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Stitz-Zeager_College_Algebra_e-book

We continue to move to the right since x is still

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Unformatted text preview: n fact, the set arg(z ) of all arguments of z can be described using set-builder notation as arg(z ) = {Arg(z ) + 2πk : k is an integer}.3 If z = 0 then the point in question is the origin, which we know can be represented in polar coordinates as (0, θ) for any angle θ. In this case, we have arg(0) = (−∞, ∞) and since there is no one value of θ which lies (−π, π ], we leave Arg(0) undeﬁned.4 It is time for an example. Example 11.7.1. For each of the following complex numbers ﬁnd Re(z ), Im(z ), |z |, arg(z ), and Arg(z ). Plot z in the complex plane. √ 1. z = 3 − i 2. z = −2 + 4i 3. z = 3i 4. z = −117 Solution. √ √ √ 1. For z = 3 − i = 3 + (−1)i, we have Re(z ) = 3 and Im(z ) = −1. To ﬁnd |z |√arg(z ) , and Arg(z ), we need to ﬁnd a polar representation5 (r, θ) with r ≥ 0 for the point P ( 3, −1) √ associated with z . We know r2 = ( 3)2 + (−1)2 = 4, so r = ±2. Since we require r ≥ 0, we choose r = 2, so |z | = 2. Next, we ﬁnd a corresponding angle θ. Since r > 0 and P lies √ −...
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