Complex number modulus z: definition, properties

In this publication, we will consider what the modulus of a complex number is, and also give its main properties.

Determining the modulus of a complex number

Let’s say we have a complex number z, which corresponds to the expression:

z = x + y ⋅ i

• x и y are real numbers;
• i – imaginary unit (i² =-1);
• x is the real part;
• y ⋅ i is the imaginary part.

The modulus of a complex number z equal to the arithmetic square root of the sum of the squares of the real and imaginary parts of that number.

Properties of the modulus of a complex number

1. The modulus is always greater than or equal to zero.
2. The domain of definition of the module is the entire complex plane.
3. Because the Cauchy-Riemann conditions are not met (relations connecting the real and imaginary parts), the module is not differentiated at any point (as a function with a complex variable).