Let’s find the absolute value of 3 −j4. In this case, a= 3 and b=−4. Squaring both of these
and adding the results gives us
32 + (– 4)^2 = 9 + 16
= 25
The positive square root of 25 is 5. Therefore, |3 −j4|= 5.
Are you confused?
Do you wonder how the set of complex numbers relates to the sets of natural numbers, integers, rational
numbers, irrational numbers, imaginary numbers, and real numbers? The answer is that every one of those
other sets is a proper subset of the set of complex numbers.
The set of complex numbers, sometimes denoted C, is the “grandmother of all number sets” as far as
most algebra is concerned. It’s as far into the universe of numbers as we’ll go. Someday, you might take
courses that take you “farther out.” Who knows? Maybe you’ll discover or invent a new realm of numbers
that nobody has worked with before.
Here’s a challenge!
Find the sum (2 +j3) and (3 +j). Then plot both of these numbers, as well as their sum, in the complex-
number plane. These three points, along with the origin, form the vertices of a quadrilateral (four-sided
geometric figure). It’s a special sort of quadrilateral. What sort? Why?
Solution
First, we should note that when j is multiplied by 1 in a complex number, there’s no need to write down
the numeral 1 after the j. That’s why the j is all alone in the second addend above. To find the sum of these
two complex numbers, we add their pure real and pure imaginary parts separately and then put the results
back together, like this:
(2 +j3)+ (3 +j)= (2 + 3) +j(3+ 1)
= 5 +j 4
Figure 21-5 shows the two complex numbers, along with their sum and the origin, as ordered pairs in the
complex-number plane.
The four points lie at the vertices of a parallelogram. Remember from geometry: A parallelogram is a
quadrilateral in which the pairs of opposite sides are parallel. To prove that the quadrilateral in Fig. 21-5 is
a parallelogram, we can show that the pairs of line segments forming opposite sides have identical slopes.
The slope m 1 of the line segment connecting (0, j 0) and (2, j 3) can be found by taking the ratio of the
difference in jb to the difference in a, like this:
m 1 =Δjb/Δa
= ( j 3 −j0)/(2− 0)
=j3/2
Real + Imaginary = Complex 359