Algebra Know-It-ALL

358 Imaginary and Complex Numbers

a complex-number division problem, you can use the following formula. If a,b,c, and d are

real numbers, and if c and d are not both equal to 0, then

(a+jb)/(c+jd)= (ac+bd)/(c^2 +d^2 )+j(bc−ad)/(c^2 +d^2 )

Complex conjugates

Every complex number has a sort of “mirror image.” Imagine two complex numbers that have

the same real coefficients, but where one is a sum and the other is a difference, like this:

a+jb

and

a−jb

These two numbers are called complex conjugates. When you add a+jb to its conjugate, you

get a pure real number:

(a+jb)+ (a−jb)= (a+a)+ (jb−jb)

= 2 a+j 0

= 2 a

When you multiply a+jb by its conjugate, you get another pure real number:

(a+jb)(a−jb)=a^2 −jab+jba−j^2 b^2

=a^2 −jab+jab+b^2

=a^2 +b^2

Complex conjugates show up together when you solve certain equations. You’ll see some

examples in Chap. 23.

Absolute value of a complex number

Theabsolute value of a complex number a+jbis the distance from the origin (0, j 0) to the point

(a, jb) on the complex-number plane. For a+j0 when a is positive, the absolute value is a. For

a+j0 when a is negative, the absolute value is −a. For a pure imaginary number 0 +jb where b

is positive, the absolute value is b. If b is negative, the absolute value of 0 +jb is equal to −b.

Suppose we want to find the absolute value of − 22 −j0. This is a pure real number. It is

the same as − 22 +j0, becausej 0 =0. Therefore, the absolute value of this complex number is

−(−22)= 22. What about the absolute value of 0 −j34? This is a pure imaginary number. The

value of b is −34, because 0 −j 34 = 0 +j(−34). Therefore, the absolute value is −(−34)= 34.

If a complex number a+jb is neither pure real or pure imaginary, the absolute value must be

found by going through a little arithmetic. First, we square both aandb separately. Then we add

the squares. Finally, we take the positive square root of that sum. We can write this as a formula:

|a+jb|= (a^2 +b^2 )1/2