Panorea: first attempt
On Friday 27th November 2009 was my first attempt to lead the group of the studio of choreography. It was the first time that I had to teach math stuff to a group of actors, dancers and choreographers all of whom were adults. It was difficult and challenging at the same time to explain my strong feeling by then – and belief at the moment – that mathematics share a lot in common with representing and creative arts. I had almost one hour and a half to practice with them. I divided my session into two parts. First we dived into the mystery of prime numbers and we investigated one of their most important properties and secondly we played with divisibility of natural numbers. For each idea first I made a short theoretical introduction and afterwards the group practiced on the related excercises.
On Friday 27th November 2009 was my first attempt to lead the group of the studio of choreography. It was the first time that I had to teach math stuff to a group of actors, dancers and choreographers all of whom were adults. It was difficult and challenging at the same time to explain my strong feeling by then – and belief at the moment – that mathematics share a lot in common with representing and creative arts. I had almost one hour and a half to practice with them. I divided my session into two parts. First we dived into the mystery of prime numbers and we investigated one of their most important properties and secondly we played with divisibility of natural numbers. For each idea first I made a short theoretical introduction and afterwards the group practiced on the related excercises.
A] Investigating natural numbers (properties & functions)
A set is any collection of numbers that belong to a defined category.
A set can be finite (e.g. the set of hours in a day) or infinite (e.g. the set of hours in the future).
Natural numbers - also called positive integers - are the set of numbers from 0 on obtained by adding 1 first to 0, and the sequence to each number so obtained, thus : 0,1,2,3,4,5,...
(the three ellipsis points “…” indicate and so on to infinity).
Natural numbers have beginning (0) but no end: there is no last natural number.
Each natural number n has an immediate successor n+1.
1st classification of natural numbers: even & odd is given by the division by 2
• Even numbers are those divisible by 2
• Odd numbers are those that are not divisible by 2.
The steady alternation of even and odd marks the sequence of natural numbers
Even are doubles (2n)
Odds (2n+1)
2nd classification of natural numbers: prime numbers
Prime numbers are the numbers that are the product of themselves and 1 – they cannot be divided except by themselves and 1. This group includes numbers 2, 3, 5, 7, 11, 13, 17 and an infinite list of others. Every integer is either a prime, or the product of prime numbers. To identify a number by its primes is called “decomposition into prime factors”.
Each number has one and only one formula of decomposition into prime factors – unique to it – a fact of great importance. Since every decomposition is unique, the collection of its prime divisors can be considered its signature e.g. 30107 = 7 x 11 x 17 x 23.
Exercise
First we choose a few numbers. Secondly, we decompose them into their prime factors. Then each person represents a prime number. And every time I show them one of the numbers we have chosen , the corresponding primes try to build up the given number.
I tried this exercise with 4 persons. Each one represented one of the first four prime numbers, thus the first was 2, the second represented 3, etc. I had already done the decomposition into prime factors for numbers which were the product of 2,3,5 and/ or 7.
We tried to build up, or - preferably to me - to construct the signature - of the following numbers: 210=2x3x5x7, 60=22x3x5, 20=22x5, 150=2x3x52, 300=22x3x52.
Is the signature something still or there is movement that is being born? What happens when a prime number is in power 2? Which is the difference between 2 and 22 in a signature? How this affects the shape or the movement of the prime? Which is the feeling or the meaning of being a number and a part of another number at the same time? Which is the difference between a prime number alone and a prime number as a factor in a decomposition into primes’ product?