COMENIUS PROJECT ACTION 1
SUBPROJECT
"PITAGORA"
MATHS
FOR EUROPE
"Progetto
Pitagora" is the name that identifies the contribution given by our Liceo of Vittorio Veneto (Italy) to
the realization of the "Projct Comenius-Action 1".
The
work we are sending you, represents the first stage of a project that is going to continue
next year. In fact, we intend to explore other topics within the mathematical thinking,
including even contributions from psychologists of the twentieth century.
We
are not sure whether we will succeed or not, but we are going to try!
All
the students from Form 1B (Liceo Classico), a student from Form 1A (Liceo Classico) and
two students of the same age from our Liceo Scientifico have participated in the
realization of the first stage of the project.
These students were : Azzalini Giulio (1983) Bortoletti
Gaetano (1983) Carlet Federica (1983) De Biasi Lara (1983) De Faveri Silvia
(1983) El Tabch Jumana (1982) Fedrigo Erika (1983) Meghini Gabriella (1983)
Michelon Elena (1982) Morandin Ivana (1983) Peruch Francesca (1983) Poser
Lorenzo (1983) Salvador Piera (1983) Scapol Lisa (1983) Stefani Alessia (1983)
Tonon Giacomo (1983)
We
are all nice people who appreciate the experience of "co-operative learning"
and we would greatly like to be involved in an exchange programme: some of you might be
guests of our family and some of us might visit Norway, Finland or Belgium. Are you
interested? Think about our proposal and let us know as soon as possible.
This
work has been co-ordinated by Mr Di Nunno(philosophy), whom you have already met, with the help of the
following teachers: Mrs Meschini Cesira(maths), Mr Spagnolo Filippo(philosophy), Mrs Da Ros
Carmen(English), Mrs
Michielini Micaela(English), Mr Da Re Gianfranco and Mrs Valentini Tommasa.
From
now on, please contact us: we are sending you our photograph, too!
"Up with United Europe!"
We
have chosen two texts to start with our classwork. The first one deals with Euclid's
elements, the second one with "Introduction to Maths" by Nichomacus of Gerasa.
Euclid
says that a number (ariqmoV)
is a multitude composed of units (ek
monadwn);
Nichomacus gives the same definition using nearly the same words: "number is a
determinate multitude or collection of units or flow of quantity made up of units".
We can see that both mathematicians think that any number is made up by adding units; that
is why the greek matematicians call the number logoV.
One of the basic concepts of Greek philosophy is that the person whoever the concept of
number owns the logoV
of universe. The Maths building went crumbling down when the so called Pythagorics found a
quantity which could not be measured by a number. As a consequence they used the word
"irrational number" (alogoi
ariqmoi).
If numbers are the reason of all things, how can we speak of numbers without reason? We
have realized that the Pythagorics had established a relationship between Mathematics and
Geometry. What is the meaning of this relationship? The set of numbers corresponds to the
set of geometrical figures (biunivocal correspondence).
Actually
they never spoke of "numbers" without qualifying them:
· triangular
numbers
· square
numbers
· rectangular
numbers
TRIANGULAR
NUMBERS
We
get this sequence adding the following terms of natural numbers sequence
S
1,3,6,10,...
As the unit (1) is the origine of triangular numbers, the point is the origine of all triangular figures. As the second triangular number is the addition of 1+2, the second triangular figure is composed of three points.
SQUARE
NUMBERS
We
get the series of square numbers adding the odd terms in the series of natural numbers. We
call S1
the series of natural numbers and S2
the series of square numbers.
S1 1,2,3,4,5,...
S2 1,4,9,16,25,...
As
the unit is the origin of all numbers, the point is the origin of square geometrical
figures.
RECTANGULAR
NUMBERS
We
get the series of rectangular numbers adding the even terms in the series of natural
numbers. We call S1
the series of natural numbers and S2
the series of rectangular numbers.
S1
1,2,3,4,5,...
S2 2,6,12, 20,...
DIFFICULTIES WITH THE BIUNIVOCAL CORRESPONDENCE
When
does this mathematical building, based on the biunivocal correspondence between numbers
and geometrical figures, fail?
We
get hints of that in one of Plato's Dialogues
"Menon". What is the mathematical problem that Socrates asks Menon's servant?
The question is: "How can we find the side of a square whose area is twice the area of the given square?" The servant tries to work out the problem and "remembers" (anamnhsiV), that he makes links.
Q(1)
Þ
S1 = 4p
l(x)
Þ
Q(2) = 2Q(1) = 8p
What
is the first link that the servant makes?
