<html><head></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><div><div>On Nov 27, 2010, at 2:15 AM, Mike Dougherty wrote:</div><br class="Apple-interchange-newline"><blockquote type="cite"><span class="Apple-style-span" style="border-collapse: separate; font-family: Verdana; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; font-size: medium; ">Imaginary numbers really do require imagination to grok i^2 = -1. I'd love to see that represented in a visually intuitive way*.</span></blockquote><div><br></div>Make the Real numbers be the horizontal axis of a graph and the imaginary numbers be the vertical axis, now whenever you multiply a Real or Imaginary number by i you can intuitively think about it as rotating it by 90 degrees in a counterclockwise direction. </div><div><br></div><div>Look at i, it sits one unit above the real horizontal axis so draw a line from the real numbers to i, so if you multiply i by i (i^2) it rotates to become −1, multiply it by i again(i^3) and it becomes −i, multiply it by i again (i^4) and it becomes 1, multiply it by i again (i^5) and you've rotated it a complete 360 degrees and you're right back where you started at i.</div><div><br></div><div>It is this property of rotation that makes i so valuable, the best example may be electromagnetism where Maxwell used it to describe how electric and magnetic fields change in the X and Y direction (that is to say in the Real and Imaginary direction) as the wave propagates in the Z direction.</div><div><br></div><div> John K Clark </div><br></body></html>