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
This preview has intentionally blurred sections. Sign up to view the full version.
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
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '09

Click to edit the document details