
# 2.1 Deﬁnition of complex numbers

We begin with the following deﬁnition.

Definition 2.1.1.  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.

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