1
Complex numbers
Complex numbers can be interpreted in two
ways:
•Analitically - as ‘special numbers’
which solve polynomial equations.
•Geometrically - as points on a 2-plane
(which can be added and multiplied).
Mysterious ‘e’.
Lecture 2
1
Complex numbers
Complex numbers can be interpreted in two
ways:
•Analitically - as ‘special numbers’
which solve polynomial equations.
•Geometrically - as points on a 2-plane
(which can be added and multiplied).
Mysterious ‘e’.
Lecture 2
2
Consider a simple equation:
x
2
+ 1= 0
Wouldn’t it be nice to be able to solve it
It doesn’t have any solutions
!!!
Lets imagine that we can solve it, and lets name the
solution ‘
i’
i
2
=
-1
1. Analytical interpretation ‘intro’
z = x + i·y
where
x
and
y
are Real Numbers
3
.
v
y
i
u
x
iv
u
iy
x
)
(
yu
xv
i
yv
xu
yv
i
iyu
ixv
xu
iv
u
iy
x
2
The addition of complex
numbers
The multiplication of complex
numbers
.
Let z = x + i·y, w = u + i·v be two complex numbers
How do we add and multiply such numbers?
Can a "thing" z = x + iy be treated like a number ?
Yes
, we call it a
complex number
.
4
5
Definition
The complex conjugate of
z = a + ib
is the
number
a – i b
.
Notation:
.
Two numbers of which one is a conjugate of the other,
are called
conjugate numbers.
ib
a
z
6
Solve the equation
i
z
i
z
3
1
)
1
(
2
Let
y
i
x
z
i
y
x
i
y
x
y
ix
iy
x
iy
x
y
i
x
i
y
i
x
z
i
z
3
1
)
(
)
3
(
2
2
)
)(
1
(
)
(
2
)
1
(
2
then
Because the real and imaginary parts on the right hand side
must be equal to the real and imaginary parts on the left hand side
3
1
3
y
x
y
x
The solution is
5
,
2
y
x
Example
7
z
z
z
Im
i
2
z
z
z
Re
2
)
4
z
)
z
(
)
3
w
z
w
z
w
z
w
z
)
2
w
z
w
z
w
z
w
z
)
1
Properties
8
z = (2,1) = 2+i
Real axis
Imaginary axis
A complex number z can be viewed as a point or a position vector
in a two-dimensional Cartesian coordinate system called
the Complex Plane or Argand diagram.
1. Geometrical ‘Intro’
The point z is described by two coordinates:
• x - the Real axis coordinate
• y - the Imaginary axis coordinate
9
10
The sum of complex numbers
z+
w
(you add them like vectors)
Analitical interpretation:
z = x+iy, w = u+iv z + w = (x+u) + i(y+v)
Vector interpretation:
z = (x,y) w = (u,v) z+w = (x+u, y+v)
11
is the mirror reflection of
z
z
Re z
Im z
y
i
x
z
y
i
x
z
12
r
b
a
z
2
2
.
5
4
3
i
4
3
2
2
Definition
The modulus (or absolute value) of a complex
number
z = a + i b, is denoted by |z| or r,
Example
Find the modulus of the complex number 3 - 4i.
Similarity: 1 = 1, i = 1, 0 = 0.
Fact
A complex number is equal to zero iff its modulus is zero:
z = 0
|z| = 0.
13
Properties of the modulus (absolute value) of
z, |z|
w
z
w
z
)
2
w
z
w
z
)
2
w
z
w
z
)
1
x +iy
2
2
y
x
z
r
14
Examples
1. Draw in the complex plane the set of points 'z' which satisfy:
|z-z
0
|=R
z
0
R
Re z
Im z
A circle with centre z
0
and radius R
2. Draw in the complex plane the set of points S
S={z: |z - z
1
| = |z - z
2
|}
z
1
z
2
Line symmetrical to segment [z
1
, z
2
]
15
Fact
(i) Conjugate numbers have the same modulus:
(ii) The product of conjugate numbers is the square of the
modulus:
bi
a
z
z
z
2
z
z
z
.
b
a
bi
a
bi
a
2
2
16
TRIGONOMETRIC REPRESENTATION
OF COMPLEX NUMBERS
17
z = (2,1) = 2+ i
Real axis
Imaginary axis
The complex point
z = x + i y
can be also described
by:
• the distance
r
from the origin of the Coordinate
System
• the angle
between the vector
z = [x,y]
and the
Real Axis
r
18
THE TRIGONOMETRIC REPRESENTATION OF COMPLEX NUMBERS
(POLAR FORM)
,
z
x
cos
,
z
y
sin
0
y
x
z
2
2
Definition
The argument of a number z = x +i y, denoted by arg
z = α is any real number α which meets the conditions:
where
The argument of number 0
is undefined.
