Complex numbers are a generalization of the real numbers. They are useful in many physics contexts, for example in the study of oscillations and in quantum mechanics. Any calculation you can make with complex numbers, you can make without them, but the calculations will be messier. The time you spend learning how to manipulate complex numbers will pay off many times over.
Definition2.1.Complex Number.
A complex number \(z\) is an ordered pair of real numbers \(x\) and \(y\) which are distinguished from each other by adjoining the symbol \(i\) to the second number:
together with a multiplication rule described in Section 2.2.
Notation2.1.Real and Imaginary Parts of a Complex Number.
The first number, \(x\text{,}\) is called the real part of the complex number \(z\) (and denoted \(\mathrm{Re}\, z\) or \(\Re\, z\)) and the second number, \(y\text{,}\) is called the imaginary part of \(z\) (and denoted \(\mathrm{Im}\, z\) or \(\Im\, z\)). A number that has zero for its imaginary part is called real or, for emphasis, pure real. A number that has zero for its real part called imaginary, or, for emphasis, pure imaginary.
Figure2.2.A pure real number \(z_1=3\text{,}\) a pure imaginary number \(z_2=2i\text{,}\) and a more generic complex number \(z_3=2-i\text{,}\) plotted as points in the complex plane.
Graphs of Complex Numbers.
It is helpful to graph a real number on the real number line. Similarly, it can be helpful to graph a complex number on a 2-d plane where \(x\) is recorded on the horizontal axis (called the real axis) and \(y\) is recorded on the vertical axis (called the imaginary axis). This plane, shown in Figure 2.2, is called the complex plane or, sometimes, an Argand diagram.
Analogy with Two-Dimensional Vectors.
There is a powerful analogy between vectors in two dimensions and complex numbers that should be obvious from the graph of the complex plane. Every operation that you can do with two dimensional vectors has an analogue with complex numbers. But the reverse is not true; as you will see in Section 2.2, complex numbers have additional structure that comes from a multiplication rule.
