# 2.1: Deﬁnition of Complex Numbers

- Page ID
- 266

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

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

Both hardbound and softbound versions of this textbook are available online at WorldScientific.com.