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CIVE2602 - Engineering Mathematics 2.2 Lecturer: Dr Duncan Borman Intro to Complex Numbers (does not fit into Limits and Sequences, but important you have.

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Presentation on theme: "CIVE2602 - Engineering Mathematics 2.2 Lecturer: Dr Duncan Borman Intro to Complex Numbers (does not fit into Limits and Sequences, but important you have."— Presentation transcript:

1 CIVE Engineering Mathematics 2.2 Lecturer: Dr Duncan Borman 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 Lecture 4 Limits, Sequences and Partial differentiation

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

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

4 What is ?

5 Adding Complex Numbers Add real parts Add imaginary parts Example

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

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

11 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 Choose A,B,C or D for each of these: What is 1) B D A C

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

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

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

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

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

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

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

23 Other representations of complex numbers Exponential form 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 The angle must be in radians!

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

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

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

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

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

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

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

31 Multiple choice 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


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