{"id":353,"date":"2012-02-25T14:03:30","date_gmt":"2012-02-25T20:03:30","guid":{"rendered":"http:\/\/linesoftangency.wordpress.com\/?p=353"},"modified":"2012-02-25T14:03:30","modified_gmt":"2012-02-25T20:03:30","slug":"two-roads-converged","status":"publish","type":"post","link":"http:\/\/blog.chrislusto.com\/?p=353","title":{"rendered":"Two Roads [Con]verged"},"content":{"rendered":"

Last week we started working with infinite geometric series<\/a>, a topic I personally love.\u00a0 First of all, it's one of the few places in a high school curriculum where deep, genuine philosophical questions bubble all the way up to the surface of a mathematical discussion.\u00a0 Second, it marks the place in my own academic life where I experienced a religious conversion to Orthodox Mathematicism:<\/p>\n

In the beginning there was a single term.\u00a0 And to that term the Teacher did add another of smaller magnitude.\u00a0 Then a third term, smaller still, appeared upon the right hand side of the chalkboard, and it was revealed to me that the terms did decrease exponentially.\u00a0 My heart saw that this shrinking and adding proceedeth forever and ever, terms without end, Amen.\u00a0 And lo, when I beheld the sum, it was finite, and I knew that it was Good.<\/p><\/blockquote>\n

If my introduction to convergent series was a baptism, then using one to demonstrate that .999... = 1<\/a> was my confirmation.\u00a0 Now, having done the same thing with my students, I think it might be even more interesting from this side of the desk.\u00a0 In particular, two of their questions\/comments highlight two very different understandings of infinity and the real numbers.<\/p>\n

First, the ingredients of a metaphor.\u00a0 If you've ever been a runner, this is easy.\u00a0 If not, I'm going to need you to go on a quick jog before you read any farther so you can appreciate the rest of this carefully crafted rhetorical device.\u00a0 I'll wait...<\/p>\n

When you drive the same stretch of road over and over again, you tend to experience it dynamically.\u00a0 You pass a landmark, anticipate a curve, accelerate over a little rise.\u00a0 The road changes in front of your eyes.\u00a0 You see the road as a <\/em>process<\/strong>.\u00a0 But when you run along the same route, it looks completely different.\u00a0 There is just this monolithic expanse of concrete laid out over the landscape.\u00a0 You can creep around and explore its different features, but you experience the road essentially as a static object<\/strong>.\u00a0 In other words, you experience the road as it actually is<\/strong>.\u00a0 Keep this in mind as you read the following two questions from my actual students.<\/p>\n

D: \"But Mr. Lusto, if .999... is exactly 1, then .999... plus .999... should equal exactly 2, but it doesn't.\u00a0 It's 1.999...8.\"<\/h1>\n

What a freaking fantastic argument!\u00a0 Here's a student who has accepted my proof, interpreted it, thought about it critically, and deduced a logical contradiction.\u00a0 My heart swelled a little bit.\u00a0 Unfortunately, the flaw in his reasoning highlights a fundamental misconception.\u00a0 D is viewing .999... like a driver.\u00a0 He sees it as a dynamic process, repeatedly appending a 9 to an ever-expanding sequence of 9s.\u00a0 He might even accept that this can theoretically go on forever, but his point-of-view still gets him into some trouble.\u00a0 When D mentally sums .999... and .999..., he's suggesting that there are two \"last 9s\" that, when added, produce a trailing 8.\u00a0 But of course there are <\/em>no \"last 9s.\"\u00a0 He's implicitly terminated the process prematurely (which is to say, at all).\u00a0 Hence his objection, though thoroughly beautiful, is ultimately illusory.<\/p>\n

J: \"But Mr. Lusto, if .999... equals 1, then doesn't 1.999... equal 2?\u00a0 Then can't we write every number in two different ways?\"<\/h1>\n

This student views .999... like a runner.\u00a0 The reason that .999... and 1 can be meaningfully thought of as equal is because they represent the same static value.\u00a0 They're just two different names for the same object.\u00a0 Here's a student who sees .999... as it actually is.\u00a0 And now, because of that, his concern is genuine.\u00a0 The fact that many real numbers have two decimal representations (one with infinite trailing 0s, one with infinite trailing 9s) is a true mathematical\/philosophical problem.\u00a0 In fact, it's an important result: those sorts of numbers turn out to be dense in the reals (in the topological sense).\u00a0 J may never care about, or even get enough math under his belt to understand,<\/em> that statement,\u00a0 but his view of the nature of infinity is already more nuanced than D's.<\/p>\n

Something to think about next time you're driving.\u00a0 Better yet, next time you're running.<\/p>\n","protected":false},"excerpt":{"rendered":"

Last week we started working with infinite geometric series, a topic I personally love.\u00a0 First of all, it's one of the few places in a high school curriculum where deep, genuine philosophical questions bubble all the way up to the surface of a mathematical discussion.\u00a0 Second, it marks the place in my own academic life […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[21,33,47,49],"class_list":["post-353","post","type-post","status-publish","format-standard","hentry","category-math-teaching","tag-infinity","tag-math-education","tag-real-numbers","tag-series"],"_links":{"self":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/353","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=353"}],"version-history":[{"count":0,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/353\/revisions"}],"wp:attachment":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=353"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=353"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=353"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}