{"id":3213,"date":"2015-09-09T13:38:21","date_gmt":"2015-09-09T17:38:21","guid":{"rendered":"http:\/\/www.goodmath.org\/blog\/?p=3213"},"modified":"2015-09-10T07:49:16","modified_gmt":"2015-09-10T11:49:16","slug":"bad-math-books-and-cantor-cardinality","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2015\/09\/09\/bad-math-books-and-cantor-cardinality\/","title":{"rendered":"Bad Math Books and Cantor Cardinality"},"content":{"rendered":"<p> A bunch of readers sent me a link to a tweet this morning from Professor Jordan Ellenberg:<\/p>\n<blockquote class=\"twitter-tweet\" lang=\"en\">\n<p lang=\"en\" dir=\"ltr\">If you&#39;re writing a math textbook and you&#39;re not sure you understand a piece of math, just don&#39;t include it. <a href=\"http:\/\/t.co\/nVe6oLvLhg\">http:\/\/t.co\/nVe6oLvLhg<\/a><\/p>\n<p>&mdash; Jordan Ellenberg (@JSEllenberg) <a href=\"https:\/\/twitter.com\/JSEllenberg\/status\/641427832200142848\">September 9, 2015<\/a><\/p><\/blockquote>\n<p><script async src=\"\/\/platform.twitter.com\/widgets.js\" charset=\"utf-8\"><\/script><\/p>\n<p> The tweet links to the following image:<\/p>\n<p><img data-recalc-dims=\"1\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/i.imgur.com\/tDSX24E.jpg?w=625\"\/><\/p>\n<p> (And yes, this is real. You can see it in context <a href=\"http:\/\/nseuntj.weebly.com\/uploads\/1\/8\/2\/0\/18201983\/2.1relations_and_functions.pdf\">here<\/a>.)<\/p>\n<p> This is absolutely infuriating.<\/p>\n<p> This is a photo of a problem assignment in a <em>math textbook<\/em> published by an imprint of McGraw-Hill. And it&#8217;s absolutely, unquestionably, <em>trivially<\/em> wrong. No one who knew anything about math looked at this before it was published.<\/p>\n<p> The basic concept underneath this is fundamental: it&#8217;s the cardinality of sets from Cantor&#8217;s set theory. It&#8217;s an extremely important concept. And it&#8217;s a concept that&#8217;s at the root of a huge amount of misunderstandings, confusion, and frustration among math students. <\/p>\n<p> Cardinality, and the notion of cardinality relations between infinite sets, are difficult concepts, and they lead to some very un-intuitive results. Infinity isn&#8217;t one thing: there are different <em>sizes<\/em> of infinities. That&#8217;s a rough concept to grasp!<\/p>\n<p> Here on this blog, I&#8217;ve spent more time dealing with people who believe that it must be wrong &#8211; a subject that I call Cantor crackpottery &#8211; than with any other bad math topic. This error teaches students something deeply wrong, and it encourages Cantor crackpottery!<\/p>\n<p> Let&#8217;s review.<\/p>\n<p> Cantor said that two collections of things are the same size if it&#8217;s possible to create a one-to-one mapping between the two. Imagine you&#8217;ve got a set of 3 apples and a set of 3 oranges. They&#8217;re the same size. We know that because they both have 3 elements; but we can also show it by setting aside pairs of one apple and one orange &#8211; you&#8217;ll get three pairs. <\/p>\n<p> The same idea applies when you look at infinitely large sets. The set of positive integers and the set of negative integers are the same size. They&#8217;re both infinite &#8211; but we can show how you can create a one-to-one relation between them: you can take any positive integer <img src='http:\/\/l.wordpress.com\/latex.php?latex=i&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='i' style='vertical-align:1%' class='tex' alt='i' \/>, and map it to exactly one negative integer, <img src='http:\/\/l.wordpress.com\/latex.php?latex=0%20-%20i&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='0 - i' style='vertical-align:1%' class='tex' alt='0 - i' \/>. <\/p>\n<p> That leads to some unintuitive results. For example, the set of all natural numbers and the set of all <em>even<\/em> natural numbers are the same size. That seems crazy, because the set of all even natural numbers is a <em>strict subset<\/em> of the set of natural numbers: how can they be the same size?<\/p>\n<p> But they are. We can map each natural number <img src='http:\/\/l.wordpress.com\/latex.php?latex=i&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='i' style='vertical-align:1%' class='tex' alt='i' \/> to exactly one even natural number <img src='http:\/\/l.wordpress.com\/latex.php?latex=2i&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='2i' style='vertical-align:1%' class='tex' alt='2i' \/>. That&#8217;s a perfect one-to-one map between natural numbers and even natural numbers.<\/p>\n<p> Where it gets uncomfortable for a lot of people is when we start thinking about real numbers. The set of real numbers is infinite. Even the set of real numbers between 0 and 1 is infinite! But it&#8217;s also <em>larger<\/em> than the set of natural numbers, which is also infinite. How can that be?<\/p>\n<p> The answer is that Cantor showed that for any possible one-to-one mapping between the natural numbers and the real numbers between 0 and 1, there&#8217;s at least one real number that the mapping <em>omitted<\/em>. No matter how you do it, all of the natural numbers are mapped to one value in the reals, but there&#8217;s at least one real number which is <em>not<\/em> in the mapping!<\/p>\n<p> In Cantor set theory, that means that the size of the set of real numbers between 0 and 1 is strictly larger than the set of all natural numbers. There&#8217;s an infinity bigger than infinity.<\/p>\n<p> I <em>think<\/em> that this is what the math book in question meant to say: that there&#8217;s no possible mapping between the natural numbers and the real numbers. But it&#8217;s not what they did say: what they said is that there&#8217;s no possible map between the integers and the fractions. And <em>that<\/em> is not true.