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.

image002.jpg (2297 bytes)

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.

 As the second square number is the addition of 1+3, the second square figure is composed of 1+3 points.

image004.jpg (4462 bytes)

 

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,...

 

image006.gif (1349 bytes)

 

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.

image008.jpg (4610 bytes)

 

Q(1) S1 = 4p

l(x) 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.

 Socrates invites Menon's servant to built this figure:

image010.jpg (4696 bytes)

 

l(x)   2 l (1) = 4p

Q(2)   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.

image012.jpg (7711 bytes)

 

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: 2.

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:

image014.jpg (4552 bytes)

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:

image016.jpg (2360 bytes)

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 ROMAN SYSTEM  

 

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)

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