Theory

Theory 1

A complex number $z=x+iy$ can be represented in the plane as the point with Cartesian coordinates $(x,y)$. The coefficient of “$i$” determines the vertical coordinate, and the coefficient of “$1$” determines the horizontal coordinate.

Let us be given a complex number $z=a+bi$.

The “real part” and “imaginary part” of $z$ can be extracted with designated functions:

$$\begin{array}{c}\mathrm{Re}(z)=a,\mathrm{Im}(z)=b,\text{for }z=a+bi\\ \\ z=\mathrm{Re}(z)+\mathrm{Im}(z)i\end{array}$$

The polar data (radius and angle) have special names and notations for complex numbers:

$$\begin{array}{cc}r& =\sqrt{a^2+b^2}\\ & \\ & =|z|\\ & \\ & =\text{“modulus” of }z\end{array}\begin{array}{cc}\theta & ={\tan }^{-1}(b/a)\stackrel{?}{+}\pi \\ & \\ & =\mathrm{Arg}(z)\\ & \\ & =\text{“argument” of }z\end{array}$$

Using this notation, we see that product with the conjugate gives square of modulus:

$$z\bar{z}=|z{|}^2$$