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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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