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CHAPTER 6 Algebra of Complex Numbers
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Thus we can present EVERY REAL NUMBER as a point on the straight line of Fig. 106, and EVERY IMAGINARY NUMBER as a point on the straight line of Fig. 107. But now consider the graphical representation of COMPLEX NUMBERS, a jb. Take, for example, any two complex numbers, such as 2 j2 and 2 j3. Note that there is NO POINT on a straight line that can exclusively represent both of these numbers. Thus the in nite number of di erent complex numbers cannot possibly be represented by points on a one-dimensional straight line. Instead, to graphically represent complex numbers a two-dimensional PLANE SURFACE is required. This requirement is met by positioning the axis of imaginaries perpendicular to the axis of reals , the two axes intersecting at their common 0, thus creating the COMPLEX NUMBER PLANE, as shown in Fig. 108.
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Note that all negative and positive REAL numbers are represented by points on the real or x-axis (X 0 ; X), with all negative and positive IMAGINARY numbers represented by points on the imaginary or y-axis (Y 0 ; Y), while ALL COMPLEX NUMBERS are represented by points on the plane, such as the general complex number (a jb), as illustrated in the gure. Thus, to locate the point that represents a given complex number a jb, we locate the real part a on the x-axis, then locate the imaginary part b on the y-axis; the desired point is at the intersection of the horizontal and vertical lines drawn from b and a, as shown in the gure. Thus each real, imaginary, and complex number is represented by a single unique point on the complex plane of Fig. 108. Problem 88 In section 6.3 we saw that two complex numbers are equal only if their real parts are equal and their imaginary parts are equal. Is this fact evident from inspection of Fig. 108
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CHAPTER 6 Algebra of Complex Numbers
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Thus far in our work we ve written complex numbers in what is called the RECTANGULAR form, that is, in the form a jb. It is also possible, and often highly desirable, to write complex numbers in what is called the POLAR or TRIGONOMETRIC form. In this form, a complex number is expressed in terms of magnitude, A, and angle, . The relationship between the rectangular and polar form of a complex number can be derived from inspection of Fig. 109.
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Fig. 109
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Thus, in the gure, a jb A cos  j sin  142
the right-hand side being the equivalent trigonometric form of a jb. The length A is called the modulus or absolute value of the complex number, and is always taken to be a POSITIVE value. The angle  (theta) is called the amplitude of the complex number, with positive angles measured in the ccw (counterclockwise) sense. From inspection of Fig. 109, p 143 A a2 b2 9 b > = tan  a 144 >  arctan b=a ; Complex numbers in the trigonometric form are often written in the abbreviated form A=, which is called the polar form.* Hence the rectangular, trigonometric, and polar forms all denote the same thing, a complex number; thus a jb A cos  j sin  A= 145
In the following two problems round all calculator values o to 3 decimal places. Problem 89 Write the following complex numbers in trigonometric form. 7 j2 (a) 2:3 j3:5 5:8 j13:9 (b) 4 j9
* The same notation is used in ordinary algebra when graphing purely real functions on real polar coordinate space. In our work, however, it will be used only in connection with the complex plane, as de ned in eq. (145).
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CHAPTER 6 Algebra of Complex Numbers
Problem 90 Write the following complex numbers in rectangular form, a jb. (a) 90=1668 (b) 400= 1268 (c) 17=458 22=2658
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