Topic 2: Complex Numbers

3 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How do we add, multiply and divide numbers that involve the square root of minus one, and how do we picture them?

Complex numbers extend the reals with i where i squared equals minus one, and are represented as points or vectors on the Argand plane.

Cartesian form of complex numbers, addition, subtraction, multiplication, the complex conjugate, division by rationalising the denominator, and the Argand plane representation with modulus.

8 min read →

How does writing a complex number by its modulus and angle make multiplication, division and powers easy?

Polar form expresses a complex number by its modulus and argument, and De Moivre's theorem raises it to integer powers.

Converting between Cartesian and polar form, the modulus and argument, multiplication and division in polar form, and using De Moivre's theorem to compute integer powers of complex numbers.

8 min read →

Why does a complex number have exactly n distinct nth roots, and how are they arranged?

Find the nth roots of a complex number using polar form and de Moivre's theorem, and represent them geometrically on the Argand plane.

Finding the n distinct nth roots of a complex number with de Moivre's theorem, why they are equally spaced on a circle, and solving polynomial equations such as z^n = w.

8 min read →