Tuesday, June 17, 2014

Touch Natural Numbers

In February 2010, the magazine "Scientific American" ISSN 0210136X number 401 in the Mathematics Games section, published the article of Agustin Rayo (philosophy professor at MIT) on: "bricks, locks and progressions." http://www.investigacionyciencia.es/files/3486.pdf There are specific items on the RSA encryption method.
And on the Theorem of Ben Green and Terry Tao
The article touched me.And what attracted me was the beautiful graphics representing natural numbers in the form of combinations of spheres and worms were colored graphic representation of primes. (See  Scientific American article in *.pdf cited above).At nummolt.com, I spent many years trying to graph the numbers. You can see my work from 1997 that I mean: http://www.nummolt.com/nummolt/numdown.htm

It is an old tool, based on the reflections made from quotations by Barbara Scott Nelson and others, read online about children's learning when to add and subtract with borrowing. 
But while it is easy to graphically represent the addition and subtraction of natural numbers is however much more difficult to make a simple representation of the multiplication, the amount of calculations involving multiplication algorithm that are used and taught in schools, after memorizing the "multiplication tables".There have been many attempts to make graphic representations of multiplication multiplication Mayan eg: Mayan Multiplication video
And myself, I also made my attempts in my programs.But the mere fact that prime numbers as the basic bricks which are built from all natural numbers, had not ever considered. 
When I read the article, I sent a letter to Agustin Rayo to tell him that I had really liked the article, and I naively I asked if the graphs corresponded to any investigation being carried out.
He politely replied that the drawings he had done in the best way that he had come to illustrate the article, and do not corresponded to any investigation. 

At this point, I already had developed for years with Wendy Petti program for Mathcats "Place Value Party": http://www.mathcats.com/explore/age/placevalueparty.html In this program we tried to show the value of the position of the numbers from birthday cakes with candles.A few years after,  Ulrich Kortenkamp published the Place Value Chart: Seeing Ulrich Kortenkamp program I wrote to MathForum saying that this program was a lesson for me, and I congratulate the author:http://mathforum.org/mathtools/tool/181488/ 
Few years later, talking with Joan Jareño about a game with primes, from Creamat they warned me about the existence of a poster with primes:
In reference to the original from John Graham-Cumming:http://blog.jgc.org/2012/04/make-your-own-prime-factorization.html 

At this time (2013), I understood that I had all the pieces to build a tool that I dreamed.
When I was looking at the internet to address program development, in addition to the well-known Ulam spiral, I had knowledge of the parabolic sieve: shown by Yuri Matiyasevich  and Boris Stechkin form the Steklov Mathematical Institute of the Russian  Academy of Sciences. 

There's an explanation for this, here: http://plus.maths.org/content/catching-primes

The first had to do was to assign a color to each prime number.Because the app is dedicated to the children, I decided to give the three basic colors first three prime numbers, and intersperse the following numbers.Therefore, red is 2, green 3 and 5 is blue, 7 yellow, magenta 11 cyan and 13. And from there, putting the colors go.The purpose of this distribution is that the resulting color of the product color is the sum of the colors of the prime factors or filtration of colors of the prime numbers.Thus, the color corresponding to 30 (2 * 3 * 5) will be white, and the color corresponding to 1001 (7 * 11 * 13) will be black.

Having decided this, the work was left to do was clear:Show prime numbers as small circles within a larger circle, usually corresponding to a composite number.Removing prime circles from the circumference, equals to divide.Add a prime circle, equivalent to multiply.In parallel, the the app displays numbers in a place value format, but in vertical, as in the  Mathcats "Place Value Party"

And adding the display parabolic sieve, and all the numbers well placed within the Ulam spiral and the representation of the module number.
In the end, the program Touch Natural Numbers is a small laboratory that can be studied composition and numbers, and make elementary operations within the set of natural numbers.
And never there is a division that is not resulting integer.
Nor has there ever the possibility of subtraction with a negative result.

Hope you like it, and especially useful for teaching math in elementary school.

Touch Natural Numbers App at Google Play:


Maurici Carbó


From here I recommend the book "You Can Count on Monsters" from Richard Evan Schwartz, with a similar approach to the natural numbers: