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Exercises for Chapter II Sections 1 and 2
For all problems calling for a graph, 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. Do NOT use a calculator for any of these problems, give only exact answers. (See Preface)
1. Let 
Give the polar form (i.e., 
form) for each of the
following products.
(a) CD; (b) CE; (c) CF; (d) C2.
2. Give an alternate representation for 
using a negative argument and one using a positive argument different
from 
3. 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.
4. Do as in problem 3 with 
and 
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 
Plot M, N, and M + N.
8. Let 
Plot M, N, and M + N.
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 
Find the polar form of a point N such that P + N = O. Show P,O, and N in a diagram.
12. Let P be as in Problem 11. Find the polar form of a point M such that PM = U. Show P, U, and M in a diagram.
13. 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 15 of Chapter 1.
14. Let P, Q, S, T be four complex numbers. Use the results of Problem 13 to show that:
(a) (P + Q)2 = P2 + 2PQ + Q2.
(b) (S + T)(S - T) = S2 - Q2.
(c) (P + Q)(S + T) = PS + QS + PT + QT.
15. Let 
Find the polar form of:
(a) EG; (b) FH; (c) FG; (d) EH.
16. Let E, F, G, and H be as in Problem 15. Find the polar form of:
(a) E + G; (b) F + H; (c) F + G; (d) E + H.
17. 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.
18. Let 
Show 
in a diagram.
19. Let 
(a) Verify that 
(b) Find in polar form a complex number B such that B5 = A.
(c) Find in polar form and plot B, CB, C2B, C3B, C4B, and C5B.
(d) Verify that 
20. Find in polar form and plot 5 fifth roots of 
What kind of geometrical figure has these 5 points as
vertices?
21. Find in polar form and plot 7 seventh roots of 
22. Let n be an integer greater than 1 and let 
be any complex number. You may assume
(a) Find the complex number F with the smallest possible positive argument such that F n = U.
(b) Find a complex number E such that E n = D.
(c) Verify that the complex numbers EF k, for k = 0, 1, 2, ..., n - 1 are distinct nth roots of D.
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