Graphing Complex FunctionsDate: 08/11/98 at 03:44:56 From: Michael Liddle Subject: Complex solutions to polynomials In my Calculus class (Paraparaumu College, New Zealand), we have just begun a new unit on "Complex Numbers." In our study so far, we have learned the significance of i and its properties and the way in which it behaves with operations and relations. We have also learned about the modulus and argument of z. My question, however relates to the complex solutions to various polynomials. Up until now we have said that, for example, the quadratic equation (y = x^2 + 5x + 12) when y = 0 has no solutions. We have now been told that in effect this equation in fact has two solutions, just that they are both "imaginary" as opposed to "real" numbers. What I want to find out is where (if anywhere), these numbers lie on the graph of this equation (x,y Cartesian form). I asked my calculus teacher this question and he said that in 20 years of teaching he has never been asked this before. My problem may well lie in the fact that my experience in this area is extremely limited, and I am simply as yet unable to comprehend (or accept) that in fact they do not lie anywhere on the graph. However I thought that maybe that there was a third "imaginary axis," or that the answer may lie in a "fourth dimension" that cannot be represented graphically. Thank you very much for your assistance in this area. Date: 08/14/98 at 14:17:34 From: Doctor Benway Subject: Re: Complex solutions to polynomials Hi Michael, The easy answer to your question is that yes, you do need four dimensions to picture where the solutions are if you want to plug complex variables such as 1 + 2i into your equations. The regular way of representing complex numbers in the plane is to let the x coordinate be the real part and the y coordinate be the imaginary part, so 1 + 2i would correspond to the point (1, 2). However, when we plot functions of complex numbers, both the "input" variable, the domain of the function, and the "output" variable, the range of the function, require two axes. Therefore you would need four dimensions: two dimensions to plot the coordinate of the point you are plugging into the equation and two dimensions to plot the output. For example, when we square (1 + 2i), our result is -3 + 4i. To plot this would require four dimensions, two to represent (1 + 2i) and two to represent (-3 + 4i). However, not all is lost! There is another way of thinking about functions (which should be instructive when you're only looking at functions of a real variable as well), and that is thinking of them as a mapping of points to other points. For example, imagine you are standing on the real line on the point corresponding to the number 3. Suddenly out of nowhere comes the function "x^2 - 1" which moves all the points around. You are swept away from 3 in a whirlwind of mathematical activity. When the dust finally settles, you find yourself standing on the point corresponding to the number 8 (3^2 - 1 = 8). Thinking of functions this way, you see that functions move points around. A little more thought shows the following: solutions to the equation x^2 - 1 = 0 can be thought of as the points which get moved to 0 when the function x^2 - 1 strikes. Had you been standing on the points corresponding to the numbers 1 or -1, then you would have ended up standing on the origin after the function hit. Now imagine yourself standing on the complex plane instead of on the number line. Say you are standing on the point corresponding to the complex number -2 + i, which in cartesian coordinates would be the point (-2,1). Along comes the function x^2 + 4x + 5, which sweeps up all the points and sets them down in different places. When the dust settles, you find yourself standing at the origin, (0,0). "Aha," you say to yourself, "I must have been standing on a solution to x^2 + 4x + 5 = 0." One way to represent these mappings is to have one set of axes for the domain and another set of axes for the range. It's a little confusing to picture what the function is doing, but if you trace the domain with one finger and the range with another, you get an idea of what's happening. I hope this helps answer your question. This problem bothered me for a long time when I was in high school too, and I was very happy when I finally found a way to think about it that made sense to me. Thanks for writing. - Doctor Benway, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 08/15/98 at 12:32:57 From: Doctor Sonya Subject: Re: Complex solutions to polynomials Hi there. My name is Dr. Sonya, and I am one of the other math doctors. I just wanted to point you towards a very interesting discussion in our archives about what the complex roots of polynomials have to do with the graph. The URL of the page is: http://mathforum.org/dr.math/problems/complex.roots.html and it is one of our best discussions. - Doctor Sonya, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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