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,
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 .