If the system of parametric equations contains

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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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