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</o:shapelayout></xml><![endif]--></head><body bgcolor=white lang=EN-US link=blue vlink=purple><div class=WordSection1><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'>Funny thing happened on the way to the forum: yesterday I was teaching my son about irrational numbers. He learned a couple weeks ago how to convert any fraction to a decimal, so every fraction either terminates or repeats. He entertained himself looking for the ratio which has the longest repeating pattern. <o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'>Yesterday I used his compass to draw a radius and a half circumference, then asked him to guess the ratio of their lengths. First he guessed 3. So I constructed half a hexagon to show him it was more than three. He next guessed 4, so I constructed a 4R path that goes way outside the half circle. So he guessed 3 and a tenth, which I found interesting in itself.<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'>I then went on to explain irrational numbers, but his reaction really surprised me: it really upset that little second grader. His world just got a bit messier. The existence of numbers that cannot be expressed as a ratio bothers the hell out of him. I was vaguely reminded of the ancient Greeks’ reaction when they were confronted with the concept. According to legend, Hippasus discovered irrationals while on a sea voyage. It is said to have upset the other Pythagoreans so badly they hurled him overboard. I don’t know if that is true, or if he was jettisoned for some other reason such as for being a generic asshole, but I find it interesting that my own son would be so bothered by the whole concept.<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'>I am open to suggestion for how to present irrationals in a more friendly way, especially since I have more bad news for that lad: most numbers are irrational. Depending on how you count them of course. There are infinitely many rationals of course, but there are infinely many more irrationals, since you can give me *<b>any</b>* two rational numbers and I can give you infinitely many irrationals that fit between them, larger than the smaller and smaller than the larger. Of course, I can also give you infinitely many rational that fit between the two as well, but between each of those between the two are infinitely many irrationals.<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'>Aint math cooool? {8-]<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'>spike</span><span style='color:#1F497D'><o:p></o:p></span></p></div></body></html>