Exercises for Section 1.4.  The Geometry of Complex Numbers, Continued

Section 1.4  

Exercise 1.  Find    for the following values of  z.  

Hint.  Recall that    is the principle value of multi-valued function    and that  .
We can use the real function  ArcTan  to evaluate provided that we use the following rules:
            

1 (a).  .  
Solution 1 (a).

1 (b).  .  
Solution 1 (b).

1 (c).  .  
Solution 1 (c).

1 (d).  .  
Solution 1 (d).

1 (e).  .  
Solution 1 (e).

1 (f).  .  
Solution 1 (f).

1 (g).  .  
Solution 1 (g).

1 (h).  .  
Solution 1 (h).

Exercise 2.  Use exponential notation   to show that  

2 (a).  .  
Solution 2 (a).

2 (b).  .  
Solution 2 (b).

2 (c).  .  
Solution 2 (c).

2 (d).  .  
Solution 2 (d).

Exercise 3.  Represent the following complex numbers in polar form  .  

3 (a).  .  
Solution 3 (a).

3 (b).  .  
Solution 3 (b).

3 (c).  .  
Solution 3 (c).

3 (d).  .  
Solution 3 (d).

3 (e).  .  
Solution 3 (e).

3 (f).  .  
Solution 3 (f).

3 (g).  .  
Solution 3 (g).

3 (h).  .  
Solution 3 (h).

Exercise 4.  Show that  ,  thus completing the proof of  Theorem 1.3.  
Solution 4.

Exercise 5.  Express the following complex numbers in    form.  

5 (a).  .  
Solution 5 (a).

5 (b).  .  
Solution 5 (b).

5 (c).  .  
Solution 5 (c).

5 (d).  .  
Solution 5 (d).

5 (e).  .  
Solution 5 (e).

5 (f).  .  
Solution 5 (f).

5 (g).  .  
Solution 5 (g).

5 (h).  .  
Solution 5 (h).

Exercise 6.  Show that     iff  ,  where  c  is a positive real constant.  
Solution 6.

Exercise 7.  Let    and  .  Show that
the equation    does not hold for these specific choice of  .  
Solution 7.

Exercise 8.  Show that the equation    is true if   and .  
Also, describe the set of points that meets this criterion.  
Solution 8.

Exercise 9.  Describe the set of complex numbers for which  .  Prove your assertion.  
Solution 9.

Exercise 10.  Establish the identity  .  
Solution 10.

Exercise 11.  Show that  .  
Solution 11.

Exercise 12.  Show that  .  
Solution 12.

Exercise 13.  Show that, if  ,  then  

13 (a).  .  
Solution 13 (a).

13 (b).    when  .  
Solution 13 (b).

Exercise 14.  Let    form the vertices of a triangle as indicated in Figure 1.16.  
Show that    is an expression for the angle at the vertex  .  

                           Figure 1.16.
Solution 14.

Exercise 15.  Let  .   Show that the polar representation    can be used to denote the displacement vector from , as indicated in Figure 1.17.  

                           Figure 1.17.
Solution 15.

Exercise 16.  Show that    iff    is not a negative real number or zero.
Solution 16.

(c) 2008 John H. Mathews, Russell W. Howell