Physics 214
1/23/07
Notes on Complex Numbers
Let
a, b, x, y, r, φ
,
θ
and
ρ
be real numbers. Defne
i
as
i
2
=

1
so, For
b
n
= 0,
ib
is a purely imaginary number and
a
+
ib
is a complex number.
Any complex number
z
can be written in 2 ways (cartesian and polar) :
z
=
x
+
iy
=
re
iθ
(1)
An important identity is the Euler Identity :
e
iθ
= cos
θ
+
i
sin
θ
(2)
Combining the above 2 equations, we obtain
z
=
x
+
iy
=
re
iθ
=
r
cos
θ
+
ir
sin
θ
(3)
so we have
x
=
r
cos
θ
and
y
=
r
sin
θ
Consider the 2dimensional
xy
plane : every complex number
z
corresponds to a point in the
xy

plane. We call this
xy
plane the complex plane. Here,
x
=
Re
(
z
) is the real part oF
z
,
y
=
Im
(
z
) is the
imaginary part oF
z
,
θ
is the phase oF
z
, and
r
=
r
x
2
+
y
2
is the magnitude oF
z
.
Let us consider the addition and the multiplication oF 2 complex numbers :
z
and
w
=
a
+
ib
=
ρe
iφ
.
Using the complex plane, it is easy to see that the addition oF
z
and
w
is given by
z
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 Economics, Complex number, Imaginary unit, zw

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