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</o:shapelayout></xml><![endif]--></head><body 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><o:p> </o:p></p><div><p class=MsoNormal>2010/11/24 Darren Greer <<a href="mailto:darren.greer3@gmail.com">darren.greer3@gmail.com</a>><o:p></o:p></p><p class=MsoNormal><span style='color:#1F497D'>…</span> I'm having trouble understanding how the square root of negative one could have a practical application beyond abstract mathematics. Or even in abstract mathematics, for that matter. <span style='color:#1F497D'>..</span>. Any takers? Remember. Key words: FIRST YEAR. An engineer might be good here. Spike?<span style='color:#1F497D'><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'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'>Darren, really are a skerjillion real world applications for imaginary numbers. It is actually very unfortunate that they were ever given that name. Calling them real and imaginary makes it sound like the imaginary numbers are somehow not *<b>real</b>*. {8^D <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'> A number with both a real and an imaginary part is called a complex number. That is another *<b>terrible</b>* term, because it scares non-mathematics types.<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'>They should have been called something different. I would propose the reals be called horizontal numbers and imaginary called vertical numbers. Then if it has both, it is an off-axis number, because it is on neither the horizontal or vertical axis.<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'>Here’s a mathematical comment that will blow your mind if you think about it hard enough, and one which also has real world applications:<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'> <o:p></o:p></span></p><div><p class=MsoNormal><span style='color:#1F497D'>e^(i*pi) = -1<o:p></o:p></span></p><p class=MsoNormal><span style='color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='color:#1F497D'>I would put a ! after that comment but for the fact that ! has its own meaning in mathematics, which makes it difficult to express one’s enthusiasm at a really exciting equation without messing up the equation. But this is a very exciting equation! It is a result of the fact that<o:p></o:p></span></p><p class=MsoNormal><span style='color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='color:#1F497D'>e^(i*theta) = cosine(theta) + i*sin(theta)<o:p></o:p></span></p><p class=MsoNormal><span style='color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='color:#1F497D'>!!<o:p></o:p></span></p><p class=MsoNormal><span style='color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='color:#1F497D'>Is this cool or what?<o:p></o:p></span></p><p class=MsoNormal><span style='color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='color:#1F497D'>spike</span><br><br><o:p></o:p></p></div></div></div></body></html>