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

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

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

What is ?

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

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

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.:

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

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

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 8 + 3 +6i -4i = 11+2i i(3 +3 -3i +3i) = 6i 1/5 (7+6i) 1 -1 +i +i = 2i

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,

We show complex numbers on an Argand diagram Imaginary Real

Complex Roots of Equations Quickly Solve

Complex Roots of Equations Now Solve

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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