Complex Modulus Calculator

This complex modulus calculator works out the modulus, or magnitude, of any complex number written in the form a + bi, along with its argument, the two figures you need to convert a complex number into polar form. Complex numbers underpin electrical engineering, signal processing, control theory and pure maths, and the modulus is simply the complex-number equivalent of absolute value: it tells you how far the point (a, b) sits from the origin on the complex plane. To use it, enter the real part, labelled a, and the imaginary part, labelled b, using positive or negative values as needed. The calculator instantly returns the modulus as its main result, plus the argument shown two ways, in degrees and in radians, so you can use whichever unit your course or software expects. Under the hood it applies Pythagoras, taking the square root of a squared plus b squared for the modulus, and the atan2 function on b and a for the argument, which keeps the angle in the correct quadrant. The worked example on this page, 3 + 4i, shows a modulus of 5 and an argument of about 53.13 degrees, a useful check that you are reading your own numbers correctly. Results are rounded for display, so treat many decimal places as illustrative precision rather than guaranteed accuracy for extreme inputs.

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Real part a

Imaginary part b

Argument (degrees)

Argument (radians)

The formula

The modulus of a + bi is the square root of (a squared plus b squared). On the complex plane it is the straight line distance from the origin to the point (a, b). The argument is atan2(b, a).

Worked example

For 3 + 4i, the modulus is the square root of (9 + 16) = 5, and the argument is atan2(4, 3), about 53.13 degrees. Enter 3 and 4 to confirm.

Frequently asked questions

What is the modulus of a complex number?

Its distance from the origin on the complex plane, the square root of the real part squared plus the imaginary part squared.

Is it the same as absolute value?

Yes. The modulus is the complex number version of absolute value.

What is the argument?

The angle the number makes with the positive real axis, found with atan2(b, a).

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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