(cos
α, sin α)
α
19
ANGLES
The angle is measured in radians, not degrees.
radius
arc
an
of
length
of
measure
radian
Notation:
We say, in geometry, that an arc or curve "subtends" an angle, (stretches under).
A line or curve AC subtends angle ABC at point B.
20
360 degrees = 2 =
6.28319...
180 degrees = =
3.14159...
90 degrees = /2 =
1.5708...
60 degrees = /3 =
1.0472...
30 degrees = /6 =
0.523599...
45 degrees = /4 =
0.785398...
1 radian = 57.2957 degrees
1 degree = 0.0174532 radians
In mathematics if you write the size of an angle as a pure number,
without the degree unit marker after it, then the angle is taken to be in
radians.
So, if someone were to write down:
= 4
Then that would mean that angle has a measurement of 4
radians and not 4 degrees.
21
α in
degrees
0
0
30
0
45
0
60
0
90
0
180
0
270
0
360
0
α in
radians
0
π/6
π/4
π/
3
π/2
π 3π/
2
2π
sin α
0
1/2
1
0
-1
0
cos α
1
1/2
0
-1
0
1
/2
2
/2
3
/2
3
/2
2
22
Definition
The
main argument
of the complex number
z
is the
argument of
z
taken from the interval
( - , ].
The
main argument
is denoted by
Arg z
,
thus
- < Argz
and
arg z = Arg z + 2k ,
k =
0, 1, ...
o
o
o
o
360
k
90
i
90
i
Arg
360
k
1
0
1
Arg
k
2
2
i
2
i
Arg
k
2
1
0
1
Arg
arg
,
,
arg
,
.
/
arg
,
/
,
arg
,
Example:
i
1
/2
23
DEFINITION
The polar form (trigonomeric representation) of a complex number
z = x + i y
, is
z = r ( cos + i sin )
,
where r is the modulus r = |z| , and
is the argument of
z
.
x = r cos
,
y = r sin
x
y
r
24
i
3
z
.
2
1
3
z
r
o
o
360
k
30
k
2
6
2
1
sin
2
3
cos
6
sin
i
6
cos
2
i
3
Example
Find the polar form of
use the main argument.
The argument of the complex number i.e. the angle
α
which satisfies:
Thus:
1.
2.
The radius (modulus)
25
.
sin
cos
6
6
2
3
i
i
3
1
/6
i
3
Re
Im
26
MULTIPLICATION IN POLAR FORM
27
Theorem
Let
z
1
= r
1
(cos α
1
+ i∙sin α
1
)
and
z
2
= r
2
(cos
α
2
+ i∙sin α
2
).
Then:
.
0
z
for
)],
sin(
i
)
[cos(
r
r
z
z
)],
sin(
i
)
[cos(
r
r
z
z
2
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
Multiplying
complex numbers we
multiply their modules
and
add their arguments
.
Dividing
complex numbers we
divide their modules
and
subtract their arguments.
28
How can we ‘multiply’
points???
Imaginary axis
z
w
Real Axis
1
z · w
The angle of ‘z·w’ is + β
β
Lets start with the
‘multiplication’ of points
located on the unit circle
29
The point X is the product of A and B.
The length of X is the length of A times length
of B.
The angle of X is the angle of A plus the angle of
B.
The product of complex numbers of arbitrary length.
Points on a 2-dimension plane with addition and
multiplication defined as above form the complex
number system.
30
Theorem ( de Moivré's)
For every complex number
z = r (cos
α
+ i sin
α
)
and natural number
n = 0, 1, 2, 3, ... , the powers of
z
are given by:
z
n
= r
n
(cos n
α
+ i sin n
α
).
The above theorem is also true for negative integers.
EXAMPLE
i
2
= ( cos(/2) + i sin (/2) )
2
= {de Moivre’s} = cos + i sin = -1
i
-1
/2
31
32
)
sin
(cos
2
i
2
r
z
2
2
A formula for
cos 2, sin 2
)
sin
(cos
i
r
z
1. Let
)
cos
sin
sin
(cos
)
sin
(cos
i
2
a
r
i
r
z
2
2
2
2
2
2
then its square is
2. On the other hand from the d'Moivre's theorem:
)
sin
(cos
cos
2
2
2
2
a
r
2
r
cos
sin
sin
2
r
2
r
2
2
We equate the real and imaginary parts:
2
2
a
2
sin
cos
cos
so
cos
sin
sin
2
2
Re
Im
so
33
N-TH ROOTS OF COMPLEX
NUMBERS
34
THE n-TH ROOT OF A COMPLEX NUMBER
is a set of numbers,
it is not a function in the strict meaning of this word.
