Algebra Know-It-ALL

In general, if b is any nonnegative real number, then

|jb|=b

and

|−jb|=b

To add, move upward

Think of upward distances on the imaginary number line as positive imaginary displace-
ments, and downward distances as negative imaginary displacements. If we have an imaginary
numberjb 1 and we want to add another imaginary number jb 2 to it, we first find the point on
the number line representing jb 1. Then we move up along the line by b 2 units. That will get us
to the point representing the sum of the two numbers, jb 1 +jb 2.
As an example, suppose b 1 =−3 and b 2 = 2. We start at the point for −j3 and move up
2 units. That gets us to the point for −j 3 +j2. It happens to be −j, as shown on the left side
of Fig. 21-2.
Now suppose that we start with j2 and travel upward by −3 units. We’re talking about
displacement here, not simple distance, so negatives can make sense! An upward displacement

Start here

Start here

Move upward

by 2 units

Finish here Finish here

Move upward

by-3 units

j 3

j 2

j

0

• j 2


• j 3


• j

Figure 21-2 On the left, a number-line rendition of

−j 3 +j2. On the right, a number-line

rendition of j 2 + (−j3). When we move

negatively upward, we move downward by

the equivalent distance.

The Imaginary Number Line 353