# Terminology and Notation

A *complex number* $z$ is a number that can be expressed in the form $x + iy,$ where $x$ and $y$
are
real numbers and $i$ is the imaginary unit, that is, $i^2 = -1.$ In this expression, $x$ is
the *real part* and $y$ is the *imaginary part* of the complex number.

The complex numbers, denoted by $\mathbb C,$ extend the concept of the one-dimensional number line to the two-dimensional complex plane (also known as Argand plane) by using the horizontal axis for the real part and the vertical axis for the imaginary part. The analogy with two-dimensional vectors is immediate. The complex number $x+iy$ can be identified with the point $(x, y)$ in the complex plane but also it can be interpreted as a two-dimensional vector.

It is useful to introduce another representation of complex numbers, namely polar coordinates $(r, \theta)$:

The radius $r$ is denoted by
$$r=\sqrt{x^2+y^2}=|z|$$
and naturally gives us a notion of the *absolute value* of $z,$ denoted by $|z|,$
that is, it is the length of the vector associated with $z.$
The value $|z|$ is often referred to as the *argument* (or *phase*)
of $z$ and is denoted
by $\textbf{arg}(z).$ When $z\neq 0,$ the values of $\theta$ can be found from (\ref{par})
via standard trigonometry:
$$\tan \theta = \frac{y}{x}$$
where the quadrant in which $x,$ $y$ lie is understood as given.

At this point it is convenient to introduce a special exponential function. The polar exponential is defined by $$\cos \theta +i\sin \theta = e^{i\theta}.$$ Hence equation (\ref{polar}) implies that $z$ can be written in the form $$z=r e^{i\theta}.$$ This exponential function has all of the standard properties we are familiar within elementary calculus and is a special case of the complex exponential function.

Finally, the *complex conjugate* of $z$ is defined as
$$\overline{z}=x-iy.$$

Addition, subtraction, multiplication, and division of complex numbers follow from the rules governing real numbers. Thus, noting $i^2=-1,$ we have

Some of the terminology and notation used to describe complex numbers is summarized in Figure 1.

I suggest you to make yourself comfortable with the concepts, terminology, and notation introduced thus far. To do so, try to convince yourself geometrically (and/or algebraically) of each of the following facts:

*generalized triangle inequality*: