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Multiplying complex numbers
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You should know how complex numbers are multiplied, to have a full understanding of their behavior. When you multiply these numbers, you only need to treat them as sums of number pairs, that is, as binomials. It s easier to give the general formula than to work with specifics here. The product of (a + jb) and (c jd) is equal to ac jad jbc jjbd. Simplifying, remember that jj 1, so you get the final formula: (a jb)(c jd) = (ac bd) j(ad bc)
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As with the addition and subtraction of complex numbers, you must be careful with signs (plus and minus). And also, as with addition and subtraction, you can get used to doing these problems with a little practice. Engineers sometimes (but not too often) have to multiply complex numbers.
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The complex number plane
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Real and imaginary numbers can be thought of as points on a line. Complex numbers lend themselves to the notion of points on a plane. This plane is made by taking the real and imaginary number lines and placing them together, at right angles, so that they intersect at the zero points, 0 and j0. This is shown in Fig. 15-2. The result is a Cartesian coordinate plane, just like the ones you use to make graphs of everyday things like bank-account balance versus time.
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On this plane, a complex number might be represented as a jb (in engineering or physicists notation), or as a bi (in mathematicians notation), or as an ordered pair (a, b). Wait, you ask. Is there a misprint here Why does b go after the j, but in front of the i The answer is as follows: Mathematicians and engineers/physicists just don t think alike, and this is but one of myriad ways in which this is apparent. In other words, it s a matter of
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The complex number plane 267
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15-2
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The complex number plane.
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notational convention, and that is all. (It s also a somewhat humorous illustration of the different angle that an engineer takes in approaching a problem, as opposed to a mathematician.)
Complex number vectors
Complex numbers can also be represented as vectors in the complex plane. This gives each complex number a unique magnitude and direction. The magnitude is the distance of the point a jb from the origin 0 j0. The direction is the angle of the vector, measured counterclockwise from the a axis. This is shown in Fig. 15-3.
Absolute value
The absolute value of a complex number a jb is the length, or magnitude, of its vector in the complex plane, measured from the origin (0,0) to the point (a,b).
268 Impedance and admittance
15-3 Magnitude and direction of a vector in the complex number plane.
In the case of a pure real number a j0, the absolute value is simply the number itself, a, if it is positive, and a if a is negative. In the case of a pure imaginary number 0 jb, the absolute value is equal to b if b (which is a real number) is positive, and b if b is negative. If the number is neither pure real or pure imaginary, the absolute value must be found by using a formula. First, square both a and b. Then add them. Finally, take the square root. This is the length of the vector a jb. The situation is illustrated in Fig. 15-4.
15-4 Calculation of absolute value, or vector length.
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Problem 15-1
Find the absolute value of the complex number 22 j0. Note that this is a pure real. Actually, it is the same as 22 j0, because j0 = 0. Therefore, the absolute value of this complex number is ( 22) 22.
Problem 15-2
Find the absolute value of 0 j34. This is a pure imaginary number. The value of b in this case is 0 j( 34). Therefore, the absolute value is ( 34) 34. 34, because 0 j34
Problem 15-3
Find the absolute value of 3 j4. In this number, a 3 and b 4, because 3 j4 can be rewritten as 3 j( 4). Squaring both of these, and adding the results, gives 32 ( 4)2 9 16 25. The square root of 25 is 5; therefore, the absolute value of this complex number is 5. You might notice this 3, 4, 5 relationship and recall the Pythagorean theorem for finding the length of the hypotenuse of a right triangle. The formula for finding the length of a vector in the complex-number plane comes directly from this theorem. If you don t remember the Pythagorean theorem, don t worry; just remember the formula for the length of a vector.
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