<\/p>\n<p> Here&#8217;s how you generate the mapping between the integers and the rational numbers (fractions) between 0 and 1, written as a pseudo-Python program:<\/p>\n<pre>\n i = 0\n for denom in Natural:\n   for num in 1 .. denom:\n      if num is relatively prime with denom:\n         print(\"%d => %d\/%d\" % (i, num, denom))\n         i += 1\n<\/pre>\n<p> It produces a mapping (0 => 0, 1 => 1, 2 => 1\/2, 3 => 1\/3, 4 => 2\/3, 5 => 1\/4, 6 => 3\/4, &#8230;). It&#8217;ll never finish running &#8211; but you can easily show that for any possible fraction, there&#8217;ll be exactly one integer that maps to it. <\/p>\n<p> That means that the set of all rational numbers between 0 and 1 is <em>the same size<\/em> as the set of all natural numbers. There&#8217;s a similar way of producing a mapping between the set of <em>all<\/em> fractions and the set of natural numbers &#8211; so the set of all fractions is the same size as the set of natural numbers. But both are <em>smaller than<\/em> the set of all <em>real<\/em> numbers, because there are many, many real numbers that cannot be written as fractions. (For example, <img src='http:\/\/l.wordpress.com\/latex.php?latex=%5Cpi&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='\\pi' style='vertical-align:1%' class='tex' alt='\\pi' \/>. Or the square root of 2. Or <img src='http:\/\/l.wordpress.com\/latex.php?latex=e&#038;bg=FFFFFF&#038;fg=000000&#038;s=0' title='e' style='vertical-align:1%' class='tex' alt='e' \/>. )<\/p>\n<p> This is terrible on multiple levels.<\/p>\n<ol>\n<li> It&#8217;s a <em>math textbook<\/em> written and reviewed by people who don&#8217;t understand the basic math that they&#8217;re writing about.<\/li>\n<li> It&#8217;s teaching children something incorrect about something that&#8217;s already likely to confuse them.<\/li>\n<li> It&#8217;s teaching something incorrect about a topic that doesn&#8217;t need to be covered <em>at all<\/em> in the textbook. This is an algebra-2 textbook. You don&#8217;t need to cover Cantor&#8217;s infinite cardinalities in Algebra-2. It&#8217;s not <em>wrong<\/em> to cover it &#8211; but it&#8217;s not necessary.    If the authors didn&#8217;t understand cardinality, they could have just left it out.<\/li>\n<li> It&#8217;s <em>obviously<\/em> wrong. Plenty of bright students are going to come up with the the mapping between the fractions and the natural numbers. They&#8217;re going to come away believing that they&#8217;ve disproved Cantor.<\/li>\n<\/ol>\n<p> I&#8217;m sure some people will argue with that last point. My evidence in support of it? <em>I<\/em> came up with a proof of that in high school. Fortunately, my math teacher was able to explain why it was wrong. (Thanks Mrs. Stevens!) Since I write this blog, people assume I&#8217;m a mathematician. I&#8217;m not. I&#8217;m just an engineer who really loves math. I was a good math student, but far from a great one. I&#8217;d guess that every medium-sized high school has at least one math student every year who&#8217;s better than I was.<\/p>\n<p> The proof I came up with is absolutely trivial, and I&#8217;d expect tons of bright math-geek kids to come up with something like it. Here goes:<\/p>\n<ol>\n<li> The set of fractions is a strict subset of the set of ordered pairs of natural numbers.<\/li>\n<li> So: if there&#8217;s a one-to-one mapping between the set of ordered pairs and the naturals, then there must be a one-to-one mapping between the fractions and the naturals.<\/li>\n<li> On a two-d grid, put the natural numbers across, and then down.<\/li>\n<li> Zigzag diagonally through the grid, forming pairs of the horizontal position and the vertical position: (0,0), (1, 0), (0, 1), (2, 0), (1, 1), (0, 2), (3, 0), (2, 1), (1, 2), (0, 3). <\/li>\n<li> This will produce every possible ordered pair of natural numbers. For each number in the list, produce a mapping between the position in the list, and the pair. So (0, 0) is 0,  (2, 0) is 3, etc.<\/li>\n<\/ol>\n<p> As a proof, it&#8217;s sloppy &#8211; but it&#8217;s correct. And plenty of high school students will come up with something like it. How many of them will walk away believing that they just disproved Cantor?<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A bunch of readers sent me a link to a tweet this morning from Professor Jordan Ellenberg: If you&#39;re writing a math textbook and you&#39;re not sure you understand a piece of math, just don&#39;t include it. http:\/\/t.co\/nVe6oLvLhg &mdash; Jordan Ellenberg (@JSEllenberg) September 9, 2015 The tweet links to the following image: (And yes, this [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[70,302,11],"tags":[],"class_list":["post-3213","post","type-post","status-publish","format-standard","hentry","category-bad-algebra","category-bad-numbers","category-cantor-crankery"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-PP","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/3213","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=3213"}],"version-history":[{"count":2,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/3213\/revisions"}],"predecessor-version":[{"id":3215,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/3213\/revisions\/3215"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=3213"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=3213"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=3213"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}