Basics of Complex Numbers (I)
1.
General
•
i
≡
√

1, so
i
2
=

1,
i
3
=

i
,
i
4
= 1 and then it starts over again (
i
5
=
i
,
i
6
=

1,.
. . ).
•
Any complex number
z
can be written as the sum of a real part and an imaginary part:
z
= [Re
z
] +
i
[Im
z
]
,
where the numbers or variables in the []’s are
real
. So
z
=
x
+
y i
with
x
and
y
real is
in this form but
w
= 1
/
(
a
+
b i
) is
not
(see ”Rationalizing” below). Thus, Im
z
=
y
, but
Re
w
6
= 1
/a
.
•
Complex Conjugate:
The complex conjugate of
z
, which is written as
z
*
, is found by
changing the sign of every
i
in
z
:
z
*
= [Re
z
]

i
[Im
z
]
so if
z
=
1
a
+
b i
,
then
z
*
=
1
a

b i
.
Note: There may be “hidden”
i
’s in the variables; if
a
is a complex number, then
z
*
= 1
/
(
a
*

b i
).
•
Magnitude:
The magnitude squared of a complex number
z
is:
zz
*
≡ 
z

2
= [Re
z
]
2

(
i
)
2
[Im
z
]
2
= [Re
z
]
2
+ [Im
z
]
2
≥
0
,
where the last equality shows that the magnitude is positive (except when
z
= 0).
Basic rule: if you need to make something real, multiply by its complex conjugate.
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 Spring '11
 Furnstahl
 Quantum Physics, Complex number, complex conjugate

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