Information Technology Reference
In-Depth Information
fact, reliable navigation is based on reliable and efficient methods of trigonomet-
ric computations (for the angles to follow along a navigation trajectory). Therefore,
logarithmic tables of sine, and cosine values were essential computational tools for
the geographic revolution of that time.
5.4.2
Polar Representation of Complex Numbers
Euler's identity suggests a geometrical interpretation of complex numbers in the
plane. Firstly the French mathematician Argand [166], and then Gauss represented
a number
a
ib
as the point of the Cartesian plane with abscissa
a
(on the real line)
and ordinate
b
(on the imaginary line). In this way, the complex number cos
x
+
i
sin
x
is a point on the unitary circle centered in the origin of the Cartesian plane where
the radius connecting it to the origin forms an angle
x
with the real line. This means
that, according to Euler's identity,
e
ix
corresponds to this point, and any point of
the complex plane has the
exponential
or
polar
representation
+
e
ix
,where
is the
distance between the point and the origin, called the
module
of the number, while
x
is the angle that the radius connecting the point to the origin forms with the real
axis. The following identity is a consequence of the previous analysis, where arcsin
is the inverse function of sin, that is arcsin
x
is the angle
ρ
ρ
α
such that sin
(
α
)=
x
:
b
2
e
i
arcsin
(
b
/
√
a
2
+
b
2
)
.
a
2
a
+
ib
=
+
This interpretation of
gives a (geometrical) meaning to the complex numbers
based on the
imaginary
unit. Moreover, it opens enormous possibilities of applica-
tions, ranging from electromagnetism to hydrodynamics, or quantum physics, which
show, in almost literal terms, the power of mathematical imagination. The sum of
two complex numbers is easily obtained by separately summing their real and imag-
inary parts, while the exponential representation implies that their multiplication can
be carried out by multiplying their modules and by summing their angles.
An important consequence of the polar representation of complex numbers is the
proof of the fundamental theorem of algebra:
Every algebraic equation of degree
n with coefficients in
C
C
has n solutions in
C
(by counting each solution with its
multiplicity)
.
Firstly, we observe that, given an equation
P
(
x
)=
0where
P
(
x
)
is a polynomial
of degree
n
,then
α
is a solution of it iff
P
(
x
)=(
x
−
α
)
Q
(
x
)
,where
Q
(
x
)
has degree
n
−
1. In fact, if
P
(
x
)=(
x
−
α
)
Q
(
x
)
, then obviously
P
(
α
)=
0. Vice versa, If
P
(
α
)=
0, then assume that
P
(
x
)=(
x
−
α
)
Q
(
x
)+
k
, with
k
=
0, then 0
=
P
(
α
)=(
α
−
α
)
Q
(
x
)+
k
,thatis,0
=
k
, which is a contradiction. This means that if
P
(
x
)
has
n
solutions
(for some
constant
a
). Therefore, if any equation has at least one solution, then any equation
of degree
n
has exactly
n
solutions (counted with their multiplicities). In fact, if
P
α
1
,
α
2
,...,
α
n
∈
C
then
P
(
x
)=
a
(
x
−
α
1
)(
x
−
α
2
)
···
(
x
−
α
n
)
(
)
x
(
)=(
−
α
)
(
)
has degree
n
and it has at least one solution, say it
α
1
,then
P
x
x
Q
x
with
1
Q
(
x
)
of degree
n
−
1, and all solutions of
Q
(
x
)=
0 are also solutions of
P
(
x
)=
0.
But, also
Q
(
x
)
has at least one solution, therefore we can apply iteratively the same
