Information Technology Reference
In-Depth Information
5.4
Complex Numbers and Real Vectors
Complex numbers are sums of a real number with an
imaginary
number. Imaginary
numbers were discovered by the Italian mathematicians of the Renaissance, during
their study of formulas to solve algebraic equations. In fact, the two solutions of
a second degree equation
ax
2
+
+
=
bx
c
0 are given by the following formula (a
solution with
+
, the other with
−
):
√
b
2
−
b
±
−
4
ac
.
2
a
b
2
When
(
−
4
ac
)
is negative we get square roots of negative numbers, or numbers
multiplied by
√
−
1. No real number exists having a negative square, therefore
√
−
1
is meaningless in the set of real numbers. However, in formulae giving solutions of
third and fourth degree equations, in some cases, real solutions are obtained by using
formulae which manipulate imaginary numbers. This discovery was shocking for
Italian mathematicians, in the same way as that of incommensurability was shock-
ing for Greek mathematicians. In fact, meaningful solutions to a problem could be
obtained by manipulating meaningless objects. Moreover, if
i
denotes
√
−
1, then in
the set of numbers
a
the arithmetical operations can be extended
in a way which is coherent with their definition on the reals. One may just deal with
these numbers as if they were algebraic expressions (sums of two reals) including
an indeterminate
i
satisfying the condition (impossible for every real number) that
i
2
+
ib
with
a
,
b
∈
R
=
−
1. Therefore
(
a
1
+
ib
1
)(
a
2
+
ib
2
)=(
a
1
a
2
−
b
1
b
2
)+
i
(
a
1
b
2
+
b
1
a
2
)
(analo-
gously for the other operations).
5.4.1
Euler's Identity
The imaginary unit
i
is related to two numbers which are crucial in mathematics.
One of them is Archimedes' constant
, the ratio between the length of a circle and
its diameter. The second one is the constant
e
, introduced by John Napier, and fur-
ther defined and fully investigated by Leonhard Euler. The Scottish mathematician
Napier in 1614 published a work where the problem is addressed of reducing the
computation of products to that of suitable sums. An arithmetical progression is a
sequence of numbers where each element (apart the first one) is obtained by the
previous one by adding a constant value to it (the common difference of the pro-
gression). Analogously, a geometric progression is a sequence of numbers where
each element (apart from the first one) is obtained from the previous one by multi-
plying it by a constant value (the common ratio of the progression). The initial idea
of Napier (suggested by mechanical analogies) was to determine geometrical pro-
gressions coinciding, with a good approximation, with an arithmetical progression
of very small common difference, for example 0
π
.
001. In this manner, a product of
two numbers
a
b
, with an approximation to the third decimal, can be obtained by
locating them in the geometrical progression and then by locating, a greater number
c
at a distance (number of elements) from
a
equal to the position of
b
.Inthisway,
≤
