mat 3730 complex variables section 1.1 the algebra of complex numbers
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MAT 3730Complex Variables
Section 1.1
The Algebra of Complex Numbers
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Definitions, Notations Some materials from section 1.2
Complex Number System
In order to solve the equation
we need to expand the real number system (R) to include
12 x
1i
Definition
A Complex Number is an expression of the form
Two numbers are the same iff
Rbabiaz ,
dicbia and
dbca and
Notations
C = collection of all complex numbers
CR
biaz
Definition
bzb
aza
)Im( part,imaginary
)Re( part, real
biaz
Definition
Complex Conjugate of z
biaz
biaz
Modulus/ Absolute Value of z
)(
22
zz
baz
Operations
dicz
biaz
2
1
iadbcbdacdicbiazz
idbcazz
idbcazz
)()())((
)()(
)()(
21
21
21
Example 1
zzz
thatShow
Operations
dicz
biaz
2
1
12 2 2 2
2
z ac bd bc adi
z c d c d
Example 2
i
i
23
2
Compute
Things we take for granted…
0, ?z
zz
Properties
Geometry of Complex Numbers
We can identify z as the ordered pair (a,b).
Thus, we can represent z as the point (a,b) in the xy-plane.
biaz
y
),( ba
xa
b
Geometry of Complex Numbers
We can identify z as the ordered pair (a,b).
Thus, we can represent z as the point (a,b) in the xy-plane (Sect. 1.2)
biaz
),( ba
x
y
a
b
z
Geometry of Complex Numbers
We can also identify z as the position vector (Sect. 1.3)
biaz
ba,
x
y
a
b
ba,
Geometry of Complex Numbers
We can also identify z as the position vector (Sect. 1.3)
biaz
x
y
a
b
ba,
baz , oflength
Geometry of Complex Numbers
In these contexts, the xy-plane is referred as the Complex Plane
biaz
),( ba
xa
b
y
x
y
a
b
ba,
Example 3
0z
1e
that Prove
.0)Re(such that Let
R
zCz
Next Class
Read section 1.2