tag:blogger.com,1999:blog-7245208048685880741.post7669471399865979549..comments2022-07-22T08:52:39.692-04:00Comments on Rational Expressions: Betty and John's Bogus JourneyMichael Pershanhttp://www.blogger.com/profile/17046644130957574890noreply@blogger.comBlogger10125tag:blogger.com,1999:blog-7245208048685880741.post-10084107047286783062014-04-09T12:55:28.869-04:002014-04-09T12:55:28.869-04:00The square roots of i are +/-(1 + i)/sqrt(2).The square roots of i are +/-(1 + i)/sqrt(2).Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-80784014375533541282014-04-01T23:02:30.754-04:002014-04-01T23:02:30.754-04:00Don't remember the details from my long-ago Hi...Don't remember the details from my long-ago History of Math course...but from what I recall there were certain cubic equations that mathematicians knew had real roots, but that they couldn't solve using regular strategies without getting this square-roots of negative numbers thing cropping up. So, they went with it, the terms cancelled out and the solutions were confirmed. <br /><br />I think this was the only "legitimate" use for imaginary numbers for a while--looking at cases in which they *didn't* cancel was another big step in mathematical thinking. Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-82089291287832118042014-03-31T17:23:39.077-04:002014-03-31T17:23:39.077-04:00Rather curious here... how did complex numbers not...Rather curious here... how did complex numbers not arise is solving the quadratic before solving the cubic? Was the cubic solved first, or was it due to the knowledge that in the cubic the "imaginary" parts cancelled out and thus had to be meaningful that inspired mathematicians to look at them more closely?Jeffrey Harthttps://www.blogger.com/profile/16011898296267164168noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-18364889924110930322014-03-30T23:36:23.179-04:002014-03-30T23:36:23.179-04:00Touche'. (Now I'll have my students read J...Touche'. (Now I'll have my students read John and Betty, and then this. Then we can discuss.)Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-12279163538503008662014-03-30T23:33:20.480-04:002014-03-30T23:33:20.480-04:00But foofoo is just sqrt2 +sqrt2*i. But foofoo is just sqrt2 +sqrt2*i. Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-4780360421709114372014-03-30T22:10:24.370-04:002014-03-30T22:10:24.370-04:00NonEuclidean geometry?NonEuclidean geometry?James Clevelandhttp://rootsoftheequation.wordpress.comnoreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-14565244823203437982014-03-30T21:16:18.765-04:002014-03-30T21:16:18.765-04:00This story makes me uncomfortable because it misre...This story makes me uncomfortable because it misrepresents the history of complex numbers. Complex numbers arose from the solution of the cubic equation. If a cubic equation with integer coefficients has three distinct real roots, and it is solved using the cubic formula, then square roots of negative numbers will occur at intermediate steps of the calculation. Although the imaginary parts cancel out in the end, the complex number system was needed to make sense of these calculations.<br /><br />Of course, mathematicians are free to ask what-if questions, and this is sometimes fruitful, but important ideas usually come from attempting to solve specific problems.Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-9664807718653784932014-03-30T20:17:19.961-04:002014-03-30T20:17:19.961-04:00I don't know... but I tell people that we can&...I don't know... but I tell people that we can't really have less than nothing, either. I explain it as taking the same "opposite" concept that means -1 times -1 is + 1, we can extend the numberline in two dimensions -- voila! a graph! and find an opposite of -1 that's even more opposite... so that it takes i * i to make -1. <br /> <br /> I'm just glad nobody's making 'em work with complicated complex numbers so that foofoo squared = i. Just for fun, you know. SiouxGeonzhttps://www.blogger.com/profile/14852040976080951492noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-37476921186675475162014-03-30T19:16:09.639-04:002014-03-30T19:16:09.639-04:00I don't want to pretend to have anything like ...I don't want to pretend to have anything like a wide-ranging knowledge of the history of mathematics and its development. But where exactly does this sort of whimsy have its place in our subject's history?<br /><br />To be clear, I'm not denying that a certain amount of whimsy is necessary in constructing and inventing proofs and solutions. That seems just part of the game of creative work -- trying on ideas, seeing what works and then progressing.<br /><br />But we're talking about something entirely different with complex numbers. The claim is that you can motivate their existence just by pure invention. What else in the history of math has been created through such what-if-itude? Michael Pershanhttps://www.blogger.com/profile/17046644130957574890noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-75344442029796517872014-03-30T16:57:24.524-04:002014-03-30T16:57:24.524-04:00"The way of mathematics is to make stuff up a..."The way of mathematics is to make stuff up and see what happens." Vi Hart.Curmudgeonhttps://www.blogger.com/profile/04323026187622872114noreply@blogger.com