Mathematics

Complex Numbers and Rotations

An exploration of representing complex numbers in polar form and the geometric interpretation of multiplication by the imaginary unit i as a 90-degree rotation.

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Representing z in Polar Form

z = \sqrt{3} - 3i

Let's represent the complex number z = sqrt(3) - 3i. We need its distance from the origin, called the modulus, and the angle it makes with the positive real axis, the argument.

Step-by-step solver

(a) Modulus and Argument

Find the modulus r using the Pythagorean theorem, and the argument theta using inverse tangent in the fourth quadrant.

r = \sqrt{(\sqrt{3})^2 + (-3)^2} = \sqrt{12} = 2\sqrt{3}, \quad \theta = \arctan(\frac{-3}{\sqrt{3}}) = -\frac{\pi}{3}

(b) Mapping P to Q

Identify P as the coordinate (sqrt(3), -3) and compute iz = 3 + sqrt(3)i, which corresponds to Q = (3, sqrt(3)).

P = (\sqrt{3}, -3), \quad Q = (3, \sqrt{3})

Observe that multiplying by i corresponds to a rotation of 90 degrees (pi/2 radians) anticlockwise about the origin.

iz = e^{i\pi/2} z

Original question

A-Level question (see attached image). Let z = sqrt(3) - 3i. (a) Write z in the form r(cos theta + i sin theta) where -pi < theta <= pi. (b) Show and label on a single Argand diagram the point P representing z and the point Q representing iz. (c) Describe the geometrical transformation that maps P onto Q.

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