1. Further Pure 1
Complex Numbers

2. Numbers What types of numbers do we already know?
Real numbers – All numbers ( 2, 3.15, π ,√2)
Rational – Any number that can be expressed as a fraction
– (4, 2.5, 1/3)
Irrational – Any number that can`t be expressed as a fraction. ( π ,√2, √3 + 1)
Natural numbers – The counting numbers (1, 2, 3, ….)
Integers – All whole numbers ( -5, -1, 6)
Complex Numbers (Imaginary numbers)

3. Complex Numbers
What is the √(-1)?
We define the √(-1) to be the imaginary number j. (Hence j2 = -1)
Note that lots of other courses use the letter i, but we are going to use j.
We can now use this to calculate a whole new range of square roots.
What is √(-144)?
Answer is +12j and -12j, or ±12j.

4. Complex Numbers
Now we can define a complex number (z) to be a number that is made up of real and imaginary parts.
z = x + y j
Here x and y are real numbers.
x is said to be the real part of z, or Re(z).
y is said to be the imaginary part of z, or Im(z).

5. Solving Quadratics
Use the knowledge you have gained in the last few slides to solve the quadratic equation
z2 + 6z + 25 = 0
Remember
Solution

6. Solving Quadratics

7. Addition and Subtraction
To Add and subtract complex numbers all you have to do is add/subtract the real and imaginary parts of the number.
(x1+y1j) + (x2+y2j) = x1 + x2 + y1j + y2j
= (x1 + x2)+ (y1 + y2)j
(x1+y1j) – (x2+y2j) = x1 - x2 + y1j - y2j
= (x1 - x2)+ (y1 - y2)j

8. Multiplying
Multiplying two complex numbers is just like multiplying out two brackets.
You can use the FOIL method.
First Outside Inside Last.
Remember j2 = -1
(x1+y1j)(x2+y2j) = x1x2 + x1y2j + x2y1j + y1y2j2
= x1x2 + x1y2j + x2y1j - y1y2
= x1x2 - y1y2 + x1y2j + x2y1j
= (x1x2 - y1y2) + (x1y2 + x2y1)j
What is j3, j4, j5?

9. Multiplying x1 y1j x2 x1x2 x2y1j y2j x1y2j - y1y2
Alternatively you could use the box method.
(x1+y1j)(x2+y2j) = (x1x2 - y1y2) + (x1y2 + x2y1)j x1
y1j x2
x1x2
x2y1j
y2j
x1y2j
- y1y2

10. Questions If z1 = 5 + 4j z2 = 3 + j z3 = 7 – 2j Find
a) z1 + z3 = j
b) z1 - z2 = 3 + 3j
c) z1 – z3 = j
d) z1 × z2 = j
e) z1 × z3 = j

11. Complex Conjugates The complex conjugate of
z = (x + yj) is z* = (x – yj)
If you remember the two solutions to the quadratic from a few slides back then they where complex conjugates.
z = j & z = -3 – 8j
In fact all complex solutions to quadratics will be complex conjugates.
If z = 5 + 4j
What is z + z*
What is z × z*

12. Activity
Prove that for any complex number z = x + yj, that z + z* and z × z* are real numbers.
First z + z* = (x + yj) + (x – yj)
= x + x + yj – yj
= 2x = Real
Now z × z* = (x + yj)(x – yj)
= x2 – xyj + xyj – y2j2
= x2 – y2(-1)
= x2 + y2 = Real
Now complete Ex 2A pg 50

13. Division
There are two ways two solve problems involving division with complex numbers.
First you need to know that if two complex numbers are equal then the real parts are identical and so are there imaginary parts.
If we want to solve a question like 1 ÷ (2 + 4j) we first write it equal to a complex number p + qj.
Now we re-arrange the equation to find p and q.
(p + qj)(2 + 4j) = 1

14. Division Expanding the equation gives 2p – 4q + 2qj + 4pj = 1
The number 1 can be written as 1 + 0j So
(2p – 4q) + (2q + 4p)j = 1 + 0j
Now we can equate real and imaginary parts.
2p – 4q = 1
4p + 2q = 0
Solve these equations
p = 1/10 & q = -1/5
Therefore 1 ÷ (2 + 4j) = 0.1 – 0.2j

15. Division
The second method is similar to rationalising the denominator in C1.
The 20 on the bottom comes from the algebra we proved a few slides back. (x + yj)(x – yj) = x2 + y2
Now see if you can find (3 - 5j) ÷ (2+9j)

16. Argand Diagrams
Complex numbers can be shown Geometrically on an Argand diagram
The real part of the number is represented on the x-axis and the imaginary part on the y. -3
-4j
3 + 2j
2 – 2j Im Re

17. Modulus of a complex number
A complex number can be represented by the position vector.
The Modulus of a complex number is the distance from the origin to the point.
|z| = √(x2+y2)
Note |x| = x Im y x Re

18. Modulus of a complex number
Find
a) |3 + 4j| = 5
b) |5 - 12j| = 13
c) |6 - 8j| = 10
d) | j| = 26

19. Sum of complex numbers
z1 + z 2 = Im
z1 + z2 z2 z1 Re

20. Difference of complex numbers
z2 - z1 = Im z2 z1
z2 – z1 Re

21. Sets of points in Argand diagram
What does |z2 – z1| represent?
If z1 = x1 +y1j
& z2 = x2 +y2j
Then z2 – z1
= (x2 – x1) + (y2 – y1)j
So |z2 – z1|
= √((x2 – x1)2 + (y2 – y1)2)
This represents the distance between to complex numbers
z1 & z2. Im
(x2,y2)
y2- y1
(x1,y1)
x2- x1 Re

22. Examples
Draw an argand diagram showing the set of points for which |z – 3 – 4j| = 5
Solution
First re-arrange the question
|z – (3 + 4j)| = 5
From the previous slide this represents a constant distance of 5 between the point (3,4) and z.
This will give a circle centre (3,4)
Now do Ex 2D pg 60 Re Im

23. Examples How would you show the sets of points for which:
i) |z – 3 – 4j)| ≤ 5
ii) |z – 3 – 4j)| < 5
iii) |z – 3 – 4j)| ≥ 5 Re Im Im Re Re Im