Modulus And Argument Calculator

Work out modulus and argument instantly with clear inputs, formula shown and shareable results.

About Modulus And Argument Calculator

The modulus is the distance from the origin in the Argand plane and the argument is the angle from the positive real axis. Using atan2 gives the principal argument in (−180°, 180°] with the correct quadrant.

Formula

Modulus and argument

|z| = √(a² + b²), arg z = atan2(b, a)

Worked example

With the default values below, Modulus And Argument Calculator returns Modulus |z| of 1.99999999. Change any input above and every figure here is recalculated in your browser.

Inputs used

Real part: -1
Imaginary part: 1.732051

Results

Modulus |z|: 1.99999999
Argument: 120.000000 °
Argument in radians: 2.09439510
Quadrant in the Argand plane: Second quadrant

Step by step

1Values used
2Modulus and argument
3Modulus |z|= 1.99999999
4Argument= 120.000000 °
5Argument in radians= 2.09439510
6Quadrant in the Argand plane= Second quadrant

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For −1 + √3·i?

The modulus is 2 and the argument is 120°, since the point lies in the second quadrant.

What is the principal argument?

The representative in (−π, π]. Adding any multiple of 2π gives the same complex number.