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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 an overview)
- Real and imaginary numbers- Working with complex numbers- Different complex number representations
Lecture 4
Limits, Sequences and Partial differentiation
What is ? 9
What is ? 1
What two numbers multiply together to give -1?
What is ? 1
i 1
Complex Numbers
A Complex number (z) has Real and Imaginary part:
For example:biaz
12 ior
Test i2 i3 i4 etc
What is ? 2i
3i
4i
5i
6i
12i 1
Adding Complex Numbers Add real parts
Add imaginary parts
iz 321 iz 42
Example
21 zz
iz 3
31 zz
2312 zzz
Multiplying Complex Numbers
Multiplying by a real number
Multiplying by an imaginary number
)32(5 i
)32(3 ii
Multiplying by a Complex number
)32()3( ii
Remember 12 i
Complex Conjugate
If we have a Complex number :
Its Complex Conjugate is:
bia
bia
When a complex number is multiplied by its Conjugate, the imaginary parts cancel out e.g.:
)25)(25( ii
Dividing by a Complex number
)51(
)21(
i
i
)51()21( ii
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
)51(
)51(
)51(
)21(
i
i
i
i
)25551(
)10521(
ii
ii
)311(26
1i
Try these:
)1(6 i
)2)(34( ii
1)
2)
3)
4)
5)
6)
7)
)2()4( ii
)43(6 ii
)52(432 iii
)33)(1( iii
2)1( i
3 +10i
)1(6 i
)2)(34( ii
1)
2)
3)
4)
5)
6)
7)
)2()4( ii
)43(6 ii
)52(432 iii
)33)(1( iii
2)1( i
Try these:
3 -2i
-6 +6i
8 + 3 +6i -4i = 11+2i
i(3 +3 -3i +3i) = 6i
)2)(2(
)2)(4(
ii
ii
14
4218 ii
1/5 (7+6i)
1 -1 +i +i = 2i
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
iz 25
Complex Roots of Equations
Quickly Solve
0232 xx
Complex Roots of Equations
Now Solve
012 xx
Multiple choice Choose A,B,C or D for each of these:
What is 1)
i43 B
i23D
A i3
C i4
)1()32( ii
Multiple choice Choose A,B,C or D for each of these:
What is 2)
8B
i68D
A 10
C i610
)3)(3( ii
Multiple choice Choose A,B,C or D for each of these:
What is 3)
i22B
2D
A i31
C i
)2)(1( iii
Multiple choice
Estimate which number is represented on the Argand diagram
4)
iz 44 B
iz 22D
A iz 44
C iz 33
Imaginary
Real
z
Multiple choice
Estimate which number is represented on the Argand diagram
5)
iz 4B
iiz 26 D
A iz 25
C iz 55
Imaginary
Real
z
Other representations of complex numbers
Modulus and Argument form iz 43Imaginary
Real
r4
3
22 43 r
3
4tan
=Modulus of Z or |Z|)mod(zr
)arg(Z =Argument Z
yixz Imaginary
Real
ry
x22 yxr
x
ytan
Other representations of complex numbers
Modulus and Argument form
sinry
also:
andcosrx
yixz so:
)sin(cos irz
yixz
Modulus and Argument form
)sin(cos irz
Q) Covert z=1+i to mod and arg format
)mod(z
)arg(z
z
22 yxr
x
ytan
Other representations of complex numbers
Exponential form
)sin(cos iriyxz
irez
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!
Mathlab week 1 task
Week 2 task is due for a week today: Use “James” this week
Multiple choice Choose A,B,C or D for each of these:
Differentiate the following wrt x:
1) xxf 3sin)(
xxf sin3)(' B
xxf 3cos3)(' D
A xxf 3cos3
1)('
C xxf 3sin3
1)('
Multiple choice Choose A,B,C or D for each of these:
Differentiate the following:
2) xxf 10ln)(
10ln)(' xfA B
0)(' xfC D
10
1)('x
xf
xxf
10
1)('
Multiple choice Choose A,B,C or D for each of these:
Differentiating more complex functions
3)
A B
C D
xxf
2
1)('
2
1)(x
xf
xxf
2)('
3
2)('
xxf
3
1)('
xxf
Multiple choice Choose A,B,C or D for each of these:
Differentiating more complex functions
4) xxexf )(
A B
C D
xxexf x )('
xx exexf )('
)1()(' xx eexf
xexf x )('
Multiple choice Choose A,B,C or D for each of these:
Differentiate the following wrt x:
5) xxxf sin)(
xxxxf sincos)(' A xxxxxf cossin)(' B
xxxf sin)(' C xxxf sin)(' D
Multiple choice Choose A,B,C or D for each of these:
Differentiate the following wrt x:
6)
)3cos()3sin(9)(' xxxf B
)3cos()3sin(3)(' xxxf D
A xxf 3cos6)('
C )3sin().3cos(6)(' xxxf
xxf 3cos)( 2
Multiple choice Choose A,B,C or D for each of these:
Differentiating more complex functions
7)
A B
C Dxx exexf )('
x
exf
x
)(
2)('
x
exexf
xx 2
)('x
xeexf
xx
2)('
x
xeexf
xx
2
..
v
dvuduv
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