He
thinks that if you double the side of the given square, you will obtain a square whose
area is twice the area of the given square.
l(x)
Þ 2 l (1) = 4p
Q(2)
Þ 16 p
What does the servant understand through the figure?
If
you double the side you will obtain a square four times the size of the given square.
What
is the second link that the servant makes? If the 8 feet square is bigger than the 4 feet
square and smaller than the 16 feet square, the side we want to measure will be bigger
than the 2 feet side and smaller than the 4 feet side.
Q(4p)
< Q(8p) < Q(16p)
l(2p)
< lx < l(4p)
Socrates
shows through a figure that a 9 feet square can be obtained this way. The servant is
discouraged because he can't find further
links and solutions.
What
does Socrates suggest?
We have to draw the diagonal line (diametroV) inside the given square and repeat the same procedure inside the other 3 squares.
Menon's
servant understands that the side of the square which is twice of the side of the given
square corresponds to the diagonal line of the given square.
Yet
the solution of the problem is geometrical and not arithmetical; indeed the diagonal line
of the given square is a irrational number: Ö2.
Why
was this number considered a irrational number (alogoV).
It
was devoid of logoV
(logoVless),
that is it was not the result of natural numbers.
This
way the Pythagoric built on the biunivocal correspondence between figures-numbers
crumbled.
The
Greeks were skilled at geometry but they had three inefficient numeration systems:
· on
sets of ten: this system was used by merchants who employed points or dashes in order to
mean the units and they grouped them by ten;
· eroedian
system:instead of symbols they used the first letter of the itself number's name;
· alphabetical
number system: they used the letters of the
alphabet made up of twenty-seven symbols;
1 2 3
4
5
6
a b g
d
e
z
alfa beta gamma delta epsilon zeta
7
8 9 10 20 30
V
h q i k
l
stigma eta theta iota kappa lambda
40 50 60
70
80 90
m n x
o
p \/\
mi ni csi omicron pi coppa
100 200
300 400
500 600
r
s
t
u
f c
rho sigma tau ipsilon phi hi
700
800
900
y
w
=)
psi omega sampi
SYSTEMS
OF NUMERATION:
how
can we write numbers?
In
the past, in order to number, men used parts of the body (hands and fingers), but there
were other ways of numbering: according to sets of two, sets of five, sets of ten. For
example in Homer’s Odissey (book 4, line 413) you find the verb pemprazein
that
means numbering by five. So we have different ways to calculate:
· the
first one called quinary (a sign every five fingers),
· the
second one called decimal (a sign every ten fingers).
If
we don't consider numbers linked to our body, we can count by dozens or by sixty... but
these systems of numbering were the result of sky’s observation. Therefore the ways
of numbering by dozens or by sixty do not originate from anatomy, but from astronomy.
Indeed the observation of the sky was the first scientific observation and it was linked
to the necessity of measuring the time in a more and more precise way. The observation of
the sky was entrusted to priests-teachers observed that the sun had a 360-day-cycle, the
moon had a 30-day-cycle. Accordingly if you divide 360 by 30, you obtain the number of the
month (12). Three-hundred and sixty is a too big a number so the Cheldees (Magni Caldei)
chose its sixth part (60) as basis of their numbering.
The
Babylonians used two ways of numbering: one on set of ten, one on set of 60 and they used
cuneiform signs.
The Egyptians used a decimal system employing the following symbols:
And in order to write the other numbers, they followed the additive rule. For example they wrote 300 by repeating 3 times the symbol corresponding to 100:
The
Greeks were skilled for geometry, but they had 3 inefficient numeration systems:
· one
set of ten: this system was used by merchants, who employed points or lines in order to
mean the units and they grouped them by ten;
· Erodian
system: instead of symbols they used the first letter of the name of the number itself;
· alphabetical
system: they used the letters of the alphabet made up of 27 symbols.
The Etruscans used a system of numbering based on set of ten, that probably came from a primitive system of numbering based on set of five. The Roman system of numbering (Roman numbers) was a modification of the Etruscan one. The great historian Theodor Mommsen in the symbol V recognized the schematic drawing of lifted (open) hand. The symbol X could be the schematic drawing of two open and joined hands: one turned upwards, the other turned downwards; this led Theoder Mommsen to think that probably there was a primitive system of numbering based on five. When the symbols came one after the other in a decreasing order from left to right, you had to add (MDV means 1000+500+5). If the lower symbol preceded (from left to right) the higher one, you had to subtract the lower one from the higher one (CM means 1000-100 = 900 IV means 5-1 = 4)