Graphing Complex and Real Numbers
Date: 02/26/2003 at 20:32:45 From: Paul Lehmbeck Subject: Graphing Complex and Real Numbers I have been finding the zeros for polynomial functions in class and I have found real, imaginary, and complex numbers as the zeros. But when I graph the zeros I run into a problem. Since on the Cartesian plane you can only graph real zeros and real solutions, I wonder if you are truly graphing the function when you omit the complex and imaginary zeros and solutions. For example, for f(x) = x^2+1 there are no real zeros, but the imaginary zeros are -i and i. MAIN QUESTION - Since you can't graph the imaginary zeros, is the function truly being represented by the graph? This has led me to wonder if it is at all possible to graph the real, imaginary, and complex input and output solutions on the same graph and still remain in the 2nd or 3rd dimension without hypothesizing about a fourth dimension? If it is possible to represent all solutions to a function, including real, complex, and imaginary numbers, on a single graph, what would that graph be if it is not a Cartesian plane or Complex plane? If it is not possible, would you then need two graphs for the same function - one for the real inputs and outputs, and another for the imaginary and complex inputs and outputs (if it is possible) to show all solutions? Thank you for your time and explanation.
Date: 02/27/2003 at 15:49:41 From: Doctor Tom Subject: Re: Graphing Complex and Real Numbers Hi Paul, Great question! At this stage, almost every function you've seen takes real values as inputs and gives real values as outputs, so when you plot such a function on the standard x-y plane, you are showing all the values of the function with real-valued inputs. For a function like y = x^2 + 1 that has no real roots, you are still plotting all the information for real inputs. The curve never passes through y = 0, so there simply are no real roots. You can't plot the output for an input value of i, since i is imaginary and not found on the real line. In our standard three-dimensional space it is impossible to plot functions, like polynomials, that have complex inputs and complex outputs. You can imagine the real-imaginary complex plane, and plot outputs above it, but to display all the information, you would need to display two output values - the real and the imaginary components. Given this restriction, here are a few things that are normally done: It is possible to have one 3-D plot that shows the real part of the function and another 3-D plot that shows the imaginary component. Both would appear to be surfaces over the complex plane whose heights above that plane correspond to the real and imaginary components of the function's values. It's a bit tricky from a pair of plots like this to find roots, however. The reason is that to find a zero, the function has to be zero both in its real AND imaginary components. In general, functions will have lines or curves on the complex plane above which the real part is zero or above which the imaginary part is zero. Only where those curves cross (and where both components are zero) is where the function has a root. The other method commonly uses plots in the third dimension the absolute value of the complex output. If z = a+bi, the absolute value of z is the square root of (a^2 + b^2), and this can be zero only if both a and b are zero. Any place this surface touches the complex plane is a root, but the plot throws away the individual real and imaginary components of the function's output. Plotting all three will give you a lot of information to help visualize what's going on. Unfortunately, you need a computer to do this quickly enough and accurately enough that you can get useful images. Remember that you are usually looking at a tiny fraction of the available polynomials. For example, could you find the roots of: (3-2i)x^2 + (7+3i)x - (15 + 43i) = 0? It is just a quadratic equation, and the only thing that makes it different from the ones you've looked at is that it has complex coefficients. There is a reasonable way to extend almost all of your favorite functions to the complex domain, although some strange things do happen. For example, it makes sense to talk about sine(3-2i) or log(-4+3i). You may have seen this: e^(it) = cos(t) + i sin(t) You have probably only used it for real values of t, but it is true for complex values as well, and by messing around with those, you could learn a lot! I hope this helps. - Doctor Tom, The Math Forum http://mathforum.org/dr.math/
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