# Complex Numbers

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,  Example 1:

Plot the number on a complex plane.

The real part of the complex number is and the imaginary part is .

Start at the origin. Move units to the left on the real axis to reach the point . Now, move units upward to reach the point . 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.

Example 2:

Plot the number on the complex plane.

The real part of the complex number is and the imaginary part is . So, the number will lie on the real axis.

Start at the origin. Move units to the right on the real axis to reach the point . Example 3:

Plot the number on the complex plane.

The real part of the complex number is zero and the imaginary part is .

Start at the origin. Move units down along the imaginary axis to reach the point . 