Unformatted text preview: s ‘modulus’, ‘argument’ and ‘principal argument’ are well-deﬁned.
Concerning the modulus, if z = 0 then the point associated with z is the origin. In this case, the
only r-value which can be used here is r = 0. Hence for z = 0, |z | = 0 is well-deﬁned. If z = 0,
then the point associated with z is not the origin, and there are two possibilities for r: one positive
and one negative. However, we stipulated r ≥ 0 in our deﬁnition so this pins down the value of |z |
to one and only one number. Thus the modulus is well-deﬁned in this case, too.2 Even with the
requirement r ≥ 0, there are inﬁnitely many angles θ which can be used in a polar representation
of a point (r, θ). If z = 0 then the point in question is not the origin, so all of these angles θ are
coterminal. Since coterminal angles are exactly 2π radians apart, we are guaranteed that only one
of them lies in the interval (−π, π ], and this angle is what we call the principal argument of z ,
Arg(z ). I...
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