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* See note 12 in Appendix.
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153
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CHAPTER 6 Algebra of Complex Numbers
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which is called EULER S FORMULA (pronounced oiler ). This is one of the most important relationships in all of mathematics and the engineering sciences.* In eq. (153) let us replace x with x; doing this, and remembering that cos x cos x and that sin x sin x, we have the additional relationship  jx cos x j sin x 154
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Equations (153) and (154) are valid for all positive and negative values of the variable x. The left-hand sides of the equations are the exponential forms of the complex numbers on the right-hand sides. Now let us return to Fig. 109 and eq. (142) in section 6.4. Comparison of eqs. (142) and (153), writing  in place of x in eq. (153), shows that a jb A j 155 Thus the complex number a jb of Fig. 109 can also be represented in the form A j , where A is the MAGNITUDE of the complex number and  is the ANGLE of A on the complex plane. Hence it is possible to represent a complex number in any of the following FOUR EQUIVALENT WAYS: rectangular form: polar form: trigonometric form: exponential form: a jb A= A cos  j sin  A j 156 157 158 159
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in which the MAGNITUDE A and ANGLE  are given by eqs. (143) and (144) in section 6.4.{ Each of the four forms (eqs. (156) through (159)) has certain advantages and disadvantages, depending upon the type of operation (addition, subtraction, multiplication, or division) that is to be performed. Let us rst take up the case of ADDITION AND SUBTRACTION as follows. In section 6.2 we showed that the real and imaginary parts of the SUM OR DIFFERENCE of two or more complex numbers is, respectively, equal to the sum or di erence of the REAL PARTS of the numbers and the sum or di erence of the IMAGINARY PARTS of the numbers. Hence, in order to carry out the operation of addition or subtraction, the real and imaginary part of each complex number must be available separately; it thus follows that, to nd the sum or di erence of complex numbers, the numbers must rst be expressed in the form of either equation (156) or (158). Problem 91 Find the algebraic sum of the following complex numbers (answer in the rectangular form of eq. (156)). 16=368 22=3158 9:15 j6:88
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* Named for the great Swiss mathematician Leonard Euler (1707 1783). { Theoretically the angle  is in radians, because the forms of the sine and cosine series in eqs. (151) and (152) are derived for the case of the angle being measured in radians. In certain operations in the calculus it is necessary that the angles be in radians; however, in the ordinary operations of addition, subtraction, multiplication, and division, the angles can be expressed in degrees if we wish, and we will usually do so.
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CHAPTER 6 Algebra of Complex Numbers
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In several of the following problems we are asked to convert the rectangular form, a jb, into the polar or exponential form; in doing this some caution is called for, as follows. The conversion of a jb into the polar or exponential form requires the use of eqs. (143) and (144). While there is no di culty in using eq. (143), care must be taken when using eq. (144), that is, in nding the correct value of the angle  by using the equation  arctan b=a . This is because, for a given complex number a jb, the correct value of  depends upon the QUADRANT in the complex plane that the point representing a jb falls in. This is illustrated in Fig. 110, in which h is the angle given by eq. (144) using the magnitude of b=a only; that is h arctan jb=aj
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