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 $(r, \theta)$: \begin{eqnarray}\label{par} x= r\cos \theta, \quad y=r\sin \theta \quad (r\geq 0) \end{eqnarray} Hence the complex number $z$ can be written in the alternative polar form: \begin{eqnarray}\label{polar} z=x+iy=r(\cos \theta + i \sin \theta). \end{eqnarray}

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 $$z_1\pm z_2=(x_1\pm x_2)+i(y_1\pm y_2)$$ and $$z_1z_2=(x_1+iy_1)(x_2+iy_2)=(x_1x_2-y_1y_2)+i(x_1y_2+x_2y_1).$$ Now, we note that $$z\overline{z}= (x + iy)(x − iy) = x^2+y^2=|z|^2.$$ This fact is useful for division of complex numbers, \[ \frac{z_1}{z_2}=\frac{x_1x_2+y_1y_2}{x_2^2+y_2^2}+i\frac{x_2y_1-x_1y_2}{x_2^2+y_2^2}. \] It is easily shown that the commutative, associative, and distributive laws of addition and multiplication hold. Geometrically speaking, addition of two complex numbers is equivalent to that of the parallelogram law of vectors.

Use the following applet to practice the basic operations with complex numbers (in cartesian or polar form). Select an operation from the list, drag points and calculate the result. Check your results by entering the real and imaginary parts in the input boxes at the bottom.

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:
\begin{eqnarray*}
\textbf{Re}(z)=\frac{1}{2}\left(z+\overline{z}\right)\quad\quad \textbf{Im}(z)=\frac{1}{2i}\left(z-\overline{z}\right)\quad \quad|z|=\sqrt{x^2+y^2}
\end{eqnarray*}
\begin{eqnarray*}
\tan\left(\textbf{arg}(z)\right)=\frac{\textbf{Im}(z)}{\textbf{Re}(z)}\quad \quad re^{i\theta}=r(\cos \theta +i \sin \theta)
\end{eqnarray*}
\begin{eqnarray*}
\overline{\overline{z}}=z\quad \quad \left|z_1z_2\right|=\left|z_1\right|\left|z_2\right|\quad \quad \left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|},\; (z_2\neq0)
\end{eqnarray*}
\begin{eqnarray*}
\overline{z_1\pm z_2}=\overline{z_1}\pm\overline{z_2}\quad \quad \quad \overline{z_1z_2}=\overline{z_1}\cdot \overline{z_2}\quad \quad \quad \overline{\left(\frac{z_1}{z_2}\right)}=\frac{\overline{z_1}}{\overline{z_2}},\; (z_2\neq0)
\end{eqnarray*}
\begin{eqnarray*}
\left|z_1\pm z_2\right|\leq \left|z_1\right|+\left|z_2\right| \quad \quad \quad \left|\left|z_1\right|-\left|z_2\right|\right|\leq \left|z_1\pm z_2\right|
\end{eqnarray*}
The following is called the *generalized triangle inequality*:
\begin{eqnarray*}
|z_1+z_2+\cdots +z_n|\leq |z_1|+ |z_2|+\cdots |z_n|
\end{eqnarray*}
When does equality hold?

NEXT: The Principal Argument