Math · Precalculus · Concept

Complex numbers: arithmetic and polar form

A complex number a + bi combines a real part a and an imaginary part b, where i² = −1. Complex numbers add and multiply like binomials, and dividing uses the conjugate. Plotted as points, they have a length, the modulus, and an angle, the argument: multiplying multiplies the lengths and adds the angles, which is how De Moivre’s theorem finds powers and roots.

The imaginary unit

i is a number whose square is −1, so √−16 = 4i and the equation x² = −1 has the solutions ±i. Powers of i repeat every four steps: i, −1, −i, 1.

i^{2} = - 1, \sqrt{- 16} = 4 i

Adding and multiplying

Add or subtract the real parts and the imaginary parts separately. Multiply as you would two binomials, then replace i² with −1.

(a + b i) (c + d i) = (a c - b d) + (a d + b c) i

\begin{array}{c} (a + b i) (c + d i) \\ = (a c - b d) + (a d + b c) i \end{array}

Dividing with the conjugate

The conjugate of a + bi is a − bi, and their product a² + b² is real. To divide, multiply the numerator and the denominator by the conjugate of the denominator.

\frac{a + b i}{c + d i} = \frac{(a + b i) (c - d i)}{c^{2} + d^{2}}

Complex solutions of quadratics

A quadratic with a negative discriminant has two complex solutions, and they form a conjugate pair. For x² + 2x + 5 = 0 the discriminant is 4 − 20 = −16, so x = (−2 ± 4i)/2 = −1 ± 2i.

x = \frac{- 2 \pm \sqrt{- 16}}{2} = - 1 \pm 2 i

The complex plane and polar form

Plot a + bi at the point (a, b). Its distance from 0 is the modulus r = √(a² + b²), and its angle from the positive real axis is an argument θ. Since a = r cos θ and b = r sin θ, every nonzero complex number has a polar form.

a + b i = r (cos θ + i sin θ)

Multiplying in polar form

To multiply, multiply the moduli and add the arguments: multiplying by a complex number scales the plane and rotates it. Repeating the multiplication gives De Moivre’s theorem for powers.

{[r (cos θ + i sin θ)]}^{n} = r^{n} (cos n θ + i sin n θ)

\begin{array}{c} {[r (cos θ + i sin θ)]}^{n} \\ = r^{n} (cos n θ + i sin n θ) \end{array}

Roots

A nonzero complex number has exactly n nth roots. They share the modulus r^(1/n), and their arguments are (θ + 2πk)/n for k = 0, 1, …, n − 1, so they sit evenly spaced around a circle. The nth roots of 1, the roots of unity, sit on the unit circle at the corners of a regular n-sided polygon.

Multiplication as a transformation

Multiplying by a + bi sends 1 to a + bi and i to −b + ai. That is the same map as the matrix whose columns are (a, b) and (−b, a): a rotation combined with a scaling by the modulus.

(\begin{array}{cccccccccc} a & - b \\ b & a \end{array})

Common mistakes

Writing i² = 1: i² = −1, so (2i)² = −4.
Dividing by the real part of the denominator alone instead of multiplying by its conjugate.
Taking the argument straight from arctan(b/a): for −2 + 2i that gives −45°, but the point is in quadrant II, at 135°.
Writing √−4 · √−9 = √36 = 6: the product is 2i · 3i = −6.

Key terms

Complex number: A number a + bi, where a and b are real and i² = −1. It can be plotted as the point (a, b) in a plane; real numbers are the ones with b = 0.
Imaginary unit: The number i, with i² = −1. It lets you take square roots of negative numbers: √−9 = 3i.
Complex conjugate: The conjugate of a + bi is a − bi, its mirror image across the real axis; their product, a² + b², is real. In algebra, A + B and A − B are also called conjugates, since their product is A² − B².
Complex plane: A plane for plotting complex numbers: the real part goes along the horizontal axis and the imaginary part up the vertical axis, so a + bi sits at (a, b).
Complex modulus: The distance of a complex number from 0: |a + bi| = √(a² + b²). When complex numbers are multiplied, their moduli multiply.
Complex argument: The angle from the positive real axis to a nonzero complex number in the complex plane. Adding a full turn gives the same number, and 0 has no argument.
De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ: raising to a whole-number power multiplies the angle. Run backwards, it gives the n nth roots of a complex number, equally spaced around a circle.
Polar coordinates: Describing a point by its distance r from the origin and its angle θ from the positive x-axis, written (r, θ). One point has many descriptions, because adding 2π to θ lands in the same place.

Work through an example

For z₁ = 3 + 2i and z₂ = 1 − 4i, find z₁z₂ and z₁/z₂ in the form a + bi.

Multiply and divide complex numbers →

Write a complex number in polar form →

Find powers and roots with De Moivre’s theorem →

Solve a quadratic with complex roots →

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