Unformatted text preview: r of polar coordinates (r, θ). Although it is not a straightforward as
the deﬁnitions of Re(z ) and Im(z ), we can still give r and θ special names in relation to z .
Definition 11.2. The Modulus and Argument of Complex Numbers: Let z = a + bi be a
complex number with a = Re(z ) and b = Im(z ). Let (r, θ) be a polar representation of the point
with rectangular coordinates (a, b) where r ≥ 0.
• The modulus of z , denoted |z |, is deﬁned by |z | = r.
• The angle θ is an argument of z . The set of all arguments of z is denoted arg(z ).
• If z = 0 and −π < θ ≤ π , then θ is the principal argument of z , written θ = Arg(z ).
Some remarks about Deﬁnition 11.2 are in order. We know from Section 11.4 that every point in
the plane has inﬁnitely many polar coordinate representations (r, θ) which means it’s worth our
1 ‘Well-deﬁned’ means that no matter how we express z , the number Re(z ) is always the same, and the number
Im(z ) is always the same. In other words, Re and Im are functions of complex numbers. 11.7 Polar Form of Complex Numbers 843 time to make sure the quantitie...
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