were to say that the difference is 0.1, I could > show that 0.999... > 0.9 so the difference is smaller. For every 0 you > added to your challenge: 0.1, 0.01, 0.001 I could provide a > counterexample with another 9: 0.9, 0.99, 0.999, ... In other words, > there is no positive number that satisfies your claim, so equality must > hold. > > Have all good days, > David Sletten > > On Oct 13, 2010, at 6:36 PM, Matt Fowles wrote: > >> Felix~ >> >> You are correct that the sequence of numbers >> >> 0.9 >> 0.99 >> 0.999 >> ... >> >> asymptotically approaches 1; however, the number 0.9999... (with an >> infinite number of 9s) is equal to 1. The formal proof of this is >> fairly tricky as the definition of the real number is usually done as >> an equivalence class of Cauchy sequences; a simplified version of the >> proof can be thought of as follows: >> >> For any two real numbers /a/ and /b/ there exists an infinite number >> of real numbers /c /such that /a < c < b/. However, there do not >> exist any numbers between 0.99999... and 1, thus they must be same number. >> >> As it turns out, it took mathematicians a long time to nail down >> formally exactly what we naively think of as "numbers". >> >> Matt >> >> On Wed, Oct 13, 2010 at 6:27 PM, Felix H. Dahlke <f...@ubercode.de >> <mailto:f...@ubercode.de>> wrote: >> >> On 13/10/10 22:28, David Sletten wrote: >> > >> > On Oct 12, 2010, at 5:44 PM, Brian Hurt wrote: >> > >> >> For example, in base 10, 1/3 * 3 = 0.99999... >> > >> > It may seem counterintuitive, but that statement is perfectly true. >> > 1 = 0.9999... >> > >> > That's a good test of how well you understand infinity. >> >> I'm clearly not a mathematician, but doesn't 0.99999... asymptotically >> approach 1, i.e. never reaching it? How is that the same as 1? >> >> >> >> -- >> You received this message because you are subscribed to the Google >> Groups "Clojure" group. >> To post to this group, send email to clojure@googlegroups.com >> <mailto:clojure@googlegroups.com> >> Note that posts from new members are moderated - please be patient >> with your first post. >> To unsubscribe from this group, send email to >> clojure+unsubscr...@googlegroups.com >> <mailto:clojure+unsubscr...@googlegroups.com> >> For more options, visit this group at >> http://groups.google.com/group/clojure?hl=en > > > > > -- > You received this message because you are subscribed to the Google > Groups "Clojure" group. > To post to this group, send email to clojure@googlegroups.com > Note that posts from new members are moderated - please be patient with > your first post. > To unsubscribe from this group, send email to > clojure+unsubscr...@googlegroups.com > For more options, visit this group at > http://groups.google.com/group/clojure?hl=en
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