What is the complex plane?

The complex plane (or Argand plane) is a 2D coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part of a complex number z = a + bi. Every complex number corresponds to a unique point (a, b) on this plane, connecting algebra with geometry.

Why does multiplying complex numbers rotate them?

In polar form, z = r(cosθ + i·sinθ). When you multiply two complex numbers, you multiply their moduli and add their arguments: |z1·z2| = |z1|·|z2| and arg(z1·z2) = arg(z1) + arg(z2). So multiplying by a complex number with modulus 1 is a pure rotation by its argument angle.

What are roots of unity?

The nth roots of unity are the n complex numbers that satisfy z^n = 1. They are given by e^(2πik/n) for k = 0, 1, ..., n-1, and are equally spaced around the unit circle. For example, the cube roots of unity are 1, e^(2πi/3), and e^(4πi/3) — three points forming an equilateral triangle on the unit circle.

Complex Number Visualizer

Explore complex arithmetic on the Argand plane. Drag z₁ and z₂ to see addition, multiplication, and more — geometrically.

Complex Operations

Presets

Operation

Add Subtract Multiply Divide

Drag z₁ and z₂ to change values.

Properties

z₁	--
z₂	--
z₁ polar	--
z₂ polar	--
Operation	--
Result	--

The Math Behind It

Complex Number Basics

Rectangular form: z = a + bi
Modulus: |z| = √(a² + b²)
Argument: θ = atan2(b, a)
Polar form: z = r(cosθ + i·sinθ)
Conjugate: z̄ = a - bi (reflect across real axis)
Euler: e^iθ = cosθ + i·sinθ

Operations Geometrically

Addition: parallelogram rule — place vectors tip to tail
Subtraction: add the negative — reverse z₂ then add
Multiplication: multiply moduli, add arguments
|z₁z₂| = |z₁|·|z₂|, arg(z₁z₂) = arg(z₁) + arg(z₂)
Division: divide moduli, subtract arguments
|z₁/z₂| = |z₁|/|z₂|, arg(z₁/z₂) = arg(z₁) - arg(z₂)

Roots of Unity

nth roots of unity: e^2πik/n for k = 0, 1, ..., n-1
Equally spaced points on the unit circle

Complex Number Visualizer

Complex Operations

Properties

The Math Behind It

Complex Number Basics

Operations Geometrically

Roots of Unity

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Polar Coordinates

2D Vectors

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