Definition:Complex Modulus

Definition
Let $z = a + i b$ be a complex number, where $a, b \in \R$.

Then the (complex) modulus of $z$ is written $\cmod z$ and is defined as the square root of the sum of the squares of the real and imaginary parts:


 * $\cmod z := \sqrt {a^2 + b^2}$

The complex modulus is a real-valued function, and, as and when appropriate, can be referred to as the complex modulus function.

Also known as
The complex modulus is also known as the complex absolute value, or just absolute value.

Others use that term only for the absolute value of a real number.

The notation $\bmod z$ is sometimes seen for $\cmod z$, but on $\cmod z$ is preferred.

Some linguistic purists object to the term complex modulus on the grounds that it is not the modulus itself which is complex.

Such schools of thought prefer the term modulus of a complex number.

Also see

 * Complex Modulus is Norm, showing that the modulus defines a norm on the field of complex numbers.
 * Modulus in Terms of Conjugate

Special cases

 * Definition:Absolute Value of Real Number, as shown at Complex Modulus of Real Number equals Absolute Value