tag:blogger.com,1999:blog-5964889903484807623.post6547539816066010853..comments2021-05-11T07:20:00.591-07:00Comments on Questions?: How Close is Close Enough?David Coxhttp://www.blogger.com/profile/06277427735527075341noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-5964889903484807623.post-26295092141514570152009-08-31T02:50:19.000-07:002009-08-31T02:50:19.000-07:00In grade school I wondered how something in the re...In grade school I wondered how something in the real world could be exactly 1/3 of an inch. I was hung up on the infinite 3s and how no ruler could measure it. Many years later I realized the answer had to do with our representation of numbers -- base 10 -- and not something inherent in numbers themselves. After all, 1/3 in base 3 is terminating - it's 0.1!Rick Reganhttp://www.exploringbinary.com/noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-67460467224197302402009-09-01T03:55:55.000-07:002009-09-01T03:55:55.000-07:00Great lesson! What the world needs now is more mat...Great lesson! What the world needs now is more math lessons like this one.<br><br>I always loved seeing the light of curiosity and thought appear in kids eyes when presented with a seeming paradox. Good on ya for bringing it out in your kids eyes.Darren Kuropatwahttp://adifference.blogspot.comnoreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-75067939504771003422009-09-01T05:58:59.000-07:002009-09-01T05:58:59.000-07:00These were the types of questions that kept me com...These were the types of questions that kept me coming to class while in college. It still bugs me that I never had any conversation like this in middle school or high school. How many fires could get sparked if we just allow for 15 minutes of a rabbit trail every now and then? <br><br>Thanks for the encouragement, Darren. BTW, I have taken the advice that many of you gave regarding my 8th grade class and have started taking a more analytical approach to the Geometry. The Exeter problems as well as some adaptations that Alison sent have been a great resource.David Coxhttp://coxmath.pbwiki.comnoreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-16825259926079493122009-09-01T08:55:34.000-07:002009-09-01T08:55:34.000-07:00This conversation comes up in every class I teach....This conversation comes up in every class I teach. The idea of the infintesimal just runs so deep through mathematics. I don't think I've ever been able to convince anyone that 0.999... = 1 though.Alison Blankhttp://axiomstoteachby.blogspot.comnoreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-31997783118883508192009-09-02T12:05:45.000-07:002009-09-02T12:05:45.000-07:00@David That's great. When I started using a si...@David That's great. When I started using a similar approach im my classes I saw more lightbulbs going on over kids heads that I had seen before. It also made teaching more interesting for me; I started to grok the connections between the seemingly disparate branches of mathematics.<br><br>@Alison Even when you used David's approach here? <br>Find the decimal representation of 4/9. <br>Find the decimal representation of 5/9.<br>Add the decimals, what do you get?<br>Add the fractions what do you get?<br>Same thing, no?<br><br>It really helps underscore that infinity is not a number; it's an idea. ;-)Darren Kuropatwahttp://adifference.blogspot.comnoreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-48079948835522775622009-09-04T04:38:53.000-07:002009-09-04T04:38:53.000-07:00I had an interesting conversation with a colleague...I had an interesting conversation with a colleague the other day about this. He approached the question from a philosophical standpoint. Are they actually =? Philosophically, I suppose we could say no. The numbers become ultimately equal (as per Newtons Lemma 1). He doesn't say that they ultimately become equal. Leave it to a philosopher, huh?David Coxhttp://coxmath.pbwiki.comnoreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-44867946690212964012009-09-05T06:20:02.000-07:002009-09-05T06:20:02.000-07:00Actually, I would say, yes, they are actually equa...Actually, I would say, yes, they are actually equal. 0.9999... out to infinity IS 1. It isn't 1 only if you stop somewhere, but if you stop somewhere then 0.9999... isn't an infinite decimal.Darren Kuropatwahttp://adifference.blogspot.comnoreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-63897897977647713492009-09-08T11:46:32.000-07:002009-09-08T11:46:32.000-07:00I agree that they're equal. It's the infi...I agree that they're equal. It's the infinite decimal that is the sticking point. His point of emphasis is "when will it reach 1 if it's infinite?" and my point of emphasis is "then give me a number between .999... and 1.". Still a fun conversation to have.David Coxnoreply@blogger.com