**(Cosθ + i Sinθ)**^{n} = Cos nθ + i Sin nθ

Take a fixed complex number on the unit circle

z = Cosθ + i Sinθ |z| = 1

Consider multiples of z by a real, positive number r.

r z = r (Cosθ + i Sinθ) |r z| = r |z| = r

As r increases from 1, point moves out along the ray with tail at the origin and passes through the point z.

As r decreases from 1 to zero, point moves inward along the same way toward the origin. The modulus of the point is r. We call the angle θ which this ray makes with the *x*-axis, the argument of the number z. All the numbers rz have the same argument.

arg rz = θ

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