Complex Numbers: A Brief Review
•
Any complex number
z
can be written in the
Cartesian form
z
=
x
+
iy
,where
x
and
y
are real and
i
is one of the square roots of

1 (the other being

i
). Here,
x
≡
Re
z
is
the
real part
of
z
and
y
≡
Im
z
is its
imaginary part
.Re
z
and Im
z
can be interpreted
as the
x
and
y
components of a point representing
z
on the
complex plane.
•
By comparing Taylor series expansions, one can verify Euler’s formula:
e
iθ
=cos
θ
+
i
sin
θ
for any real
θ
(expressed in radians). This leads to an alternative
polar form
z
=
re
iθ
for any complex number in terms of two reals:
r
≡
z
≥
0isthe
magnitude
or
modulus
of
z
,and
θ
≡
arg
z
is its
argument
or
phase
.
•
We can convert between Cartesian and polar forms for
complex number
z
using
r
=
p
x
2
+
y
2
x
=
r
cos
θ
θ
=atan(
y/x
)
y
=
r
sin
θ
Arguments
θ
and
θ
+2
πn
for integer
n
describe the same
point on the complex plane.
The
principal argument
Arg
z
is deﬁned to satisfy

π<
Arg
z
≤
π
.
r
x
y
z
z
Re
Im
z
θ
•
The real and imaginary parts of complex numbers are added/subtracted separately:
z
1
±
z
2
=(
x
1
+
iy
1
)
±
(
x
2
+
iy
2
)=(
x
1
±
x
2
)+
i
(
y
1
±
y
2
)
.
•
Complex numbers are multiplied/divided by multiplying/dividing their magnitudes
and adding/subtracting their arguments:
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 Fall '07
 Field
 mechanics, Complex Numbers, Complex number, Complex Plane

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