Complex Numbers and Exponentials
Definition and Basic Operations
A complex number is nothing more than a point in the
xy
–plane. The sum and product of two complex
numbers (
x
1
, y
1
) and (
x
2
, y
2
) is defined by
(
x
1
, y
1
) + (
x
2
, y
2
) = (
x
1
+
x
2
, y
1
+
y
2
)
(
x
1
, y
1
) (
x
2
, y
2
) = (
x
1
x
2

y
1
y
2
, x
1
y
2
+
x
2
y
1
)
respectively. It is conventional to use the notation
x
+
iy
(or in electrical engineering country
x
+
jy
) to stand
for the complex number (
x, y
). In other words, it is conventional to write
x
in place of (
x,
0) and
i
in place
of (0
,
1). In this notation, the sum and product of two complex numbers
z
1
=
x
1
+
iy
1
and
z
2
=
x
2
+
iy
2
is
given by
z
1
+
z
2
= (
x
1
+
x
2
) +
i
(
y
1
+
y
2
)
z
1
z
2
=
x
1
x
2

y
1
y
2
+
i
(
x
1
y
2
+
x
2
y
1
)
The complex number
i
has the special property
i
2
= (0 + 1
i
)(0 + 1
i
) = (0
×
0

1
×
1) +
i
(0
×
1 + 1
×
0) =

1
For example, if
z
= 1 + 2
i
and
w
= 3 + 4
i
, then
z
+
w
= (1 + 2
i
) + (3 + 4
i
) = 4 + 6
i
zw
= (1 + 2
i
)(3 + 4
i
)
= 3 + 4
i
+ 6
i
+ 8
i
2
= 3 + 4
i
+ 6
i

8 =

5 + 10
i
Addition and multiplication of complex numbers obey the familiar algebraic rules
z
1
+
z
2
=
z
2
+
z
1
z
1
z
2
=
z
2
z
1
z
1
+ (
z
2
+
z
3
) = (
z
1
+
z
2
) +
z
3
z
1
(
z
2
z
3
) = (
z
1
z
2
)
z
3
0 +
z
1
=
z
1
1
z
1
=
z
1
z
1
(
z
2
+
z
3
) =
z
1
z
2
+
z
1
z
3
(
z
1
+
z
2
)
z
3
=
z
1
z
3
+
z
2
z
3
The negative of any complex number
z
=
x
+
iy
is defined by

z
=

x
+ (

y
)
i
, and obeys
z
+ (

z
) = 0.
Other Operations
The complex conjugate of
z
is denoted ¯
z
and is defined to be ¯
z
=
x

iy
. That is, to take the complex
conjugate, one replaces every
i
by

i
. Note that
z
¯
z
= (
x
+
iy
)(
x

iy
) =
x
2

ixy
+
ixy
+
y
2
=
x
2
+
y
2
is always a positive real number. In fact, it is the square of the distance from
x
+
iy
(recall that this is the
point (
x, y
) in the
xy
–plane) to 0 (which is the point (0
,
0)). The distance from
z
=
x
+
iy
to 0 is denoted

z

and is called the absolute value, or modulus, of
z
. It is given by

z

=
p
x
2
+
y
2
=
√
z
¯
z
December 17, 2007
Complex Numbers and Exponentials
1
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Since
z
1
z
2
= (
x
1
+
iy
1
)(
x
2
+
iy
2
) = (
x
1
x
2

y
1
y
2
) +
i
(
x
1
y
2
+
x
2
y
1
),

z
1
z
2

=
p
(
x
1
x
2

y
1
y
2
)
2
+ (
x
1
y
2
+
x
2
y
1
)
2
=
q
x
2
1
x
2
2

2
x
1
x
2
y
1
y
2
+
y
2
1
y
2
2
+
x
2
1
y
2
2
+ 2
x
1
y
2
x
2
y
1
+
x
2
2
y
2
1
=
q
x
2
1
x
2
2
+
y
2
1
y
2
2
+
x
2
1
y
2
2
+
x
2
2
y
2
1
=
q
(
x
2
1
+
y
2
1
)(
x
2
2
+
y
2
2
)
=

z
1

z
2

for all complex numbers
z
1
, z
2
.
Since

z

2
=
z
¯
z
, we have
z
(
¯
z

z

2
)
= 1 for all complex numbers
z
6
= 0 . This says that the multiplicative
inverse, denoted
z

1
or
1
z
, of any nonzero complex number
z
=
x
+
iy
is
z

1
=
¯
z

z

2
=
x

iy
x
2
+
y
2
=
x
x
2
+
y
2

y
x
2
+
y
2
i
It is easy to divide a complex number by a real number. For example
11+2
i
25
=
11
25
+
2
25
i
In general, there is a trick for rewriting any ratio of complex numbers as a ratio with a real denominator.
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 Exponential Function, Complex Numbers, Complex number, Exponentials

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