628 CHAPTER 9 Polar Coordinates; Vectors Plotting a Point in the Complex Plane Plot the point corresponding to = − z i 3 in the complex plane. Solution EXAMPLE 1 The point corresponding to = − z i 3 has the rectangular coordinates ( ) − 3, 1 . This point, located in quadrant IV, is plotted in Figure 39. Figure 39 Imaginary axis Real axis 22 2 2 22 z 5 3 2 i O DEFINITION Magnitude or Modulus Suppose = + z x yi is a complex number. The magnitude or modulus of z, denoted z , is the distance from the origin to the point ( ) x y , . That is, = + z x y 2 2 (1) = z zz (2) See Figure 40 for an illustration. This definition for z is consistent with the definition for the absolute value of a real number: If = + z x yi is a real number then = + z x i0 and = + = = z x x x 0 2 2 2 For this reason, the magnitude of z is sometimes called the absolute value of z. Recall that if = + z x yi, then its conjugate , denoted z, is = − z x yi. Because = + zz x y 2 2 is a nonnegative real number, it follows from equation (1) that the magnitude of z can be written as Figure 40 y Imaginary axis Real axis x z 5 x 1 yi O ƒ z ƒ 5 x 2 1 y 2 2 Convert a Complex Number between Rectangular Form and Polar Form or Exponential Form When a complex number is written in the standard form = + z x yi, it is in rectangular , or Cartesian , form , because ( ) x y , are the rectangular coordinates of the corresponding point in the complex plane. Suppose that θ ( ) r, are polar coordinates of this point. Then x r y r cos sin θ θ = = DEFINITION Polar Form of a Complex Number If ≥ r 0 and θ π ≤ < 0 2 , the complex number = + z x yi can be written in polar form as z x yi r r i r i cos sin cos sin θ θ θ θ ( ) ( ) = + = + = + (3) See Figure 41. If z r i cos sin θ θ ( ) = + is the polar form of a complex number,* the angle ,θ θ π ≤ < 0 2 , is called the argument of z. *Some texts abbreviate the polar form using z r i r cos sin cis . θ θ θ ( ) = + = Figure 41 y r z Imaginary axis Real axis x z 5 x 1 yi 5 r(cos u 1 i sin u), r $ 0, 0 # u , 2p O u
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