Exercise 1 - Standard Form
Write these Complex numbers in Standard Form a.
b.
c.
d.
Exercise 2 - Addition and Subtraction and the Complex Plane
Perform the addition or subtraction and draw the new Complex number a.
b.
c.
d.
Exercise 3 - Multiplication, Modulus and the Complex Plane
Perform the multiplication, draw the new Complex number and find the modulus a.
b.
c.
d.
e.
Exercise 4 - Powers of (1+i) and the Complex Plane
Perform the following calculations on
and state the position in the plane, how much you rotated and the modulus at each step a.
b.
c.
d.
e.
f.
g.
Exercise 5 - Opposites, Conjugates and Inverses
Find
and
(if they exist) for each of the following a.
b.
c.
d.
e.
f.
Exercise 6 - Reference Angles
Find the fractions that make the equations true a. What is
in the equation
? b. What is
in the equation
? c. What is
in the equation
?
Exercise 7- Division
Perform the division a.
b.
c.
d.
e.
Exercise 8 - Special Triangles and Arguments
Use special triangles to find a Complex number that has each of these arguments a.
b.
c.
d.
e.
f.
g.
Exercise 9 - Polar Form of Complex Numbers
Determine the Polar Form for each of these Complex numbers
a.
b.
c.
d.
e.
f.
g.
Exercise 10 - Roots of Equations
Solve for the roots of these equations a.
b.
c.
Exercise 11 - Powers of a Complex Number
Calculate the following numbers a.
b.
Expand
to find c.
d.
If 
find the explicit answer for
e.
f.
Exercise 12 - Complex Roots
a. Calculate the 4th roots of unity
b. Calculate the 6th roots of unity
c. Calculate
d. Calculate
Solutions for Exercises 1-12
Solutions for Exercise 1 - Standard Form
Write these Complex numbers in Standard Form a.
then
b.
c.
d.
Solutions for Exercise 2 - Addition and Subtraction and the Complex Plane
Perform the addition or subtraction and draw the new Complex number a.
Solutions for Exercise 3 - Multiplication, Modulus and the Complex Plane
Perform the multiplication, draw the new Complex number and find the modulus a.
modulus
modulus
modulus
modulus
modulus
Solutions for Exercise 4 - Powers of (1+i) and the Complex Plane
Perform the following calculations on
and state the position in the plane, how much you rotated and the modulus at each step 
rotation and position:
rotation to the positive y-axis at point
all of the rotations will be
radians modulus:
b.
location: halfway between the positive y-axis and negative x-axis at the point
angle:
modulus:
c.
location: on the negative x-axis at the point
modulus:
d.
location: halfway between the negative x-axis and the negative y-axis at the point
modulus:
e.
location: on the negative y-axis at the point
modulus:
f.
location: halfway between the negative y-axis and the positive x-axis at the point
modulus:
g.
location: the positive x-axis at the point
modulus:
Solutions for Exercise 5 - Opposites, Conjugates and Inverses
Find
and
(if they exist) for each of the following a.
b.
c.
d.
e.
f.
Solutions for Exercise 6 - Reference Angles
Find the fractions that make the equations true a. What is
in the equation
?
b. What is
in the equation
?
c. What is
in the equation
?
Solutions for Exercise 7 - Division
Perform the division a.
b.
c.
d.
e.
Solutions for Exercise 8 - Special Triangles and Arguments
Use special triangles to find a Complex number that has each of these arguments a.
Quadrant II with x negative and y positive a
triangle for odd multiples of
and
,
etc. any number with
b.
Negative y-axis with y negative
,
etc. any number with
and
Quadrant I with x and y both positive a
triangle with
and
,
etc. d.
Quadrant IV with
and
triangle with
and
,
etc. e.
Quadrant III with both
triangle with
and
,
etc. f.
Quadrant III equivalent to angle
with both
triangle with
and
,
etc. g.
Quadrant III equivalent to angle
with
triangle with
and
,
Solutions for Exercise 9 - Polar Form
Determine the Polar Form for each of these Complex numbers
a.
b.
c.
d.
e.
f.
g.
or
or
Solutions for Exercise 10 - Roots of Equations
Solve for the roots of these equations a.
and
or
and
solutions are
b.
Expand
Set the Real parts and the Imaginary parts of each side equal to each other
and
If we take the modulus of each side, we can obtain a expression for
Now
and
Then
and
or
so
with
or
with
solutions
and
with
or
c.
then
and
with
with
then
with
or
with
solutions
and
with
or
Solutions for Exercise 11 - Powers of a Complex Number
Calculate the following numbers a.
which is equivalent to
b.
with
and
with
or
and
Expand
to find Binomial expansion for
:
c.
d.
If find the explicit answer for
e.
f.
Solutions for Exercise 12 - Complex Roots
where
b.
and
6th Roots of Unity:
c. Calculate
because
and
Calculate the roots of these equations
Use the Complex version of the Quadratic Formula to obtain the roots to the equation
Quadratic Equation:
Quadratic Formula:
with
and
then
and
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