Some Basics in Operations with Complex Numbers
Consider two complex numbers
a
and
b
:
a
=
α
+
i
β
b
=
δ
+
i
γ
where
i
=
1
.
Addition
a
+
b
=
+
i
( )+
+
i
( )
=
+
i
+
( )
Subtraction
a
−
b
=
+
i
( )
−
+
i
( )
=
−
i
−
( )
Multiplication
ab
=
+
i
( )
+
i
( )
=
αδ
+
i
αγ
+
i
βδ
+
i
2
βγ
=
−
+
i
+
( )
since
i
2
=
−
1
.
Division
a
b
=
+
i
+
i
=
+
i
+
i
−
i
−
i
=
+
+
i
−
+
( )
2
+
2
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View Full DocumentThe complex exponential
If we make a plot of the complex number
a
in the complex plane, we have:
From this plot, we see that:
a
=
α
2
+
β
2
tan
φ
=
or, alternately,
=
a
cos
=
a
sin
and
a
=
+
i
=
a
cos
+
i a
sin
=
a
cos
+
i
sin
( )
=
a
e
i
where Euler’s identity (
e
i
=
cos
+
i
sin
) has been used in the last equality.
The complex exponential form of complex numbers often times will simplify the
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 Fall '09
 Complex Numbers, Complex number, Euler's formula, α, tan φ

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