Complex Numbers

A  complex number  is a number of the form  $a+bi$ , where  $a$  and  $b$  are real numbers and  $i$  is the imaginary unit , the square root of $-1$ .

In a complex number  $z=a+bi$ , $a$ is called the "real part" of  $z$  and  $b$  is called the "imaginary part." If  $b=0$ , the complex number is a real number; if  $a=0$ , then the complex number is "purely imaginary."

We can graph a complex number on the Cartesian plane , using the horizontal axis as the real axis and the vertical axis as the imaginary axis. When we use the Cartesian plane this way, we call it the complex plane .

The complex number  $a+bi$ can be plotted as the ordered pair $\left(a,b\right)$ on the complex plane.

The absolute value or modulus of a complex number $z=a+bi$ can be interpreted as the distance of the point $\left(a,b\right)$ from the origin on a complex plane.

Using the Distance Formula,

$\begin{array}{l}\left|z\right|=\left|a+bi\right|\\ =\sqrt{{\left(a-0\right)}^2+{\left(b-0\right)}^2}\\ =\sqrt{a^2+b^2}\end{array}$

If the real part of a complex number is zero, the number lies on the imaginary axis. Similarly, if the imaginary part of a complex number is zero, the number lies on the real axis.