Complex Phase Calculator

Calculator Form

Use rectangular or polar input. Large screens show three columns, medium screens show two, and mobile shows one.

Input Mode

Real Part (a)

Imaginary Part (b)

Magnitude (r)

Angle Value (θ)

Polar Input Unit

Output Unit

Principal Argument Range

Decimal Places

Example Data Table

Input Form	Value	Magnitude	Principal Phase	Quadrant / Axis
Rectangular	3 + 4i	5.000000	53.130102°	Quadrant I
Rectangular	-5 + 5i	7.071068	135.000000°	Quadrant II
Rectangular	-2 - 2i	2.828427	-135.000000°	Quadrant III
Rectangular	6 - 6i	8.485281	-45.000000°	Quadrant IV
Polar	10∠210°	10.000000	-150.000000°	Quadrant III

Formula Used

Rectangular form: For z = a + bi, the phase or argument is:

arg(z) = atan2(b, a)

Magnitude:

|z| = √(a² + b²)

Polar conversion:

a = r cos(θ) and b = r sin(θ)

Principal argument normalization:

The calculator normalizes the raw angle into either (-π, π] or [0, 2π).

Coterminal angles: Any valid phase can be written as θ + 2kπ in radians or θ + 360k in degrees, where k is any integer.

How to Use This Calculator

Choose Rectangular if you know the real and imaginary parts.
Choose Polar if you know the magnitude and angle.
Pick the polar input unit when entering an angle directly.
Select the output unit you want for the final phase.
Choose the principal range for your preferred branch.
Set decimal places for cleaner or more precise output.
Press Calculate Phase to see the result block above the form.
Use the CSV and PDF buttons to save the result summary.

Frequently Asked Questions

1) What does the complex phase represent?

The complex phase is the angle made by the complex number’s vector with the positive real axis. It shows direction in the complex plane and is often written as the argument of the number.

2) Why does this calculator use atan2 instead of atan?

The atan2 function correctly handles all quadrants and axis cases. A plain arctangent can lose sign information and return the wrong phase when the real part is negative or zero.

3) What is the principal argument?

The principal argument is the normalized phase chosen from one standard interval. This calculator supports (-π, π] and [0, 2π) so you can match your convention.

4) Why is the phase undefined for zero?

The zero complex number has no direction from the origin, so no unique angle exists. That is why the calculator stops and shows a validation message for zero magnitude or 0 + 0i.

5) Can I enter negative magnitude in polar form?

Yes. The calculator converts a negative magnitude into an equivalent positive magnitude and shifts the angle by π radians or 180 degrees before normalizing the principal phase.

6) What are coterminal angles?

Coterminal angles describe the same direction in the complex plane. They differ by full turns, which means adding or subtracting 2π radians or 360° repeatedly.

7) What does the Plotly graph show?

The graph plots the complex number as a vector from the origin to the point (a, b). It also draws a small phase arc so the angle can be understood visually.

8) When should I use degrees or radians?

Use degrees when you want familiar angle values like 45° or 135°. Use radians when working with calculus, signal processing, engineering formulas, or theoretical mathematics.