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**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

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

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

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**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

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What is ?

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**Adding Complex Numbers**

Add real parts Adding Complex Numbers Add imaginary parts Example

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**Multiplying Complex Numbers**

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

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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 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

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**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,

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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 1) A B C D What is**

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

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**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

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**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

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**Multiple choice 4) B A C D Imaginary**

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

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**Multiple choice 5) B A C D Imaginary**

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

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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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**Other representations of complex numbers Modulus and Argument form**

Imaginary Real y x 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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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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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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Section 2-5 Complex Numbers.

Section 2-5 Complex Numbers.

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