Complex Number Polar Form Calculator
Convert a complex number between rectangular and polar form.
Enter your values
ⓘ Live results
Results
- Modulus (r)
- 5
- Argument (degrees)
- 126.87°
- Argument (radians)
- 2.2143
- Polar form
- 5(cos 126.9° + i sin 126.9°)
Quick answer
The modulus is r = sqrt(a^2+b^2) and the argument is theta = atan2(b,a), giving the polar form r(cos θ + i sin θ).
r = sqrt(a^2+b^2), θ = atan2(b,a)
How to use the Complex Number Polar Form Calculator
- 1Enter the real part a and imaginary part b of the complex number.
- 2The modulus is the distance from the origin, found with the Pythagorean theorem.
- 3The argument is the angle from the positive real axis, found with atan2 to handle all quadrants.
Worked example
Using the values the calculator loads with:
Inputs
- Real part a-3
- Imaginary part b4
Results
- Modulus (r)5
- Argument (degrees)126.87°
- Argument (radians)2.2143
- Polar form5(cos 126.9° + i sin 126.9°)
Frequently asked questions
Why use atan2 instead of atan?
atan2 correctly determines the quadrant, avoiding sign ambiguities that plain arctan has.
What is polar form used for?
It simplifies multiplication, division and powers of complex numbers via De Moivre's theorem.
About this calculator
- Built and reviewed by James Whitfield · How we build and test calculators
- Report an error
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.