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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

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

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

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What is ?

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Adding Complex Numbers Add real parts Add imaginary parts Example

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

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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.:

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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

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

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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

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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,

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We show complex numbers on an Argand diagram Imaginary Real

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Complex Roots of Equations Quickly Solve

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Complex Roots of Equations Now Solve

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

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

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

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

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

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

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

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

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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!

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

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

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

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

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

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

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

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

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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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