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CHAPTER 6 Algebra of Complex Numbers
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Eq. (141) says that the quotient of two complex numbers, c jd divided by a jb, is equal to a SINGLE COMPLEX NUMBER, which we show as A jB in eq. (141). The PROBLEM is, given the complex numbers c jd and a jb, nd the values of A and B in eq. (141). Fortunately, this can be done by multiplying the numerator and denominator of the fraction by the CONJUGATE OF THE DENOMINATOR. To show why this is true, let us apply the rule to the fraction on the left-hand side of eq. (141); doing this, and making use of eq. (140), we nd that eq. (141) becomes (on the left-hand side) c jd a jb ac bd j ad bc a2 b2 a2 b2 Notice that now the common denominator, a2 b2 , is a positive real number, which therefore allows us to separate the fraction into its real and imaginary parts; thus ac bd ad bc j 2 A jB a2 b 2 a b2 hence showing that multiplying the numerator and denominator by the conjugate of the denominator does convert a given fraction into the form A jB. Furthermore, in the above example we see that, by inspection,* A Example
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Find the value of the quotient
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Solution By nd the value of we mean resolve the fraction into its real and imaginary parts. To do this we multiply the numerator and denominator by the conjugate of the denominator ; thus 5 j4 2 j3 10 j15 j8 12 22 j7 1:6923 j 0:5385; 13 13 13 2 2 32 answer:
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The above procedure can be referred to as rationalizing the fraction, which enables us to simplify the ratio of two complex numbers into the form of a single complex number. Problem 85 Find the values of each of the following: a Problem 86 Find the value of 4 j 2 j 1 j 3 j2
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* Two complex numbers are de ned to be equal IF AND ONLY IF their real parts are equal and their imaginary parts are equal.
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CHAPTER 6 Algebra of Complex Numbers
Problem 87 Simplify the expression j12 1 j 4 Answer: j3
To close this section, suppose a complex number is equal to zero; that is, suppose a jb 0 j0 0 Since two complex numbers can be equal only if the real parts are equal and the imaginary parts are equal, it follows that the above can be true only if a 0 and b 0; that is:
A complex number can be equal to zero only if the real and imaginary parts are BOTH equal to zero.
Graphical Representation of Complex Numbers
We are familiar with the fact that real numbers can be represented as POINTS ON A STRAIGHT LINE. This is illustrated in Fig. 106, in which X 0 X (line X prime, X) represents such a line. The point 0, called the origin, represents the number zero.
Fig. 106
As shown, all POSITIVE real numbers are represented as points to the RIGHT of the origin, while all NEGATIVE real numbers are represented by points to the LEFT of the origin. We accept the statement that TO EVERY REAL NUMBER there corresponds ONE AND ONLY ONE POINT on line X 0 X. We accept the reciprocal statement that TO EVERY POINT on line X 0 X there corresponds ONE AND ONLY ONE real number. Fig. 106 is called the AXIS OF REAL NUMBERS, or simply the axis of reals. Now consider the corresponding representation of imaginary numbers. As we know, imaginary numbers have the form jb, where j is the imaginary unit (eq. (133)) and where b can be any positive or negative real number. From the description of Fig. 106 it follows that the same idea can be applied to imaginary numbers; that is, IMAGINARY NUMBERS can also be represented as points on a straight line. This is illustrated in Fig. 107, in which Y 0 Y (line Y prime, Y), is the AXIS OF IMAGINARY NUMBERS (or axis of imaginaries ), in the same way that Fig. 106 is the axis of reals.