Complex Argument Calculator

This complex argument calculator finds the argument, or phase angle, of a complex number a plus bi. Enter the real and imaginary parts and the calculator returns the argument in degrees and radians, along with the modulus. The argument is the angle the number makes with the positive real axis, measured anticlockwise, and the calculator uses the two argument arctangent so the angle comes out in the correct quadrant. Together with the modulus, the argument gives the polar form of a complex number, which makes multiplication, division and powers much easier. The formula, a worked example and the assumptions are below.

Real part a

Imaginary part b

Argument (radians)

Modulus

The formula

The argument of a + bi is atan2(b, a), the angle from the positive real axis. Using atan2 rather than plain arctan gives the correct quadrant. It is usually taken in the range minus 180 to 180 degrees.

Worked example

For 3 + 4i, the argument is atan2(4, 3), about 53.13 degrees or 0.927295 radians, and the modulus is 5. Enter 3 and 4 to confirm.

Frequently asked questions

What is the argument of a complex number?

The angle it makes with the positive real axis on the complex plane, found with atan2 of the imaginary over the real part.

Why use atan2?

It returns the correct quadrant and works when the real part is zero, unlike plain arctan.

What range is it in?

The principal value runs from minus 180 to 180 degrees, or minus pi to pi radians.

Who this calculator is for

This calculator is for algebra students and anyone needing a quick, reliable result.

What this calculator assumes

You enter valid numbers.
The standard method is applied.
Results are rounded for display.

Formula and sources

Related calculators

Quadratic Formula Calculator: solve a quadratic.
Scientific Calculator: general maths.
Matrix Calculator: matrix operations.

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