Modulus of Complex Numbers

The modulus of complex numbers represents the distance of any number from its origin, which appears to be always positive in value. Mods are denoted as ‘| |’: therefore, a function can be written as y = |x|.

This material talks about the modulus of complex numbers. You will find brief information on the concept of modulus of complex numbers, a thorough explanation on properties of modulus of complex numbers, modulus of complex numbers formula, and so on.  So, let’s start by the definition of the modulus of complex numbers.

Definition of Modulus of Complex Numbers

Modulus of the complex numbers can be defined as the absolute value, as it is the distance complex numbers from the origin and distance will always have an absolute value, hence proved. Modulus of complex numbers can be determined with the square root of the sum of the squares of the real and imaginary numbers. As we know modulus function is denoted as |y|.

Formulation of Modulus of Complex Numbers

 Modulus of a complex number  Z = a + bi can be formulated as  | z| = √(a2 + b2) can also be written as  | Z | = √ ( Re (z) )2 + ( Im (z ) ) 2,

                   | Z | = √ a2 + b2

where Re = Real numbers and

Im = Imaginary numbers.

Properties of Modulus of Complex Numbers

In this section, you will understand the modulus of complex numbers more effectively. So, let’s walk through its properties.

1. Modulus of complex numbers complex | z | and conjugate of complex numbers

 |ˉz| are equal.

  This could be denoted as| z| =|ˉz|

2. The real part of the complex number Re (z) is ≤ to the modulus of the complex numbers  | z |. These properties are as follows:

. Re (z) <  | z |

. Re (z) = | z |

3. The imaginary unit Im (z) is ≤ modulus of complex numbers  | z |. This can be denoted as, Im (z) ≤  | z |.

4. Modulus of complex number | zn |  =  | z |n, where n is an integer. This can be expressed as,  | zn |  =  | z |n.

5. The modulus of the products of the complex number I z1 z2 I is equal to the product of their individual modulus I z1 I I z2 I. This can be expressed as,  I z1 z2 I =  I z1 I I z2 I

6. Modulus of the coefficient of the complex numbers (I z1 / z2 I) is equal to the modulus of the individual coefficient (I z1 I / I z2 I). This can be written as,

        I z1 ÷ z2 I =  I z1 I ÷ I z2 I.

 7. Modulus of the sum of complex numbers (|z1 + z2 | less than or equal to the sum of individual modulus of complex numbers (I z1 I + I z2 I). Mathematical equation, I z1 + z2 I ≤  I z1 I + I z2 I.

8. Modulus of the subtraction of complex numbers (|z1 – z2|) greater than or equal to the modulus of individual subtraction of complex numbers ( ||z1| -|z2||). Mathematical equation, (|z1 – z2|) ≥  (||z1| -|z2||).

With this discussion, you must have a sound idea of complex numbers, modulus of a complex number, and properties of complex numbers. We have also defined formula and modulus calculation methods for each of them to help you out.

Conclusion

In this article describing the modulus of complex numbers, we studied the modulus of complex numbers in detail. We covered several other topics such as the functions that can be performed with help of the modulus of complex numbers, modulus of a complex number formula, and other related topics along with the solved examples. We hope you find this study material helpful for a better understanding of the modulus of complex numbers.