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Chapter 2: Complex Numbers and Exponenti... > 2.2. Complex Multiplication - Pg. 10

10 Chapter 2 Imag (Y) axis j Z = X + jY -1 1 Real (X) axis -j Figure 2.1 Scaling a complex number is simply scaling each component: 4 Á ð3 þ j4Þ ¼ 12 þ j16 2.2 Complex Multiplication Multiplication gets a little trickier and is harder to visualize graphically. Here is the way the mechanics of it work: ðA þ jBÞ Á ðC þ jDÞ ¼ A Á C þ jB Á C þ A Á jD þ jB Á jD ¼ AC þ jBC þ jAD þ j 2 BD Now remember that j 2 is, by definition, equal to ­1. After collecting terms, we get AC þ jBC þ jAD þ j 2 BD ¼ AC þ jBC þ jAD À BD ¼ ðAC À BDÞ þ jðBC þ ADÞ The result is another complex number, with AC ­ BD being the real part and BC þ AD being the imaginary part (remember, imaginary just means the vertical axis, while real is the horizontal axis). This result is just another point on the complex plane. The mechanics of this arithmetic may be simple, but we need to be able to visualize what is really happening. To do this, we need to introduce polar (R, O) representation. Until now, we have been using Cartesian (X, Y) coordinates, which means each location on the complex number plane is specified by the distance along each of the two axes (like longitude and latitude on the earth's surface). Polar representation replaces these two parameters, which can specify any point on the complex plane, with another set of two parameters, which also can specify any point on the complex plane. The two new parameters are the magnitude and angle. The magnitude is simply the length of the line or vector from the origin to the point. The angle is defined as the angle of this line starting at the positive X axis and arcing counterclockwise.