Chapters

- Exercise 1 - Standard Form
- Exercise 2 - Addition and Subtraction and the Complex Plane
- Exercise 3 - Multiplication, Modulus and the Complex Plane
- Exercise 4 - Powers of (1+i) and the Complex Plane
- Exercise 5 - Opposites, Conjugates and Inverses
- Exercise 6 - Reference Angles
- Exercise 7- Division
- Exercise 8 - Special Triangles and Arguments
- Exercise 9 - Polar Form of Complex Numbers
- Exercise 10 - Roots of Equations
- Exercise 11 - Powers of a Complex Number
- Exercise 12 - Complex Roots
- Solutions for Exercises 1-12
- Solutions for Exercise 1 - Standard Form
- Solutions for Exercise 2 - Addition and Subtraction and the Complex Plane
- Solutions for Exercise 3 - Multiplication, Modulus and the Complex Plane
- Solutions for Exercise 4 - Powers of (1+i) and the Complex Plane
- Solutions for Exercise 5 - Opposites, Conjugates and Inverses
- Solutions for Exercise 6 - Reference Angles
- Solutions for Exercise 7 - Division
- Solutions for Exercise 8 - Special Triangles and Arguments
- Solutions for Exercise 9 - Polar Form
- Solutions for Exercise 10 - Roots of Equations
- Solutions for Exercise 11 - Powers of a Complex Number
- Solutions for Exercise 12 - Complex Roots

## 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

e. Use the Complex version of the Quadratic Formula to obtain the roots to the equation

## 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.

b.

c.

d.

## Solutions for Exercise 3 - Multiplication, Modulus and the Complex Plane

Perform the multiplication, draw the new Complex number and find the modulus

a.

modulus

b.

modulus

c.

modulus

d.

modulus

e.

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

a.

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 and

b.

*Negative y-axis* with *y *negative

, etc. any number with and * *

c.

*Quadrant I *with *x *and *y* both positive

a triangle with and

, etc.

d.

*Quadrant IV* with and

a triangle with and

, etc.

e.

*Quadrant III *with both and

a triangle with and

, etc.

f.

*Quadrant III* equivalent to angle with both and

a triangle with and

, etc.

g.

*Quadrant III *equivalent to angle with and

a 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

6th Roots of Unity:

c. Calculate

because

and

Calculate the roots of these equations

d.

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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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