Complex Numbers · Mathematics

Imaginary and complex numbers

The imaginary unit is i, where i² = −1. A complex number is z = x + iy, with real part x and imaginary part y. Its conjugate is z̄ = x − iy. Complex numbers are added, subtracted and multiplied like algebraic expressions, replacing i² by −1.

Multiplication and the reciprocal.

The Argand diagram, modulus and argument

A complex number x + iy is plotted as the point (x, y) on an Argand diagram, with the real axis horizontal and the imaginary axis vertical. The modulus |z| is the distance from the origin and the argument arg z is the angle from the positive real axis.

Modulus and argument.

Worked exampleModulus and argument

Find the modulus and argument of z = 1 + i√3.

Modulus.
Argument.

Answer: |z| = 2, arg z = 60° (π/3).

Polar form

Polar form and Euler’s formula e^(iθ) = cos θ + i sin θ.

In polar form, multiplication multiplies the moduli and adds the arguments: |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg z₁ + arg z₂.

De Moivre’s theorem

De Moivre’s theorem.

Worked exampleA power of a complex number

Find (1 + i)⁸.

Polar form: r = √2, θ = 45°.
Evaluate.

Answer: (1 + i)⁸ = 16.

De Moivre’s theorem also gives the n nth roots of a complex number, equally spaced around a circle on the Argand diagram, and Euler’s identity e^(iπ) + 1 = 0.

Try itModulus, argument and powers

Choose z = x + iy and a power n. De Moivre: the modulus is raised to n and the argument multiplied by n.

Check yourself i² =

Check yourself (2 + 3i)(2 − 3i) =

Check yourself The modulus of 3 + 4i is

Key terms

Imaginary unit: i, where i² = −1.
Complex conjugate: x − iy for z = x + iy.
Modulus: The distance from the origin on the Argand diagram.
Argument: The angle from the positive real axis.

Summary

A complex number is x + iy with i² = −1.
Modulus √(x² + y²); argument tan⁻¹(y/x).
Polar form r(cos θ + i sin θ) = re^(iθ).
De Moivre: (r cis θ)ⁿ = rⁿ cis nθ.