Dividing by Complex NumbersDate: 05/04/2003 at 23:07:51 From: Ryan Subject: Rational expressions with imaginary numbers Hi, I'm difficulty with problems that have an imaginary number but don't cancel. Example: Divide each pair of complex numbers: (8+4i)/(1+2i) Any help would be great. Thanks. Date: 05/05/2003 at 05:02:42 From: Doctor Luis Subject: Re: Rational expressions with imaginary numbers Hi Ryan, Good question. There's a trick for dividing by complex numbers, and to use it you need to understand something called the conjugate complex number. Essentially, the conjugate of a complex number is the number you get when you replace (i) by (-i). For example, the conjugate of 1+3i is 1+3(-i)=1-3i, and also, the conjugate of -3-2i is -3-2(-i)=-3+2i Now, something funny happens when you multiply a complex number by its conjugate. The answer turns out to be a real number. I'll illustrate with 1+3i and its conjugate 1-3i (1+3i)*(1-3i) = 1^2 - (3i)^2 = 1 - (9 * (-1)) = 1 + 9 = 10 Here, I used the algebraic formula (a-b)(a+b) = a^2 - b^2 to multiply them. (Don't forget that i^2 = -1) You should verify for yourself that for any complex number z=x+iy and its complex conjugate z'=x-iy, the product z*z' is a real number (and that it equals x^2+y^2). Knowing this fact about complex numbers, to divide you simply multiply and divide by the conjugate of the denominator. Here's how: 20 + 30i 20 + 30i -1 - 2i --------- = ---------- * ---------- (conjugate trick) -1 + 2i -1 + 2i -1 - 2i (20 + 30i)(-1 - 2i) = ------------------- (multiplying bottom) (-1)^2 + (2)^2 = (40 - 70i)/5 (after multiplying top) = 8 - 14i (final answer) That's all there is to it. You make the denominator into a number you can divide by (that is, a real number), using complex conjugates. With this background, you should be able to solve the division you asked about. Let us know if you have any more questions. - Doctor Luis, The Math Forum http://mathforum.org/dr.math/ |
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