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2.1 Definition of complex numbers

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    22:29, 5 Dec 2015
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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.