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> 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 > 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 -- 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
