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
http://myhome.spu.edu/lauw
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Preview
Definitions, Notations Some materials from section 1.2
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Complex Number System
In order to solve the equation
we need to expand the real number system (R) to include
12 x
1i
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Definition
A Complex Number is an expression of the form
Two numbers are the same iff
Rbabiaz ,
dicbia and
dbca and
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Notations
C = collection of all complex numbers
CR
biaz
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Definition
bzb
aza
)Im( part,imaginary
)Re( part, real
biaz
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Definition
Complex Conjugate of z
biaz
biaz
Modulus/ Absolute Value of z
)(
22
zz
baz
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Operations
dicz
biaz
2
1
iadbcbdacdicbiazz
idbcazz
idbcazz
)()())((
)()(
)()(
21
21
21
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Example 1
zzz
thatShow
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Operations
dicz
biaz
2
1
12 2 2 2
2
z ac bd bc adi
z c d c d
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Example 2
i
i
23
2
Compute
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Things we take for granted…
0, ?z
zz
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Properties
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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
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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
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Geometry of Complex Numbers
We can also identify z as the position vector (Sect. 1.3)
biaz
ba,
x
y
a
b
ba,
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Geometry of Complex Numbers
We can also identify z as the position vector (Sect. 1.3)
biaz
x
y
a
b
ba,
baz , oflength
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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,
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Example 3
0z
1e
that Prove
.0)Re(such that Let
R
zCz
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Next Class
Read section 1.2