For real numbers
2
4
2
4
2
2
,
This means that
2
2
4
2
,
n
z
w
n
k
2
sin
i
n
k
2
cos
r
z
n
k
k
2
1
0
n
z
z
z
z
z
...
,
,
,
where
1
n
2
1
0
k
...
,
,
,
For complex numbers
35
1
2
1
0
n
k
...
,
,
,
The roots
0
, ....,
n-1
are the vertices of a
regular n – gon inscribed in a circle of radius 1
and centre at (0, 0).
d'Moivre's theorem
The set of n-th roots of 1
The number 1 is written as a complex number
1 = 1 + 0 i = 1(cos 0 + i sin 0).
The successive n-th roots of 1 are denoted by
0
,
1
, ...,
n-1
.
From the d'Moivre's theorem we obtain
n
k
2
i
n
k
2
k
sin
cos
k = 0,1,...,n-1.
REP.
n
k
2
sin
i
n
k
2
cos
r
z
n
k
36
i
1
2
1
i
i
1
2
1
1
i
1
2
1
i
i
1
2
1
1
1
8
,
,
,
,
,
,
,
The roots of degree 8 of unity:
8
1
Each root is a vertex of a regular polygon. In the above case an
octagon.
Re
Im
37
If we have some n-th root of a complex number
:
n
s
z
z
z
then the other roots can be expressed as
s
k
k
z
z
some root of unity
i
1
2
1
i
1
2
1
i
2
,
Example
1
1
1
2
,
i
1
2
1
i
2
Note
Then, because one of the square roots of
i
is:
38
EXPONENTIAL FORM
39
w
z
w
z
e
e
e
.
1
w
z
w
z
e
e
e
.
2
n
z
n
z
e
e
.
3
,
.
PROPERTIES OF THE EXPONENTIAL FUNCTION
1
e
.
4
0
40
Argz
i
i
e
z
e
r
z
The exponential form of a complex number
z = r(cos α + i sinα)
The exponential form is the ‘friendliest’ form of the complex number!!!
sin
i
cos
e
i
Let
Example: e
iπ
= cosπ + i sinπ = -1, e
iπ/2
= cos π/2 + i sin π /2 = i
41
i
)
k
2
(
i
n
i
n
i
)
(
i
i
i
)
(
i
i
i
e
e
e
)
e
(
e
e
/
e
e
e
e
42
,
.
Examples:
1) -1·i = e
i
· e
i/2
= e
i3/2
= -i
2)
Find the real and imaginary parts of
e
-
i
:
e
-i
=cos(-1) + i sin(-1) = 0.540 + i
0.841
(in
radians)
3) 1= e
2kπ
43
0.20788...
e
e
e
i
2
/
2
/
i
i
i
2
/
i
i
44
)
(
i
2
1
2
1
i
ik
k
k
i
2
2
2
)
(
i
i
2
1
e
r
r
z
z
.
7
e
r
1
z
1
.
6
e
r
z
.
5
e
r
z
.
4
e
r
z
.
3
e
r
z
.
2
r
z
.
1
45
.
9
,
)
(
:
3
4
4
z
z
z
z
C
z
S
Use the exponential representation of a complex number to
sketch the following set.
Example
3
r
9
r
r
9
r
r
9
z
9
z
r
2
4
7
4
5
4
4
4
3
4
2
4
4
0
4
k
e
1
e
e
r
e
r
1
re
z
re
z
Let
2
3
3
3
k
2
i
i
8
i
4
4
i
4
4
i
i
)
,...,
,
,
,
,
,
)
,
46
3
Re
Im
47
EULER FORMULAS
sin
cos
i
2
e
e
2
e
e
i
i
i
i
e
i
e
-i
cosφ
the sum of two vectors
e
i
+ e
-i
= 2 cosφ
48
Euler's identity is remarkable for its mathematical beauty.
Three basic arithmetic functions are present exactly once:
addition, multiplication, and exponentiation.
The identity links five fundamental mathematical constants:
•The number 0.
•The number 1.
•The number π,.
•The number e, the base of natural logarithms,
•The number i, imaginary unit of the complex numbers
.
Euler's identity
0
1
i
e