Unformatted text preview: s ‘modulus’, ‘argument’ and ‘principal argument’ are welldeﬁned.
Concerning the modulus, if z = 0 then the point associated with z is the origin. In this case, the
only rvalue which can be used here is r = 0. Hence for z = 0, z  = 0 is welldeﬁned. 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 welldeﬁned 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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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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