Math Garden

Spring Math: Math Garden

In this app, the Natural Numbers are represented as Vegetables (plants): 

Fundamental Theorem of Arithmetic:

(From Wikipedia)

In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is prime itself or is the product of prime numbers, and that this product is unique, up to the order of the factors.

Therefore, if each decomposition of a number into its prime factors is unique, each number will have a unique form when it is shown as a plant.

But it is not exactly like that. Depends on the order in which the factors are multiplied.

This is shown in the following app videos:

There are two versions of Math Garden (Android) 

Kids Math Garden:

https://play.google.com/store/apps/details?id=com.nummolt.kinder.garden.math

Video: 

 

Math Garden:

https://play.google.com/store/apps/details?id=com.nummolt.kinder.garden.math_serialized

Video:

Details:

The general operation is similar to that of plants:
Seeds fall from the sky
They should be planted. And then grow the plant corresponding to the seed (and the number chosen).
When the time comes to harvest dandelions, the plant is plucked from the ground.
Then the seeds that it has generated are released.
And the seeds return to heaven.

The particular operation is similar to that of the numbers:
If two unseeded seeds overlap, they add up.
If two seeds are planted, the plants multiply.

The plants are structured in branches, forking in function of the prime numbers that compose the factorization of the number.

Plants multiplied underground, often have a structure different from plants planted in a single blow.
(Multiplied plants do not have the usual order of a well-made factorization)
Each plant generates as many little dandelions as the number that indicates its seed, Having any of the structures that may have.

The program has 30 furrows one behind the other to be able to plant.

I hope it will be useful to teach maths. 

Math Garden:
Pythagorean Garden:
3²+4²=5²
5²+12²=13²
8²+15²=17²
7²+24²=25²

Pythagorean Garden

105 Plant In the Math Garden
Variants: The 6 subspecies:
105=3*5*7
105=3*7*5
105=5*3*7
105=5*7*3
105=7*3*5
105=7*5*3
(In invented mathematical plants, of course!!!) Order of multiplication  vs. Factorization

 105 Plant in the Math Garden

UPDATE:  (August 2017)
Fractions: 

 


Origin of Math Garden:
Jessica's Drawing of 96
(From Simon Gregg' work)

And Nummolt's:
"Touch Natural Numbers"  
And
"Touch Integers Z"

Dancing numbers: (Math Circus)

Dancing numbers: Numbers Circus: Modulo choreography.
( And:  Circus for to tame numbers, too )

After the post dedicated to dancing sort algorithms,  I think it's time to show a program made and published in 2004, which it could be useful in teaching elementary mathematics.


Is the "Dancing numbers" 

Originally made for MathCats.com , it never has been published. Finally I posted it on my page so do not be missed for the moment.

The program allows you to manipulate a lot of integers (up to 2000) with some simple rules: decide the number of columns that are to line up neatly from lowest to highest.
(Also possible to arrange smaller amounts of numbers)

The program allows for reflection on the divisibility of numbers and remainders:

It is also advisable to use the mode 9 columns, that shows the rule of divisibility for 9.
(The sum of all members of any number equals the heading number of the column)

And all the rules explained in https://en.wikipedia.org/wiki/Modular_arithmetic
(related to addition, subtraction, multiplication and division (sometimes), and congruence).

Here is a sample of how the program works:

(Dancing numbers: youtube)

You can use the application here: http://www.nummolt.com/obbl/dancingnumbers/modular_xtd_01.html

I hope its use will be useful to teach elementary mathematics.

Dancing Sort Algorithms

Post, only to illustrate the most common types of sorting algorithms:
Graphically, Dance, and Code:

(Graphic from http://www.sorting-algorithms.com/ )
(Videos from: https://www.youtube.com/user/AlgoRythmics )
(Pseudocode: http://visualgo.net/ )
(Pseudocode Shell: https://courses.cs.washington.edu/courses/cse326/03wi/lectures/RaoLect14.pdf )
(Main Links (titles): Wikipedia: https://en.wikipedia.org/wiki/Sorting_algorithm ) 

Sorting Algorithms general animated image:
(Algorithms race)

( from: http://www.sorting-algorithms.com/ )

