Graphing Complex and Imaginary NumbersDate: 10/23/97 at 09:13:39 From: Liberal high School Subject: Algebra II How do you graph imaginary numbers? Date: 11/23/97 at 18:10:42 From: Doctor Mark Subject: Re: Algebra II Helloooooo, Liberal! As you probably know, imaginary numbers are just a particular case of complex numbers. A complex number is written as a + ib, where i is the imaginary unit (the square root of - 1), and a and b are real numbers. An imaginary number is one where the real part (the "a") is zero, so an imaginary number is written as just ib, where b is a real number. So I'm assuming that you really want to know how to graph complex numbers, not just imaginary ones. (Once you know how to graph complex numbers, you automatically know how to graph imaginary numbers: how's that for a good deal?) This is done using the *complex plane*. Here's how it goes: Since any complex number z = a + ib has two real numbers associated with it, we can associate with each complex number z = a + ib an ordered pair of real numbers (a,b). Why is it an ordered pair? Because the order is important: the ordered pair (2,3), for instance, corresponds to the complex number 2 + 3i, and the ordered pair (3,2) corresponds to the complex number 3 + 2i, which is different from 2 + 3i. Does that remind you of something? I hope it reminds you of the regular old xy plane, where you have two coordinate axes (the x and y axes), oriented perpendicular to one another, and a point in the plane is associated with an ordered pair of numbers (a,b), where the number a gives the x coordinate of the point, and the number y gives the y coordinate of the point. Now notice that if we have a complex number z = a + ib, we can associate with it the ordered pair of numbers (a,b), and that is associated with the point whose x coordinate is a, and whose y coordinate is b. This allows us to associate with every complex number a point in the xy plane. Similarly, every point in the xy plane is associated with an ordered pair of numbers (a,b), and we can associate with this ordered pair the complex number z = a + ib. Thus, every complex number is associated with a point in the xy plane, and every point in the xy plane is associated with a complex number. This allows us to represent every complex number as a point in the xy plane, and vice-versa. When we do this, we call the xy plane the *complex plane*. For instance, the complex number 2 - 3i (= 2 + (-3)i) can be represented as the point (2,-3). The complex number 4i = 0 + 4i can be represented as the point (0,4). The *real* number 2 = 2 + 0i (since it is still true that 0 times any number (real *or* complex) is 0, the number 0i is just equal to 0) can be represented by the point (2,0). Do you see that real numbers (numbers of the form a + 0i) correspond to points on the "x" axis, and that purely imaginary numbers (numbers of the form 0 + bi, where b is real) correspond to points on the "y" axis? As a result, when we talk about the complex plane, we rename the x and y axes: the x axis is now called the "real axis" (since that is where all the points correspond to real numbers), and the y axis is now called the "imaginary axis" (since that is where all the points correspond to purely imaginary numbers). The complex plane is an interesting place, and you can find out more about it in any precalculus book. For instance, if you note that any point in the complex plane can be represented by the distance of the point from the origin (the number 0 + 0i) and an angle that the line between the origin and the point makes with the x (Real) axis, you can use complex numbers to do trigonometry, and to find easy ways of understanding various trigonometric identities. The complex plane can also be used to do some calculations in calculus (though you wouldn't get this until about 5 courses after AP calculus!). It can also be used to understand how to find the solutions to cubic and quartic equations (which you never learn about in high school - or even in college - because the formulas which are the analogues of the quadratic formula for solving a quadratic equation are pretty complicated), and how to find all seven of the seventh roots of, say, 6. Hope this has been of help. Write back if there's anything I said that was confusing. -Doctor Mark, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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