Exercises for Chapter 2 Sections 4, 5, and 6.
 

1. Convert the following polar forms to rectangular form a + bi. Do each part both by hand (exact answer) and on the calculator (three decimal approximation) and compare results. For the calculator part, enter the number in polar form then switch to rectangular form.

2. Convert the following rectangular forms to polar form  Do each part both by hand (exact answer) and on the calculator (three decimal approximation) and compare results. For the calculator part, enter the numbers in rectangular form then switch to polar form. Give angles in radians.

(a) 1 + i;         (b) 7i;         (c)          (d) -3;         (e) -4 - 4i.
 

3. Find the rectangular form a + bi of  for

4. Do as in Problem 3 for 
 

5. Find the rectangular form for the conjugate  of P = -7 + 8i. Do by hand and on the calculator.
 

6. Find the rectangular form for the conjugate  of  where a and b are real numbers.
 

7. Let  Find both the polar and the rectangular form of the point symmetric to S with respect to:

(a) the real axis;         (b) the imaginary axis         (c) the origin

(d) the straight line through O and 1 + i.
 

8. Do the same as in Problem 7 for  instead of S.
 

9. Let  Find in both polar and rectangular form and in each case give both the exact answer and a four decimal approximation from the calculator:

(a) -A;      (b) -B      (e) AB;     (f) 
 

10. For A and B of Problem 9, give the rectangular form of A - B and plot O, A, A - B, and -B. What kind of quadrilateral are these four points the vertices of?

11. Let  Find the rectangular form of :

12. Let  Find  in rectangular form both as an exact value and as a 4 decimal calculator approximation.
 

13. Given that  use the Pythagorean Theorem to find r.
 

14. Given that  and b < 0, find b.
 

15. Explain why  for all real a and b and find |11 - 8i| both as an exact value and as a four decimal calculator approximation.
 

16. Let  Show that :

17. Let  Find in both polar and rectangular form:

(a) iA;         (b) -A;         (c) -iA;          (f) A²;         (g) A³.
 

18. Let  Find the polar form (in terms of ) for :

(a) -20 - 21i;         (b) 21 - 20i;         (c) 20 + 21i;         (d) -21 - 20i;         (e) 20 - 21i.
 

19. Let  Find  in both polar and rectangular form.
 

20. Let  Find  in both polar and rectangular form.
 

21. Given that  find in rectangular form:

22. Let  For each of the following angles find r, a, and b such that 

23. Express  in a + bi form by first converting  to polar form, raising to powers and multiplying in polar form, and finally converting back.
 

24. Do as in Problem 23 for (5 + 5i)¹¹(-7i)⁸.
 

25. Let 

(a) Find in polar form a complex number B such that B⁵ = A.

(b) Find in polar form and plot B, CB, C²B, C³B, C⁴B, and C⁵B.
 

26. Find in polar form and plot 5 fifth roots of  What kind of geometrical figure has these 5 points as vertices?
 

27. LetQ = 13, and S = P + Q.

(a) Explain why O, Q, S, P are the vertices of a rhombus (i.e., a parallelogram with all four sides equal).

(b) Explain why 

(c) Find the absolute value s of S and then find  in a + bi form.

(d) Let R be as in (c). Explain why R and -R are the 2 square roots of P.
 

28. Let Q = r, S = P + Q, and s = |S|. Explain why each of the following is true.

(a) O, Q, S, and P are the vertices of a rhombus.

(b) 

(c) 

(d) The square roots of P are 
 

29. Let  with b > 0. Use Problem 28 to show that the 2 square roots of P are 
 

30. Let  with b < 0. Use Problem 28 to show that the 2 square roots of P are 
 

31. Let  Show that the square roots of V are

where like signs are used if s > 0 and unlike signs if s < 0.
 

32. Use Problem 31 to find  in a + bi form.
 

33. Let D = B - A. See figure 14. (a) Explain why  and  have the same magnitude and hence |B - A| equals the distance between A and B.    (b) If A = u + vi and B = x + yi, show that

Figure 14

34. Let A = 5 + 2i, B = 9 + 5i, and C = 5 + 5i.

(a) Explain why  is a right angle.

(b) Find sides AC and CB of  and then use the Pythagorean Theorem to find side AB (i.e., the distance between A and B.)

(c) Find |B - A| and |B| - |A| and tell which equals the distance between A and B.
 

35. Sketch and identify the locus of all points A in the Argand Plane such that |A - i| = 3. (That is, give the graph in the Argand Plane of the equation |A - i| = 3.)
 

36. Do as in Problem 35 for each of the following equations.

(a) |A - 4| + |A - 3i| = 5.

(b) |A - 2i| - |A + 2i| = 1.
 

37. Let be two complex numbers with the same argument  Also let r > 0 and s > 0. Use similar right triangles to explain why the following are true:

38. Let P be as in Problem 27. With the calculator in rectangular coordinates key in P and press the  key. Compare to the results from Problem 27.
 

39. Let A and C be as in Problem 25.

(a) Use the calculator in degree mode to find B = A^1/5 by making use of the y^x key.

(b) Compare your answer to part (a) with the value of B from Problem 25 (a).

(c) Use the calculator to repeat the computations (but not the plots) from Problem 25 (b).
 

40. Use the calculator to repeat the computations, but not the plots, from Problem 26.
 

41. Let A, B, and C be as in Problem 34. Use the m-ABS function in the complex number menu to find the distances between points A and B, between A and C, and between B and C.