1.-  Insert:

mark first element as sorted

for each unsorted element

  'extract' the element

  for i = lastSortedIndex to 0

    if currentSortedElement > extractedElement

      move sorted element to the right by 1

    else: insert extracted element

2.- Select:

repeat (numOfElements - 1) times

  set the first unsorted element as the minimum

  for each of the unsorted elements

    if element < currentMinimum

      set element as new minimum

  swap minimum with first unsorted position

3.- Bubble:

do

  swapped = false

  for i = 1 to indexOfLastUnsortedElement

    if leftElement > rightElement

      swap(leftElement, rightElement)

      swapped = true

while swapped

4.- Shell:

                                # Start with the largest gap and work down to a gap of 1

                                     foreach (gap in gaps){

                                # Do a gapped insertion sort for this gap size.
                                                     # The first gap elements a[0..gap-1] are already in gapped order

                                                     # keep adding one more element until the entire array is gap sorted  

                                     for (i = gap; i < n; i += 1){

                                            temp = a[i]

                                            for (j = i; j >= gap and a[j - gap] > temp; j -= gap){

                                                  a[j] = a[j - gap]

                                            }

                                            a[j] = temp

                                      }

                                }

5.- Merge:

split each element into partitions of size 1

recursively merge adjancent partitions

  for i = leftPartStartIndex to rightPartLastIndex inclusive

    if leftPartHeadValue <= rightPartHeadValue

      copy leftPartHeadValue

    else: copy rightPartHeadValue

copy elements back to original array

7.- Quick:

for each (unsorted) partition

  set first element as pivot

  storeIndex = pivotIndex + 1

  for i = pivotIndex + 1 to rightmostIndex

    if element[i] < element[pivot]

      swap(i, storeIndex); storeIndex++

  swap(pivot, storeIndex - 1)

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:
http://esquemat.es/algebra/factores-primos-por-colores
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" program.

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:
http://play.google.com/store/apps/details?id=com.nummolt.number.natural.touch
 

 

Touch Natural Numbers at Youtube:

 

Maurici Carbó

www.nummolt.com

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

RAE (Real Academia Española): Answer about Quintillón to Centillón

En relación con su consulta, le remitimos la siguiente información:

      Ni el DRAE ni el DPD, ni tampoco otros escritos gramaticales de la Real Academia recogen ningún numeral superior al cuatrillón. El Diccionario del español actual de M. Seco trae uno más, quintillón. Por otra parte cabe decir que tampoco se documentan en nuestros bancos de datos, lo cual es indicativo del escaso uso que tendrían en español. En páginas de Internet pueden localizarse series más amplias de numerales, formados por analogía: sextillón, septillón, octillón, nonillón, decillón, undecillón, duodecillón, tridecillón/tredecillón, cuatridecillón/cuatordecillón, quindecillón, sexdecillón, septidecillón/septendecillón, octodecillón, nonidecillón/novendecillón, vigillón/vigintillón.       Lamentamos no poder ayudarle, pero nuestro cometido se limita a resolver dudas concretas sobre el uso normativo del español, y su consulta se refiere a hipotéticas voces que no tienen ningún uso en nuestra lengua.

     Reciba un cordial saludo.
__________
Departamento de «Español al día»
Real Academia Española

************

Despues de esta carta, pudimos desarrollar:
Calculadora natural


Muchas gracias!!!

Versión inglesa:

How looks a centillion?

Versión española:

Calculadora natural:

Letter: ToRAE_Quintillion_Centillion.txt

For the: 'Real Academia Española' (RAE)
 
In http://www.nummolt.com we are developing an application that writes numbers in Castilian.
Specialized in writing extremely large numbers.
We did not find any official confirmation of the name of the numbers from 10 ^30 to 10^ 594 
We need confirmation of our proposal:

