Download presentation

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

Similar presentations

Presentation is loading. Please wait....

OK

PHYS33010 Maths Methods Bob Tapper

PHYS33010 Maths Methods Bob Tapper

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on ideal gas law units Ppt on healthy diet india Ppt on forex market Ppt on area of parallelogram and triangles in real life Ppt on infosys company profile Ppt on going places by a.r.barton Make a ppt on unity in diversity and organic farming Ppt on natural numbers vs whole numbers Download ppt on transformation of energy Ppt on viruses and anti viruses