Exercises for Chapter 2 Sections 1, 2, and 3.
 

Make all diagrams neat and accurate with the unit of length at least half an inch. It may be helpful to use graph paper and a protractor.
 

1. Let  Give the polar form (i.e.,  form) for each of the following products.

(a) CD;         (b) CE;         (c) CF;         (d) C².
 

2. Do each part of Problem 1 on the calculator (four decimal places). Compare these results with your answers to Problem 1 and explain any apparent discrepancies.
 

3. Give an alternate representation for  using a negative argument and one using a positive argument different from 
 

4. Let 

(a) Give the polar form for AB.

(b) Plot A, B, and AB.

(c) Construct S = A + B given that OASB is a parallelogram.
 

5. Let  Plot G, H, and G + H and find |G + H|, i.e., the distance form O to G + H.
 

6. Let  Plot K, L, and K + L and find |K + L|.
 

7. Let 

(a) Plot M, N, and M + N.

(b) Find M + N using complex numbers on the calculator (four decimal places).

(c) Use geometric reasoning similar to Calculator Example 2.3.2 to justify the calculator's answer.
 

8. Let 

(a) Plot M, N, and M + N.

(b) Find M + N using complex numbers on the calculator (four decimal places).

(c) Use geometric reasoning to justify the calculator's answer.
 

9. The points  for which r is any nonnegative real number and is in  form the ray OU. Give similar geometric characterizations for the following sets of points:

(a) The set R of all the points  with r any nonnegative real number and  in 

(b) The set I of all the points  with s any nonnegative real number and  in 
 

10. Can every point P of the Argand Plane be expressed as P = A + B with A in the set R of Problem 9 (a) and B in the set I of Problem 9 (b)? Explain.
 

11. Let O be the origin and U be the unity in the Argand Plane and let P, Q, and R be any complex numbers. Verify that the complex number system has each of the following properties:

(a) Commutativity of addition: P + Q = Q + P.

(b) Associativity of addition: (P + Q) + R = P + (Q + R).  HINT: See Problem 13 of Chapter 1.

(c) Additive identity: O + P = P + O = P.

(d) Additive inverse: For each P there exists a complex number N such that N + P = P + N = O.

(e) Commutativity of multiplication: PQ = QP.

(f) Associativity of multiplication: (PQ)R = P(QR).

(g) Zero multiplication: OP = PO = O.

(h) Multiplicative identity: UP = PU = P.

(i) Multiplicative inverse: For each  there exists a complex number M such that MP = PM = U.

(j) Distributive law: P(Q + R) = PQ +PR.  HINT: See Problem 14 of Chapter 1.
 

12. Let  Find the polar form for each of the following:

(a) EG;         (b) FH;         (c) FG;         (d) EH.
 

13. Do Problem 12 on the calculator and compare with the previous results.
 

14. Let E, F, G, and H be as in Problem 12. Find the polar form of:

(a) E + G;      (b) F + H;      (c) F + G;      (d) E + H.
 

15. Do Problem 14 on the calculator and compare with the previous results.

16. Let P and Q both be in the set R of Problem 9 (a).

(a) Is the product PQ also in the set R? Explain.

(b) Is the sum P + Q also in R? Explain.

(c) Is  in the set I of Problem 9 (b)? Explain.
 

17. Let  Find the polar form of a point N such that P + N = O. Show P,O, and N in a diagram.
 

18. Let 

(a) Show  in a diagram.

(b) Find  on the calculator and compare with your diagram from (a).