Complex Number Polar Form Calculator
Convert a complex number between rectangular and polar form.
Enter your values
Enter the real part a you are working with.
Enter the imaginary part b you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- 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)
At a glance
| What it does | Convert a complex number between rectangular and polar form. |
|---|---|
| Category | Math |
| Inputs needed | Real part a, Imaginary part b |
| Main output | Modulus (r) |
| Formula | r = sqrt(a^2+b^2), θ = atan2(b,a) |
| Cost | Free — no sign-up, no download |
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.
Inputs explained
- Real part a
- Enter the real part a you are working with.
- Imaginary part b
- Enter the imaginary part b you are working with.
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.
What do I need to enter into the Complex Number Polar Form Calculator?
Just 2 values: real part a, imaginary part b. Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Complex Number Polar Form Calculator work out the 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). It applies the formula r = sqrt(a^2+b^2), θ = atan2(b,a) and shows the working so you can check each step by hand.
Is the Complex Number Polar Form Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.