2.1: Definition of Complex Numbers

We begin with the following definition.

Definition 2.1.1: complex numbers

The set of complex numbers C is defined as

$$\mathbb{C} = \{ (x, y) \ | \ x, y \in \mathbb{R} \}$$

Given a complex number $z = (x, y)$, we call $\text{RealPart}(z) = x$ the $\textbf{real part}$ of $z$ and $\text{ImaginaryPart}(z) = y$ the $\textbf{imaginary part}$ of $z$.

In other words, we are defining a new collection of numbers $z$ by taking every possible ordered pair $(x, y)$ of real numbers $x, y \in \mathbb{R}$, and $x$ is called the real part of the ordered pair $(x,y)$ in order to imply that the set $\mathbb{R}$ of real numbers should be identified with the subset $\{ (x, 0) \ | \ x \in \mathbb{R} \} \subset \mathbb{C}$. It is also common to use the term $\textbf{purely imaginary}$ for any complex number of the form $(0, y)$, where $y \in \mathbb{R}$. In particular, the complex number $i = (0, 1)$ is special, and it is called the $\textbf{imaginary unit}$. (The use of $i$ is standard when denoting this complex number, though $j$ is sometimes used if $i$ means something else. E.g., $i$ is used to denote electric current in Electrical Engineering.)

Note that if we write $1 = (1, 0)$, then we can express $z= (x, y)$ in $\mathbb{C}$ as

$$z=(x,y)=x(1,0) + y(0,1)=x 1+y i=x + y i.$$ It is often significantly easier to perform arithmetic operations on complex numbers when written in this form, as we illustrate in the next section.

Contributors

• Isaiah Lankham, Mathematics Department at UC Davis
• Bruno Nachtergaele, Mathematics Department at UC Davis
• Anne Schilling, Mathematics Department at UC Davis

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