10^3="mil","miles"
10^6="millón","millones"
10^12="billón","billones"
10^18="trillón","trillones"
10^24="cuatrillón","cuatrillones"
10^30="quintillón","quintillones"
10^36="sextillón","sextillones"
10^42="septillón","septillones"
10^48="octillón","octillones"
10^54="nonillón","nonillones"
10^60="decillón","decillones"
10^66="undecillón","undecillones"
10^72="duodecillón","duodecillones"
10^78="tredecillón","tredecillones"
10^84="quattuordecillón","quattuordecillones"
10^90="quindecillón","quindecillones"
10^96="sexdecillón","sexdecillones"
10^102="septendecillón","septendecillones"
10^108="octodecillón","octodecillones"
10^114="novemdecillón","novemdecillones"
10^120="vigintillón","vigintillones"
10^126="unvigintillón","unvigintillones"
10^132="duovigintillón","duovigintillones"
10^138="trevigintillón","trevigintillones"
10^144="quattuorvigintillón","quattuorvigintillones"
10^150="quinvigintillón","quinvigintillones"
10^156="sexvigintillón","sexvigintillones"
10^162="septenvigintillón","septenvigintillones"
10^168="octovigintillón","octovigintillones"
10^174="novemvigintillón","novemvigintillones"
10^180="trigintillón","trigintillones"
10^186="untrigintillón","untrigintillones"
10^192="duotrigintillón","duotrigintillones"
10^198="tretrigintillón","tretrigintillones"
10^204="quattuortrigintillón","quattuortrigintillones"
10^210="quintrigintillón","quintrigintillones"
10^216="sextrigintillón","sextrigintillones"
10^222="septentrigintillón","septentrigintillones"
10^228="octotrigintillón","octotrigintillones"
10^234="novemtrigintillón","novemtrigintillones"
10^240="quadragintillón","quadragintillones"
10^246="unquadragintillón","unquadragintillones"
10^252="duoquadragintillón","duoquadragintillones"
10^258="trequadragintillón","trequadragintillones"
10^264="quattuorquadragintillón","quattuorquadragintillones"
10^270="quinquadragintillón","quinquadragintillones"
10^276="sexquadragintillón","sexquadragintillones"
10^282="septenquadragintillón","septenquadragintillones"
10^288="octoquadragintillón","octoquadragintillones"
10^294="novemquadragintillón","novemquadragintillones"
10^300="quinquagintillón","quinquagintillones"
10^306="unquinquagintillón","unquinquagintillones"
10^312="duoquinquagintillón","duoquinquagintillones"
10^318="trequinquagintillón","trequinquagintillones"
10^324="quattuorquinquagintillón","quattuorquinquagintillones"
10^330="quinquinquagintillón","quinquinquagintillones"
10^336="sexquinquagintillón","sexquinquagintillones"
10^342="septenquinquagintillón","septenquinquagintillones"
10^348="octoquinquagintillón","octoquinquagintillones"
10^354="novemquinquagintillón","novemquinquagintillones"
10^360="sexagintillón","sexagintillones"
10^366="unsexagintillón","unsexagintillones"
10^372="duosexagintillón","duosexagintillones"
10^378="tresexagintillón","tresexagintillones"
10^384="quattuorsexagintillón","quattuorsexagintillones"
10^390="quinsexagintillón","quinsexagintillones"
10^396="sexsexagintillón","sexsexagintillones"
10^402="septsexagintillón","septsexagintillones"
10^408="octosexagintillón","octosexagintillones"
10^414="novemsexagintillón","novemsexagintillones"
10^420="septuagintillón","septuagintillones"
10^426="unseptuagintillón","unseptuagintillones"
10^432="duoseptuagintillón","duoseptuagintillones"
10^438="treseptuagintillón","treseptuagintillones"
10^444="quattuorseptuagintillón","quattuorseptuagintillones"
10^450="quinseptuagintillón","quinseptuagintillones"
10^456="sexseptuagintillón","sexseptuagintillones"
10^462="septseptuagintillón","septseptuagintillones"
10^468="octoseptuagintillón","octoseptuagintillones"
10^474="novemseptuagintillón","novemseptuagintillones"
10^480="octogintillón","octogintillones"
10^486="unoctogintillón","unoctogintillones"
10^492="duooctogintillón","duooctogintillones"
10^498="treoctogintillón","treoctogintillones"
10^504="quattuoroctogintillón","quattuoroctogintillones"
10^510="quinoctogintillón","quinoctogintillones"
10^516="sexoctogintillón","sexoctogintillones"
10^522="septoctogintillón","septoctogintillones"
10^528="octooctogintillón","octooctogintillones"
10^534="novemoctogintillón","novemoctogintillones"
10^540="nonagintillón","nonagintillones"
10^546="unnonagintillón","unnonagintillones"
10^552="duononagintillón","duononagintillones"
10^558="trenonagintillón","trenonagintillones"
10^564="quattuornonagintillón","quattuornonagintillones"
10^570="quinnonagintillón","quinnonagintillones"
10^576="sexnonagintillón","sexnonagintillones"
10^582="septnonagintillón","septnonagintillones"
10^588="octononagintillón","octononagintillones"
10^594="novemnonagintillón","novemnonagintillones"
10^600="centillón","centillones"

Confirmation of these data would be very important to us.
We would provide a right product version in the Castilian language.
 
