Operations of addition and multiplication of complex numbers are commutative, associative and distributive. If and are complex numbers, then their addition, subtraction and product are defined as:
Example 1. Calculate
Division of complex numbers
If is any complex number, then the number is called the complex conjugate of a complex number .
The pair of complex numbers and are called the pair of complex conjugate numbers. That is, this is the pair which real parts are equal, and imaginary parts are opposite real numbers.
The product of two complex conjugate numbers is a positive real number:
For the division of complex numbers we will use the rationalization of fractions. Let and be two complex numbers. Then:
Example 2. Calculate