Complex Number Polar Form Calculator
Convert a complex number between rectangular (a + bi) and polar (r∠θ) form.
- Free to use
- Accurate results
- No registration required
- Works on all devices
Choose a direction and enter either the real/imaginary parts or the magnitude/angle to convert a complex number's form.
Your result
5
Magnitude (r)
Angle (θ, degrees)53.13
Real part3
Imaginary part4
AI explanation
Formula
To polar: r = √(a² + b²), θ = arctan(b ÷ a). To rectangular: a = r·cos(θ), b = r·sin(θ)Worked example
3+4i to polar form
| Field | Value |
|---|---|
| Convert | to-polar |
| Real part | 3 |
| Imaginary part | 4 |
| Magnitude (r) | 5 |
| Angle (θ, degrees) | 53.13010235 |
| Real part | 3 |
| Imaginary part | 4 |
Assumptions
- The angle is measured counterclockwise from the positive real axis, in degrees.
Frequently asked questions
What is polar form?
A way to represent a complex number by its distance from the origin (magnitude, r) and the angle from the positive real axis (θ), instead of separate real and imaginary parts.
How do I convert rectangular to polar form?
Magnitude r = √(a² + b²) (the Pythagorean theorem); angle θ = arctan(b ÷ a), adjusted for the correct quadrant.
Why do the input fields change based on direction?
Converting to polar form needs the real and imaginary parts; converting back to rectangular form needs the magnitude and angle instead — so only the relevant fields are shown.