Complex numbers

Maths · Number systems. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A complex number is z = a + bi, where a and b are real and i² = −1. Here a is the real part and b is the imaginary part. Add, subtract and multiply like ordinary algebra, then replace i² by −1.

The conjugate of z = a + bi is z̄ = a − bi. To divide, multiply the top and the bottom by the conjugate of the bottom, which makes the denominator real.

In the Argand plane z is the point (a, b). Its distance from the origin is the modulus |z|. Two complex numbers are equal only when their real parts are equal and their imaginary parts are equal.

Rules to remember

Powers of i repeat in fours: i¹ = i, i² = −1, i³ = −i, i⁴ = 1. Divide the power by 4 and use the remainder.
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
Modulus: |z| = √(a² + b²); also z·z̄ = a² + b² = |z|².
Multiplicative inverse of a + bi is (a − bi)/(a² + b²).
|z₁z₂| = |z₁| × |z₂| and |z₁/z₂| = |z₁| ÷ |z₂|.
Polar form: z = r(cos θ + i sin θ), where r = |z|, a = r cos θ and b = r sin θ. Choose θ from the quadrant in which the point (a, b) lies.
If a + bi = c + di then a = c and b = d.

Common mistake

In a product, i² is often left as it is or taken as +1. Always replace i² by −1, so for example (2i)(3i) = 6i² = −6.

Worked example

Simplify (3 + 2i)/(1 + i) to the form a + bi.

Multiply top and bottom by the conjugate 1 − i.
Top: (3 + 2i)(1 − i) = 3 − 3i + 2i − 2i² = 3 − i + 2 = 5 − i.
Bottom: (1 + i)(1 − i) = 1 − i² = 1 + 1 = 2.
So (3 + 2i)/(1 + i) = (5 − i)/2.

Answer: 5/2 − (1/2)i

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