1. CIVE2602 - Engineering Mathematics 2.2
Limits, Sequences and Partial differentiation
Lecture 4
Intro to Complex Numbers
(does not fit into Limits and Sequences,
but important you have an overview)
Real and imaginary numbers
Working with complex numbers
Different complex number representations
Lecturer: Dr Duncan Borman

2. What two numbers multiply together to give -1?
What is ?
What is ?
What two numbers multiply together to give -1?

3. A Complex number (z) has Real and Imaginary part:
Complex Numbers
What is ? or
A Complex number (z) has Real and Imaginary part:
For example:
Test i2 i3 i4 etc

4. What is ?

5. Adding Complex Numbers
Add real parts
Adding Complex Numbers
Add imaginary parts
Example

6. Multiplying Complex Numbers
Remember
Multiplying by a real number
Multiplying by an imaginary number
Multiplying by a Complex number

7. Complex Conjugate
If we have a Complex number :
Its Complex Conjugate is:
When a complex number is multiplied by its Conjugate, the imaginary parts cancel out e.g.:

8. Dividing by a Complex number
This is a bit trickier. We need to “get rid” of the imaginary part from the bottom line.
Multiply top and bottom by the Complex Conjugate

9. Try these: 1) 2) 3) 4) 5) 6) 7)

10. 3 +10i 3 -2i -6 +6i 8 + 3 +6i -4i = 11+2i i(3 +3 -3i +3i) = 6i
Try these: 1) 2) 3) 4) 5) 6) 7)
3 +10i
3 -2i
-6 +6i
i -4i = 11+2i
i( i +3i) = 6i
1/5 (7+6i)
1 -1 +i +i = 2i

11. Why should we care about complex numbers
Why should we care about complex numbers? They allow us to describe real physical effects and phenomena. In fact there are a huge range of applications. -They turn up all over the place in physics or engineering. For example: -to describe phase differences in electrical circuits -fluid flow (2D potential flow) -stress analysis -signal processing, -image processing,

12. We show complex numbers on an Argand diagram
Imaginary
Real

13. Complex Roots of Equations
Quickly Solve

14. Complex Roots of Equations
Now Solve

15. Multiple choice 1) A B C D What is
Choose A,B,C or D for each of these: 1)
What is A B C D

16. Multiple choice 2) A B C D What is
Choose A,B,C or D for each of these: 2)
What is A B C D

17. Multiple choice 3) A B C D What is
Choose A,B,C or D for each of these: 3)
What is A B C D

18. Multiple choice 4) B A C D Imaginary
Real 4)
Estimate which number is represented on the Argand diagram B A C D

19. Multiple choice 5) B A C D Imaginary
Real 5)
Estimate which number is represented on the Argand diagram B A C D

20. Other representations of complex numbers Modulus and Argument form
Imaginary
Real 4 3
=Modulus of Z or |Z|
=Argument Z

21. Other representations of complex numbers Modulus and Argument form
Imaginary
Real y x
also:
and
so:

22. Modulus and Argument form
Q) Covert z=1+i to mod and arg format

23. The angle must be in radians!
Other representations of complex numbers
Exponential form
The angle must be in radians!
We need to cover Taylor series to see proof of this - we do this in next 2 lectures
Q) Covert z= (3+2i)(1-i) to both modulus and argument form and exponential form

24. Week 2 task is due for a week today: Use “James” this week
Mathlab week 1 task
Week 2 task is due for a week today: Use “James” this week

25. Multiple choice 1) A B C D Choose A,B,C or D for each of these:
Differentiate the following wrt x: 1) A B C D

26. Multiple choice 2) B A D C Choose A,B,C or D for each of these:
Differentiate the following: 2) A B D C

27. Multiple choice 3) B A D C Choose A,B,C or D for each of these:
Differentiating more complex functions 3) A B C D

28. Multiple choice 4) B A D C Choose A,B,C or D for each of these:
Differentiating more complex functions 4) A B C D

29. Multiple choice 5) A B C D Choose A,B,C or D for each of these:
Differentiate the following wrt x: 5) A B C D

30. Multiple choice 6) A B C D Choose A,B,C or D for each of these:
Differentiate the following wrt x: 6) A B C D

31. Multiple choice 7) B A D C Choose A,B,C or D for each of these:
Differentiating more complex functions 7) A B C D

32. Examples sheet – attempt Q1 and Q2 for tomorrow
Examples class 11am (Tuesday)
Task will be available today
Problem sheet 1 available on VLE (5%)
Hand in 27/10/08
MathLab problems –please see me at the end