ÿþÿþ<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 TRANSITIONAL//EN"> <html> <head> <title> An infinite nomber of Jewes expelled out of Spaine </title> </head> <link rel="shortcut icon" href="qed1.jpg"> <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> <body> <table width="400" border="0"> <td> <h1><HEADLINE> An infinite nomber of Jewes expelled out of Spaine </HEADLINE></h1> <!-- Insert Article Content --> <!-- Article Details --> <p> by Burkard Polster and Marty Ross </p> <p> The Age, 21 November 2011 </p> <img src="cfb8d304-1250-11e1-9437-005056b06a0e-8123342.jpg" border="0" width="400" height="79"/></div><div id="cmscontentitem_68977" class="content_html"><p><br />That is according to English merchant Robert Thorne, writing in 1527. However, it seems that Robert was exaggerating a trifle. In fact, <a href="http://www.fordham.edu/halsall/jewish/1492-jews-spain1.asp" target="_blank">the Spanish expulsion of 1492</a> affected about 200,000 Jews: a big number, but a fair way short of infinity.&nbsp;</p> <p>Of course, Robert Thorne is simply employing the word &ldquo;infinite&rdquo; as a synonym for &ldquo;bloody huge&rdquo;. Such usage was and still is very common. But what about real, true infinity? How can we make sense of that?</p> <p><a href="http://education.theage.com.au/cmspage.php?intid=147&amp;intversion=95" target="_blank">We wrote about infinity recently</a>. At the time, we declared that our single greatest goal is for students to believe and to understand that the infinite repeating decimal 0.9999&hellip; is equal to 1. And in this context, we really do mean equals, in the exact same sense that 1+1 equals 2. No fudging, no remainders, no almosts: equals means equals.</p> <p>A reader, Andy, responded to our column, offering a simple proof: beginning with the equation 0.3333&hellip; = 1/3 and then multiplying both sides by 3, we find that 0.9999&hellip; = 1.&nbsp;</p> <p>We&rsquo;ll discuss Andy&rsquo;s proof in a moment, but we first want to give our own favourite proof, <a href="http://www.qedcat.com/archive/Chasing.html" target="_blank">an algebraic argument that we&rsquo;ve presented previously.</a></p> <p>Since we don&rsquo;t know what 0.9999&hellip; is, let&rsquo;s call it <em>X</em>. Then, multiplying by 10, we have 9.9999&hellip; = 10<em>X</em>. Now we subtract:</p></div><div id="cmscontentitem_68979" class="cmsimg_center"><img src="320f705a-1254-11e1-9437-005056b06a0e-9343082.jpg" border="0" width="408" height="202"/></div><div style="clear:both;" class="cmsnewline"></div><div id="cmscontentitem_68981" class="content_html"><p>&nbsp;</p> <p>Since 9 = 9<em>X</em>, we can divide both sides by 9 to conclude that <em>X</em> = 1. QED!</p> <p>So now we have two simple proofs that 0.9999&hellip; = 1. But is either proof valid, or is there hidden trickery? Well, a bit of both.</p> <p>First of all, notice that both Andy&rsquo;s argument and ours simply assume that infinite decimals behave like everyday numbers: multiplying an infinite decimal by 10 or by 3, we&rsquo;re just hoping that the arithmetic works in a familiar manner. That&rsquo;s hardly obvious.&nbsp;</p> <p>However, what we really like about our algebraic argument is that the &ldquo;arithmetic still works&rdquo; assumption is the <em>only</em> article of faith. That is, the argument completely demonstrates that if 0.9999&hellip; makes any sense as a number then it must equal 1.</p> <p>Is the same true for Andy&rsquo;s proof? Does it also require just the one article of faith?</p> <p>Recall that Andy began with the equation 0.3333&hellip; = 1/3. Andy didn&rsquo;t suggest how we know this, but another reader, the mysterious S, obliged: long division of 1 by 3 might do the trick. So, we&rsquo;ll give it a shot.</p> <p>Let&rsquo;s divide 3 into 1 over and over, say ten times. We find that</p></div><div id="cmscontentitem_68983" class="cmsimg_center"><img src="627ed824-1255-11e1-9437-005056b06a0e-6794113.jpg" border="0" width="420" height="27"/></div><div id="cmscontentitem_68985" class="content_html"><p><br />We have that annoying remainder (which, to be precise, is 0.0000000001), but it can be divided further. So, we give it a really serious go, and divide 3 into 1a total of 200,000 times. We find that</p></div><div id="cmscontentitem_68987" class="cmsimg_center"><img src="7d210094-1255-11e1-9437-005056b06a0e-6706774.jpg" border="0" width="420" height="52"/></div><div id="cmscontentitem_68989" class="content_html"><p><br />At this stage, we should admit defeat: no matter how many times we divide, there will still be a tiny remainder left over. In reality, we&rsquo;re no closer to obtaining the infinite decimal 0.3333&hellip; than Robert Thorne was to counting an infinite nomber of Jewes.</p> <p>It is for this reason that we prefer our algebraic proof: we may not be totally comfortable with just accepting that infinite decimals work, but our proof doesn&rsquo;t attempt to smuggle infinity in through the back door.</p> <p>So what then is to be done with infinity? Does it just have to be accepted as an article of faith? Pretty much. We can think of finite processes going further and further, but to actually attain that infinite goal, we must first assume the goal is there to be attained.&nbsp;</p> <p>Not that there aren&rsquo;t clearer and more careful assumptions we can make about infinity. Much of the brilliant 20th century work on mathematical foundations was devoted to figuring out the simplest assumptions to make about the infinite, the clearest <em>axioms</em> with which to begin.</p> <p>Then, given these foundational assumptions, one can go on to prove properties of the infinite, which are just as true as 1+1 = 2 is true. And, a major tool to learning and understanding these truths is to capture the infinite as the end result of finite processes. In fact, the long division argument suggested by the mysterious <em>S </em>is a very important step in proving that the arithmetic of infinite decimals works just the way one hopes.</p> <p>So, it turns out that the finite really can help us understand the infinite. But, it remains the case that no one, not even mathematicians, can <em>create</em> the infinite out of the finite: the infinite is still an article of faith. We like to think we&rsquo;re resourceful, but we know our limits.</p></div><div id="cmscontentitem_68991" class="cmsimg_center"><img src="ac8bb0c2-1255-11e1-9437-005056b06a0e-8605789.jpg" border="0" width="182" height="177"/></div><div id="cmscontentitem_68993" class="content_html"><p><strong><br />Puzzle to Ponder:</strong> The great mathematician Gottfried Leibniz pondered the infinite sum 1 &ndash; 1 + 1 &ndash; 1 + 1 &hellip;, and he decided that it was equal to 1/2. Was he correct?</p> <p><em>Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator.&nbsp;</em></p><p><em>Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.</em></p> <p><a href="http://www.qedcat.com/">www.qedcat.com</a></p> <p align="center">Copyright 2004-&#8734; <img src="qed.jpg" width="47" height="46"> All rights reserved. </p> </body> </html>