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MATH 1200
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House Keeping
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Definitions
Definition. A complex number is an expression of the form a + bi, where
i 2 = −1 and a and b are real numbers; a is called the real part of z and is
denoted by Re(z), b is called the imaginary part of z and is denoted by
Im(z).
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Review
In an Argand diagram, a complex number z = a + bi can be represented
graphically as the vector from the origin to the point (a, b). Every complex
number
√ z = a + ib has the polar form z = r (cosθ + isinθ), where
r = a2 + b 2 and θ is the angle between real arrow and the vector from
origin to the point (a, b) in Argand diagram.
Imaginary
z
b
θ
a
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Real
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Exponential Form
First we need Euler’s Formula
e iθ = cos(θ) + isin(θ).
With Euler’s formula we can rewrite the polar form of a complex
number z = r (cos(θ) + isin(θ)),
z = r (cos(θ) + isin(θ)) = re iθ .
re iθ is called the the exponential form of z.
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Features of Polar Form
Let z1 = r1 e iθ1 and z2 = r2 e iθ2 . Then
Multiplication:
z1 z2 = (r1 e iθ1 )(r2 e iθ2 ) = (r1 r2 )e iθ1 +iθ2 = (r1 r2 )e i(θ1 +θ2 )
Inverse:
Division:
MATH 1200
1
1
1
=
= e i(−θ1 )
iθ
z1
r1
r1 e 1
z2
r2 e iθ2
r2
r2
=
= e iθ2 −iθ1 = e i(θ2 −θ1 )
iθ
1
z1
r1
r1
r1 e
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Solution to the Last Question
Question. Find all of the solution of z 3 = 8i.
Proof.
Let z = r (cos(θ) + isin(θ)). Then
z 3 = 8i ⇒ r 3 (cos(3θ) + isin(3θ)) = 8i = 8(0 + i)
⇒ r 3 = 8, cos(3θ) = 0, and sin(3θ) = 1
r3 = 8 ⇒ r = 2
π
cos(3θ) = 0 ⇒ 3θ = 2kπ + , k = 1, 2, 3
2
π
sin(3θ) = 1 ⇒ 3θ = 2kπ + , k = 1, 2, 3
2
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Proof.
π 5π 9π
, ,
6 6 6
π
π
Solutions : z1 = 2(cos( ) + isin( ))
6
6
5π
5π
z2 = 2(cos( ) + isin( ))
6
6
9π
9π
z3 = 2(cos( ) + isin( ))
6
6
r =2
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and
⇒θ=
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Imaginary
z2
z1
Real
z3
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Thank you ...
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