Convert complex numbers between rectangular form a + bi and polar/trigonometric form r(cos θ + i sin θ) = r·cis θ. Multiply and divide in polar form. Visualize on an interactive Argand diagram.
Input
Enter the complex number z = a + bi
Quick examples
Distance from origin (r ≥ 0)
Angle unit
Quick examples
Result
Multiply or divide two complex numbers using polar form rules: z₁·z₂ = r₁r₂ · cis(θ₁+θ₂) and z₁/z₂ = (r₁/r₂) · cis(θ₁−θ₂)
z₁ (sky blue)
z₂ (gold)
Result (coral)
Why Polar Form?
z₁·z₂ = r₁r₂ · cis(θ₁ + θ₂)
z₁/z₂ = (r₁/r₂) · cis(θ₁ − θ₂)
In rectangular form, multiplication requires FOIL and careful bookkeeping. In polar form, it becomes beautifully simple: multiply the moduli and add the angles.
Division is equally elegant: divide the moduli and subtract the angles. This geometric interpretation — rotation and scaling — reveals the true nature of complex multiplication.
Euler's Formula
e^(iθ) = cos θ + i sin θ
z = r·e^(iθ) = r·cis θ
Euler's formula connects the exponential function to trigonometry through complex numbers. It means every complex number on the unit circle can be written as e^(iθ), tracing a path as θ increases.
The famous identity e^(iπ) + 1 = 0 follows directly: at θ = π, cos π = −1, sin π = 0, so e^(iπ) = −1.
The cis notation is shorthand: cis θ = cos θ + i sin θ = e^(iθ). All three forms are equivalent.
Our tutors walk through polar form, Euler's formula, and De Moivre's theorem step by step — so the big picture clicks, not just the formula.