Free · Complex number calculators

Complex Number Trigonometric Amplitude & Phase

Obtain the radius and phase used in z = r(cos φ + i sin φ), with degree, radian and π-radian displays.

Calculator inputs

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Enter the known values in the units shown. Results update as you type.
Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
Use the worked example to check the method. Reset restores the starting fields.

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

For z = −1 + i, r = √2 and φ = 135°, giving z = √2(cos 135° + i sin 135°).

Magnitude = √(a² + b²); direction = atan2(b, a) for a nonzero vector

Enter these known values and leave the other values blank.

Use a shared scale for the two components. The magnitude uses that scale and is nonnegative.
The angle is counterclockwise from the positive horizontal or real axis, with principal values from −180° through 180° (−π through π). Equivalent directions differ by complete turns.
At the origin, magnitude is zero and direction is undefined; Bookify leaves the direction blank. atan2 uses both component signs so it preserves the quadrant.
This page calculates magnitude and angle from the two components. The captured model is forward-only; computed outputs are not additional independent inputs.

A quotient loses quadrant information and divides by zero on the vertical axis. atan2 uses both signed components.

The zero vector has no unique direction. Its length is zero regardless of which angle is assigned.

Bookify leaves the direction undefined at the zero vector while retaining its zero magnitude and the source’s principal-angle convention.