Complex Modulus & Argument Calculator
Convert a complex number a + bi to polar form: its modulus |z| and its argument (phase angle) in both radians and degrees.
How it works
|z| = √(a² + b²) · arg(z) = atan2(b, a)
The modulus is the distance from the origin to the point (a, b) in the complex plane — a straight application of the Pythagorean theorem. The argument is the angle that ray makes with the positive real axis, computed with atan2 so it lands in the correct quadrant across the full −180° to 180° range. Together they give the polar form r·(cos θ + i·sin θ), the natural setting for multiplication and powers.
Worked example
For 3 + 4i, the modulus is √(3² + 4²) = √25 = 5. The argument is atan2(4, 3) ≈ 0.9273 rad ≈ 53.13°, so the polar form is 5·(cos 53.13° + i·sin 53.13°).
Frequently asked questions
Why use atan2 instead of arctan(b/a)?
Plain arctan(b/a) loses the sign information and cannot tell which quadrant the number is in — it maps −1−i and 1+i to the same angle. atan2(b, a) inspects both parts separately and returns the correct angle anywhere in the plane.
What range does the argument fall in?
This calculator returns the principal value in (−180°, 180°], i.e. (−π, π] radians. You can add or subtract full turns (360°) to reach any coterminal angle.
What is the argument of a positive real number?
Zero. A positive real number lies on the positive real axis, so its argument is 0°. A negative real number has an argument of 180°.
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This calculator is for educational purposes. Double-check important results before acting on them.