Operations of Complex Numbers

 

Consider the following two complex numbers:

We can apply to them the following operations, as shown:

 

Addition of Complex Numbers

The complex numbers must be in Cartesian form. We add the real parts together and the imaginary parts together.

 

Subtraction of Complex Numbers

The complex numbers must be in Cartesian form. We subtract the real parts together and the imaginary parts together.

 

Multiplication of Complex Numbers

In Cartesian form:

More useful is in the polar form:

 

Division of Complex Numbers

In Cartesian form:

More useful is in the polar form:

 

Powers of Complex Numbers

According to De Moivre's therem, the nth power of a complex number is:

 

Roots of Complex Numbers

The nth root of a complex number is:

 

Natural Logarithm of Complex Numbers

The natural logarithm of a complex number is:

Where k=0, 1, 2,…  usually only used k=0.