The set of complex numbers, C, can be considered as a mathematical field, and some basic operations can be performed within this structure. Before delving into these operations, let’s start by explaining the mathematical term “field.”

The term “field” refers to a mathematical structure that satisfies a certain set of rules (axioms). These rules specify how addition and multiplication operations should be performed and what properties these operations will have. For a set to be a field, it must satisfy all of these axioms. We will discuss the term “field” in more detail later, but for now, this explanation will suffice for this topic.

After stating that the set of complex numbers can be considered a mathematical field, let’s discuss the basic operations that can be performed.

c. Identity Elements: The identity element for addition is 0 + 0i, i.e., zero. Adding this number to any complex number will result in the original complex number. The identity element for multiplication is 1 + 0i, i.e., one. Multiplying this number by any complex number will again result in the original complex number.

a. Addition and Subtraction: Any two complex numbers can be added or subtracted. These operations are performed independently on the real and imaginary components of the numbers.

b. Multiplication and Division: Any two complex numbers can be multiplied. In addition, division can be performed on any non-zero complex number. This operation is carried out using the conjugate and the absolute value.