Those two opposite sides are also parallel. This proves that the quadrilateral in Fig. 21-5 is a paral-
lelogram.
Here’s an extra-credit challenge!
Whenever you add two complex numbers and diagram the process after the fashion of Fig. 21-5, you’ll get
a parallelogram, or else all four points will lie along a single straight line (a “squashed parallelogram”). If
you’re ambitious, prove this. You’re on your own. Here’s a hint: Call the two complex numbers a+jb and
c+jd, where a, b, c, and d are real numbers.
Practice Exercises
This is an open-book quiz. You may (and should) refer to the text as you solve these problems.
Don’t hurry! You’ll find worked-out answers in App. C. The solutions in the appendix may
not represent the only way a problem can be figured out. If you think you can solve a particu-
lar problem in a quicker or better way than you see there, by all means try it!
- The laws of arithmetic for real numbers also apply to imaginary numbers. On that
basis, how can we determine the value of j^0?
- What is the value of j−^2? The value of j−^4? The value of j−^6? The value of j−^8? What
happens as this trend continues?
- Determine the value of j−^1 in two ways. First, use the difference of powers law. Here’s
a hint: Note that j−^1 =j^3 −^4. Second, use the law of cross multiplication. Again, here’s a
hint: Find the value of an unknown (call it z) when 1/j=z/1.
- Using the difference of powers law and all the other things we’ve learned, determine the
values of j−^3 ,j−^5 , and j−^7. Here are some hints:
j−^3 =j^1 −^4
j−^5 =j−^1 −^4
j−^7 =j−^3 −^4
What happens as this trend continues?
- Using what we’ve learned in the chapter text and so far in this set of exercises, create a
table that shows what happens when j is raised to any integer power.
- Find the following:
(a) (4 +j5)+ (3 −j8)
(b) (4 +j5)− (3 −j8)
(c) (4 +j5)(3−j8)
(d) (4 +j5)/(3−j8)
- Find the difference between the complex conjugates (a+jb) and (a−jb). First, subtract
the second from the first. Then subtract the first from the second.
Practice Exercises 361
