Complex Polar Form Calculator

Enter Complex Number Details

Use rectangular values a and b for z = a + bi. Then calculate polar form, argument, powers, roots, and plot points.

Real Part a

Imaginary Part b

Preferred Angle Display

Decimal Precision

Normalize Angle

Power n

Root Index n

Plotly Graph

The chart below shows the original complex number, its conjugate, and the nth roots on the complex plane.

Example Data Table

Example input values and expected output patterns help learners verify direction, modulus, and argument placement.

Real Part	Imaginary Part	Modulus	Argument	Polar Form
3	4	5	53.1301°	5 ∠ 53.1301°
-5	5	7.0711	135°	7.0711 ∠ 135°
0	-7	7	270°	7 ∠ 270°
6	0	6	0°	6 ∠ 0°

Formula Used

These formulas convert rectangular form into polar form and extend results to powers and roots.

Rectangular form: z = a + bi

Modulus: r = √(a² + b²)

Argument: θ = atan2(b, a)

Polar form: z = r(cosθ + i sinθ) = r∠θ

Exponential form: z = re^iθ

Power formula: zⁿ = rⁿ(cos(nθ) + i sin(nθ))

Nth roots: z_k = r^1/n [cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)], for k = 0,1,2,...,n−1

How to Use This Calculator

Enter the real part and imaginary part of the complex number.
Select whether you prefer to view angles in degrees or radians.
Choose decimal precision for cleaner classroom or engineering output.
Set the power value n for De Moivre power expansion.
Set the root index n to generate all nth roots.
Click Calculate Polar Form to show results above the form.
Review modulus, argument, conjugate, polar form, exponential form, powers, roots, and graph.
Download the generated output as CSV or PDF.

Frequently Asked Questions

1. What does polar form represent?

Polar form expresses a complex number using magnitude and angle. It shows distance from the origin and direction from the positive real axis.

2. Why is atan2 used instead of arctan?

atan2 uses both coordinates, so it places the angle in the correct quadrant. Standard arctan alone can miss sign direction.

3. What is the modulus of a complex number?

The modulus is the distance from the origin to the point on the complex plane. It equals √(a² + b²).

4. What happens when the real and imaginary parts are zero?

The modulus becomes zero, and the angle is typically treated as zero for display. All roots also collapse to the origin.

5. Why can the same complex number have multiple angle values?

Angles differ by full rotations. Adding or subtracting 360° or 2π radians still represents the same direction on the complex plane.

6. How are powers easier in polar form?

Polar form makes multiplication cleaner. Raise the modulus to the power and multiply the angle by that same power.

7. Why are nth roots spread evenly around a circle?

Each root adds an equal angle step of 2π/n. This distributes the roots symmetrically on a circle with radius r^(1/n).

8. Can I use this calculator for homework checking?

Yes. It is useful for checking conversions, verifying quadrant logic, testing powers and roots, and reviewing graph placement visually.