z2 = z2

1 + i

i 1

1 i 3

3 + i

2i

4i

1

3

2i 2

2 2i

2 cos 6 + i sin 6

4 cos 2 3 i sin 2 3

cos 3 2 + i sin 3 2

2 cos 4 + i sin 4

cos i sin

5(cos 0 + i sin 0)

2 ei/4

3 ei/2

5(cos 20 + i sin 20)

7(cos 110 i sin 1100)

1 z + 1

1 z i

1 + z 1 z

z/z

Prove that the conjugate of the quotient of two complex numbers is the quotient of the conjugates. Also prove the corresponding statements for difference and product. Hint: It is easier to prove the statements about product and quotient using the polar coordinate rei form; for the difference, it is easier to use the rectangular form x + iy

2i 1 i 2

2+3i 1 i

z z

(1 + 2i) 3

3i i 3

5 2i 5+2i

(2 3i) 4

25 3+4i

1 + i 1 i 5

x + iy = 3i 4

2ix +3= y i

x + iy = 0

x + iy = 2i 7

x + iy = y + ix

x + iy = 3i ix

(2x 3y 5) + i(x + 2y + 1) = 0

(x + 2y + 3) + i(3x y 1) = 0

(x + iy) 2 = 2ix

x + iy = (1 i) 2

(x + iy) 2 = (x iy)

x + iy x iy = i

(x + iy) 3 = 1

x + iy +2+3i 2x + 2iy 3 = i + 2

|1 (x + iy)| = x + iy

|x + iy| = y ix

|z| = 2

Re z = 0

|z 1| = 1

|z 1| < 1

z z = 5i

angle of z = 2

Re (z2 )=4

Re z > 2

|z + 3i| = 4

|z 1 + i| = 2

Im z < 0

z + 1| + |z 1| = 8

z2 = z2

z2 = z 2

Show that |z1 z2| is the distance between the points z1 and z2 in the complex plane. Use this result to identify the graphs in Problems 53, 54, 59, and 60 without computation.

Find x and y as functions of t for the example above, and verify for this case that v and a are correctly given by the method of the example.

Find v and a if z = (1 it)/(2t + i).

Find v and a if z = cos 2t + i sin 2t. Can you describe the motion?