Information Technology Reference
In-Depth Information
It can be shown that the function
∞
i
=
1
1
i
k
ζ
(
k
)=
converges for any
k
is extremely important in number theory.
Here we want to recall, from Sect. 5.4.3, the relationship, discovered by Euler, be-
tween
>
1. The function
ζ
ζ
(
2
)
and
π
:
∞
i
=
1
2
6
.
i
2
=
π
1
ζ
(
2
)=
We remark that the name of Euler is strictly connected to almost all important spe-
cial numbers of mathematics. His crucial role in the definition of
e
(called Euler
number or Napier constant) and his famous formula connecting
e
with
i
were al-
ready explained. However, apart from the constant
mentioned above, which fre-
quently occurs in many contexts, he was also the mathematician who popularized
the symbol
γ
which, maybe,
is the irrational (trascendental) number of which we know the largest number of dec-
imal digits (about 10
12
digits). This possibility is due to the BaileyBorweinPlouffe
formula (BBP formula) discovered in 1995, allowing for the computation of the
n
th
digit of
π
for the famous Archimedes' number 3
.
1415926535
...
π
without calculating the first
n
−
1 digits.
5.7
Scales of Time, Space, and Matter
Any mathematization of a natural phenomenon is based on the measurement of
quantities and on the evaluation of their orders, that is, the powers, with respect to
a given base, that better approximate their values. Table 5.23 provides the decimal
scale of powers with their corresponding symbols and prefixes (it is interesting to
realize that the tenth power of 2 is 1024 which is close to the third power of 10).
Let us evaluate some measure orders.
What is the magnitude order of a solar
year?
A year is consists of 365 days of 24 hours of 3600 seconds. Therefore, one
year is 365
10
3
10
7
seconds, that is only a few tens of millions
×
24
×
3
,
6
×
<
3
,
2
×
of seconds.
What is the magnitude order of universe life?
Assuming the universe started 15
billions of years ago, the number of seconds from the so-called “big-bang” are less
than 5
10
17
.
How many corn grains can be put on a chessboard where its 64 squares are
numbered and where one grain is put on the first square, two on the second one,
and in general the double of the number of grains placed in a square is placed on
the next square (chessboard puzzle)?
It is easy to evaluate this number as 2
64
×
−
1.
If we approximate 2
10
with 10
3
,and2
4
with 15, then the overall evaluation gives
2
64
10
19
. This means that if a grain is one gram, we need about
15000 billions of tons of corn to cover the chessboard according to this procedure.
You can never find such a corn quantity on Earth.
2
10
6
≈
(
)
×
15
=
1
.
5
×
