Algebra Know-It-ALL

352 Imaginary and Complex Numbers

Relative and absolute values

Imagine a “number reflector” plane, like a mathematical mirror, perpendicular to the

imaginary number line and passing through the point for j0, which is identical to the real

number zero. (Zero is the only real number that’s also imaginary.) The definitions of the

positive and negative imaginary numbers, as well as the definitions for “larger than” and

“smaller than,” are analogous to the definitions for the real numbers. If you go upward

on the line, the value of the imaginary number increases. If you go downward, the value

decreases.

In Fig. 21-1, the distance of an imaginary number from the point for j0 is defined as

itsabsolute value. The absolute value of an imaginary number is equal to the nonnegative

real number you get if you remove the j, and also remove the minus sign (if there is one).

To denote the absolute value of an imaginary number or imaginary-number expression, we

enclose it between vertical lines, just as we do with real numbers and real-number expressions.

For example,

|j3|= 3

and

|−j3|= 3

Larger according

to the traditional

definition

Smaller according

to the traditional

definition

Smaller

negatively

Larger

positively

Smaller

positively

Larger

negatively

j 3

j 2

j

j0 = 0

• j 2


• j 3


• j

“Number

reflector” plane

Figure 21-1 The imaginary number line. The imaginary values are

defined according to the values of the real-number

multiples of j.