Absolute Value - Definition and Properties - Complex Numbers

Complex Numbers

Since the complex numbers are not ordered, the definition given above for the real absolute value cannot be directly generalized for a complex number. However the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalized. The absolute value of a complex number is defined as its distance in the complex plane from the origin using the Pythagorean theorem. More generally the absolute value of the difference of two complex numbers is equal to the distance between those two complex numbers.

For any complex number

where x and y are real numbers, the absolute value or modulus of z is denoted | z | and is given by

When the complex part y is zero this is the same as the absolute value of the real number x.

When a complex number z is expressed in polar form as

with r ≥ 0 and θ real, its absolute value is

.

The absolute value of a complex number can be written in the complex analogue of equation (1) above as:

where is the complex conjugate of z.

The complex absolute value shares all the properties of the real absolute value given in equations (2)–(11) above.

Since the positive reals form a subgroup of the complex numbers under multiplication, we may think of absolute value as an endomorphism of the multiplicative group of the complex numbers.