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Complex Numbers

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

In a complex number  z = a + b i , 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 + b i can be plotted as the ordered pair ( a , b ) on the complex plane.

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

Using the Distance Formula,

| z | = | a + b i | = ( a 0 ) 2 + ( b 0 ) 2 = a 2 + b 2

Example 1:

Plot the number 5 + 6 i on a complex plane.

The real part of the complex number is 5 and the imaginary part is 6 .

Start at the origin. Move 5 units to the left on the real axis to reach the point ( 5 , 0 ) . Now, move 6 units upward to reach the point ( 5 , 6 ) .

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 6 on the complex plane.

The real part of the complex number is 6 and the imaginary part is 0 . So, the number will lie on the real axis.

Start at the origin. Move 6 units to the right on the real axis to reach the point ( 6 , 0 ) .

Example 3:

Plot the number 4 i on the complex plane.

The real part of the complex number 4 i is zero and the imaginary part is 4 .

Start at the origin. Move 4 units down along the imaginary axis to reach the point ( 0 , 4 ) .