Complex Numbers

A  complex number  is a number of the form  , where   and   are real numbers and   is the imaginary unit , the square root of .

In a complex number  , is called the "real part" of   and   is called the "imaginary part." If  , the complex number is a real number; if  , 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  can be plotted as the ordered pair on the complex plane.

The absolute value or modulus of a complex number can be interpreted as the distance of the point from the origin on a complex plane.

Using the Distance Formula,

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.