1. CIE Centre A-level Pure Maths
P3 Chapter 16
CIE Centre A-level Pure Maths
© Adam Gibson

2. The “depressed cubic”; other cubics can be expressed in this form.
COMPLEX NUMBERS
Girolamo Cardano:
A colourful life!
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
The “depressed cubic”;
other cubics can be expressed
in this form.

3. Cardano “stole” the method from Tartaglia. Finding x you must write:
COMPLEX NUMBERS
Cardano “stole” the method from
Tartaglia. Finding x you must
write:
but the solution
is a real number!
can be less than zero
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
Square roots of negative numbers can be useful,
just as negative numbers themselves.

4. By introducing one number
COMPLEX NUMBERS
By introducing one number
we can solve lots of new problems, and make
other problems easier.
It can be multiplied, divided, added etc. just as any
other number; but the equation above is the only
extra rule that allows you to convert between i and
real numbers.
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
Start by noticing that
So the square root of
any negative number
can be expressed in terms
of i.

5. Therefore we can solve equations like:
COMPLEX NUMBERS
Therefore we can solve equations like:
The answer is
But what is
The two types of number cannot be “mixed”.
Numbers of the form
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
are called imaginary numbers (or “pure imaginary”)
Numbers like 1, 2, -3.8 that we used before are called
real numbers.
When we combine them together in a sum we have
complex numbers.

6. COMPLEX NUMBERS
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.

7. a is the “real part” of z; Re(z) b is the “imaginary part” of z; Im(z)
COMPLEX NUMBERS
To summarize,
a and b are real numbers
a is the “real part” of z; Re(z)
b is the “imaginary part” of z; Im(z)
The sum of the two parts is called a “complex number”
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.

8. Adding and subtracting complex numbers:
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
For addition and subtraction the real and imaginary
parts are kept separate.

9. Multiplying and dividing complex numbers:
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
Notice how, for multiplication, the real and imaginary
parts “mix” through the formula i2 = -1.

10. Multiplying and dividing complex numbers:
Remember
this trick!!
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
Read through Sections
16.1 and 16.2 to make
sure you understand the basics.

11. Now there are always two solutions, albeit they can be
COMPLEX CONJUGATES
Now that we have introduced complex numbers, we can view the quadratic solution differently.
Now there are always two solutions, albeit they can be
repeated real solutions.
If the equation has no real roots, it must have two
complex roots.
If one complex root is
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
what is the other?
These two numbers are called
“complex conjugates”.

12. What are the solutions to ?
COMPLEX CONJUGATES
What are the solutions to ?
* means conjugate
If we write
Then the complex conjugate is written as
Calculate the following:
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
This will be discussed
later.

13. square a complex number z? And then square its conjugate, z*:
COMPLEX POWERS
What happens if we
square a complex number z?
And then square
its conjugate, z*:
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
Compare the two results;
they are complex conjugates!
Later we will understand
this result geometrically

14. Continuing these investigations further (you may study
COMPLEX POWERS
Continuing these investigations further (you may study
in your own time if you wish):
This will be easy to justify
later.
Examine the argument on page 228.
This is a key idea, although you don’t
have to understand the proof.
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
Non-real roots of polynomials
with real coefficients
always occur in conjugate pairs.

15. Find all the roots of the following two polynomials:
COMPLEX POWERS
Tasks
Find all the roots of the following two polynomials:
The first example can be attacked using the factor
theorem.
Examining +/-1,+/-5,+/-13 gives one root as -5.
Equating coefficients therefore gives:
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.

16. The roots are therefore -5, 2-3i, 2+3i.
COMPLEX POWERS
The roots are therefore -5, 2-3i, 2+3i.
The second example looks simpler but is, in a way,
more difficult.
First set w = z2.
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
It seems we have
to find the square
root of a complex
number!

17. Algebra is not the best way to do it, but let’s try anyway.
COMPLEX POWERS
Algebra is not the best way to do it, but let’s try anyway.
The next step is important to
understand. It is called
“equating real and imaginary
parts”.
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
Simultaneous
equations.
Let’s apply it
to our problem.

18. COMPLEX POWERS
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.

19. Special properties of complex conjugates:z
How do we know
it must be a real
number? z*
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
What is ?

20. This is a very important result.
COMPLEX CONJUGATES
This is a very important result.
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.

21. How many complex roots do the following polynomials have?
COMPLEX CONJUGATES
How many complex roots do the following polynomials
have? 10 3 5
See page 229. We always have n roots for a polynomial of degree n. If the coefficients are real numbers, then we also know that any non-real roots occur in complex conjugate pairs.
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
If 1-8i is a root of polynomial B, what are the other roots?

22. POLAR COORDINATE FORM z r
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
The modulus is the length of the line from 0+0i to the number z, i.e. r.
The argument is the angle between the positive real axis and that line, by convention we use

23. Find, to 3 s.f. the modulus and argument of the following
POLAR COORDINATE FORM
Find, to 3 s.f. the modulus and argument of the following
complex numbers:
modulus
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
To find θ we have two equations:

24. POLAR COORDINATE FORM
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.

25. Which function does this?
EXPONENTIAL FORM or
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
Which function does this?

26. So (not proof but good enough!)
EXPONENTIAL FORM
So (not proof but good enough!)
If you find this incredible or bizarre, it means you
are paying attention.
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
Substituting
gives “Euler’s jewel”:
which connects, simply, the 5 most
important numbers in mathematics.

27. We can write any complex number in this form
EXPONENTIAL FORM
We can write any complex number in this form
As before, r is the modulus and θ is the argument.
Examples:
Do you see how easy it
is to calculate powers?
Find
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.