tag:blogger.com,1999:blog-30226356.post6817662381564888170..comments2024-02-16T23:32:12.073-08:00Comments on The Exponential Curve: Next Lesson: Intro to Complex NumbersDan Wekselgreenehttp://www.blogger.com/profile/08696028020767073620noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-30226356.post-11351428639678862442007-05-18T21:18:00.000-07:002007-05-18T21:18:00.000-07:00It looks like wikipedia has a graphic on their pag...It looks like wikipedia has a graphic on their page for <A HREF="http://en.wikipedia.org/wiki/Euler%27s_identity" REL="nofollow">Euler's Identity</A> that's based on the same thing.<BR/><BR/>I like what you wrote about leaving space for the discovery. I recently had a <A HREF="http://understanding.mindtangle.net/?p=226" REL="nofollow">similar experience</A> where a student showed me a new way to see something familiar.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-30226356.post-88323418908749756862006-12-04T23:26:00.000-08:002006-12-04T23:26:00.000-08:00It is very useful, yes, to see multiplication by c...It is very useful, yes, to see multiplication by complex numbers as rotations; since you were also fiddling with matrices and solving linear systems, you might point out that rotation by an angle theta can be seen as a 2x2 matrix:<br /><br />cos theta | -sin theta<br />sin theta | cos theta<br /><br />This is fun stuff to fiddle around with. A good linear algebra book like Axler's "Linear Algebra Done Right" (aimed at undergraduates) or Paul Halmos' "Finite Dimensional Vector Spaces," will give you more fun things about the spatial relationships between complex numbers and linear algebra.<br /><br /> -- xnAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-30226356.post-33184960652395886152006-12-02T15:53:00.000-08:002006-12-02T15:53:00.000-08:00Darren,
I'll check those out. The Lakoff book is...Darren, <br />I'll check those out. The Lakoff book isn't really targeted to hardcore mathematicians either (otherwise, I wouldn't be able to read it!). It's written by a linguist and a cognitive scientist. I wouldn't call it "conversational", however. :)<br /><br />MRC, <br />I thought about putting in a vertical axis to start with, but decided not to. Actually, on the diagram I gave the students, I left off the imaginary numbers too - I just gave them a number line with the circles to guide the rotations. I wanted them to create the vertical axis themselves by using the rotation metaphor. I think it was a good choice - some students thought that a 90 degree rotation would be the same as multiplying by 0, since you land above 0 on the real number line. But then other students realized that you were no longer on the real number line, so maybe it wasn't a real number at all. I actually hadn't thought of it that way myself before our discussion. I think that giving the vertical axis might have stifled some of the initial discussion.<br /><br />After this piece of it, I gave them an actual complex plane to practice plotting complex numbers. I think I did a better job this year of getting them to understand that a complex number has a real and an imaginary component; graphically, any number on one of the axes is purely real or purely imaginary, and anything in the grid area is a combination of the two, and is therefore complex. I think I should have done a better job of connecting the circle diagram to the complex plane to reinforce where the plane comes from. I'll do this more explicitly when we go back to do the powers of i lesson.<br /><br />The book has a lot of great stuff in it. At the end, there are a couple of case studies, where they explain how their theories can be used to understand both the number e and the equation e^pi*i + 1 = 0.Dan Wekselgreenehttps://www.blogger.com/profile/08696028020767073620noreply@blogger.comtag:blogger.com,1999:blog-30226356.post-37755286362628647452006-12-02T13:43:00.000-08:002006-12-02T13:43:00.000-08:00I love this metaphor / way of explaining things. ...I love this metaphor / way of explaining things. Your diagram might benefit from a vertical axis. And it looks like I am going to need to read Lakoff's book!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-30226356.post-68932894637506027452006-12-01T19:25:00.000-08:002006-12-01T19:25:00.000-08:00I just introduced i this last week. I like the "r...I just introduced i this last week. I like the "rotation" idea!<br /><br />If you're interested in "math" books, I recommend 2 by former UC Davis professor Sherman Stein--Strength in Numbers, and How The Other Half Thinks. They're written in a clear, conversational style, and the average Joe, not the hardcore mathematician, is the target audience.Darrenhttps://www.blogger.com/profile/15730642770935985796noreply@blogger.com