Sincerely:
Maurici Carbó 
R&D Manager at: 
http://www.nummolt.com

This letter was written after developing:
Whole Calculator

"Whole Calculator"

Whole Calculator

(Android app) 

    This week (after a month of work) we started a new project : "Double Struck Capital ".
    And for the first time we can post about it on this blog.
    The first visible result of this project is the calculator of natural numbers: "Whole Calculator" 
    It works in the set of the Natural Numbers (ℕ0 in the mathematical text books)
    'ℕ' is the character named 'Double-struck Capital N'  
  
    Our new calculator has the particularity to work with arbitrarily large numbers (up to centillion). And it can show the numbers written in words.

    One purpose of this program is to support the teaching of reading whole numbers, and to be able to read and operate with numbers and huge numbers too.

    And has another intention:
    In the attempt to understand the numbers and its elementary operators in a first stage of education, it is relatively easy to explain the sum , and multiplication.
    But subtraction and division operations are another level :
    Subtraction and division are operations whose result that is not restricted into the set of natural numbers:
    Therefore, you can only partially explain those operations :
    Subtraction, only when the result is always positive.
    The division only when a provided that the dividend is a multiple of the divisor.
    The Whole Calculator , tries to show this same state of bewilderment :
    While it is capable of displaying numbers to barely imaginable figures (is able to show correct results in numbers and letters up to 10 ^ 303) however it is not able to operate with negative numbers, and if the divisions are not exact , the result has a remainder .
    "Whole calculator " is a coherent and 'trustable' product . But, when results are not within the set of natural numbers , warns the user that the " this calculator of natural numbers is not able to give the correct result" , because it is outside the scope of the natural numbers, and recommends the use (and "download" from anywhere) a better calculator: integer calculator (ℤ) or rational (ℚ) as appropriate: As if the app were a damaged or a incomplete product.
    We want to show that the "whole calculator" is a damaged product, by fault of the Natural Numbers.
     
    In turn, all this can be seen as a joke by program users .
    But in "Double Struck Capital - nummòlt " we think it is a way to teach the numbers from the beginning:
    Understand that in a first step , we learn to live with the numbers used to count and to list . But when we operate with them, and as we operate, we have not enough with these numbers (natural numbers) and we need to expand the range of numbers to use, to meet the needs that appear when we afford the subtraction and the division.
    At the time of making this app, we had to deal with the construction of the natural numbers from zero :Internally, the program works with the natural numbers as a String variable , so that the numbers do not remain limited to the small dimensions of the variables of type ' int ' or ' long' .This has forced the developer to implement the algorithm of addition, subtraction, multiplication and division from text strings.
    And we have faced the same problem as has a child when learns to add, subtract, multiply and divide.
    Recovering these skills almost lost in time for us, we has discovered that we never fully understood the algorithms of subtraction, or multiplication or division. 
Specifically:

•     We have not clear enough the mechanism of "I carry" in the addition or "borrowing" in subtraction .
•     We don't know subtract if the result will not be positive .
•     We do not understand what we do when we multiply numbers: ( The needed zeros are obviated in the mechanism we were taught in school)
•     We do not understand almost nothing of the procedure that we follow when we divide .

    In the development of the app, we had to overcome these difficulties. And we had to return to deduct procedures , modifying them to make the program work properly.
In the development of this program, we felt like little kids , having to learn the most basic things.    
    And at the same time , we felt like kids happy when it was finished.
    We expect that when the program will be downloaded, the users will can get to enjoy it as we have enjoyed making it .
    The program can be downloaded at:
http://play.google.com/store/apps/details?id=com.nummolt.whole.calculator
(There's only the Android App, no version in other operative systems)

Whole Calculator Youtube:

How looks a Centillion?:

The app spells the natural numbers from 1 to the Centillion (10^303) in english, working with the Short Scale
There's also a paid version in Castilian ( Spanish ) spelling the natural numbers from 1 to the "Centillón" (1^600) working with the Long Scale:

Calculadora Natural
http://play.google.com/store/apps/details?id=com.nummolt.calculadora.natural

Video of the Update (Version 1.0.4):

Cyclic numbers:

Video of the Update (version 1.0.7) added Catalan:

Video about Factorials: From 2! to 50!:

 
Thanks for the help:
Wendy Petti (mathcats.com)
Joan Jareño (CREAMAT)

Maurici Carbó (nummolt webmaster)