Reflections about the maths daily in the world around us.
The nummolt - mathcats materials, the mathematics behind them and related to them.
Math education and apps.
Primes as the basic building blocks of numbers

Just a note: My app "Touch Integers ℤ (+ - × ÷)" is useful to explore the Mersenne primes: Here there's a tiny exploration from 2^2 -1 to 2^23 -1. We found all numbers prime, except when p==11 and p==23 Like everybody knows:

The search is made with: "Touch Integers ℤ (+ - × ÷)" Android app:

In an innocent way, and mainly based on my ignorance, in 2001 I developed an online toy about building tiny computing objects with logic blocks: Is my OBBLOG Ignoring the equivalences of the Karnaugh table, OBBLOG contains all the possible logic gates with two inputs and one output. This is the partial living OBBLOG Truth Table:

For to build a operating toy, I added forks, crossings at different levels and bit visualizers. (Similar to a light bulb) However, the toy did not have flip-flops, so you could not make a numbers counter, although small binary calculators 1 to 16 bits. The toy allows to build little and simple animated toys:

But from the initial basic logic gates, I discovered that it was possible to build a flip-flop: (And consequently a counter of binary numbers):

(The obblog allows a closed loop that continues

to circulate indefinitely if not obstructed)

Thanks to this, the obblog could become able to build calculators calculating in binary system and be able to show results in decimal system. But being only in 2 dimensions can never bequeath to do what you can get to perform with Minecraft. (Koala steamed) (see: official Minecraft logic gates)

I posted all this, because yesterday I found similar artifacts along the history of computing: The article published in Scientific American the December of 1964: Fluid logic devices, all based in the Coanda effect:

Coding and computation in microfluidics. http://cba.mit.edu/events/07.05.fluid/ And the conference from Manu Prakash, http://cba.mit.edu/events/07.05.fluid/Prakash.pdf The manu Prakash Thesis: (About "microfluidic bubble logic"), and inside this beautiful Truth table about logic gates and its equivalents in transistors, valves, electrical devices, and fluidics:

Some examples of projected fluidic devices: (from Prakash thesis) Integrated
fluidic logic gates with a schematic integrated control circuit

The main patents about fluidics, come from the sixties: https://www.google.com/patents/US4854176 (from wikipedia) And schema about Coanda effect from Popular Science (Jun 1967 Pag 118) :

And the: Fluidics: Basic components and applications

APPLIED HYDRAULICS AND PNEUMATICS U5MEA23 Prepared by Mr. Jayavelu.S & Mr. Shri Harish Assistant Professor, Mechanical. http://slideplayer.com/slide/5675986/

I would like to see these companions of my program, running.

Anyway, everybody can see them inside the Bowles products:

And inside of the Theranos patented devices: (Lab-on-a-chip)

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:

This is just a small observation, based on my own stats: As the numbers that I show are not too high, observation may not have enough value. Typically, the math apps downloads have a proportion of users fairly well distributed among all countries. However, there is always a predominance of the US downloads: Example graph of a usual app of active users statistics:

(humble numbers related to a calculator)

In my Google Play page, there is one of the oldest applications that I've done, devoted to linear algebra and the Gauss-Jordan elimination as a method of solving two equations with two unknowns.

For the application to be a little more attractive, I called it "Adding apples and oranges", referring to an equation are getting really different variables to give a result.

It was my first attempt of to show an interactive representation of Rational numbers as fibres in (Z * Z {0}) and arrangement in Q.
As the Rational representation in Wikipedia: Rational Numbers:

In 2013 I made the new version of the app: The translation of the original executable to the Android OS.

The interaction became extended to the fraction:

The pie fraction to navigate across the fractions, and the rational representation to navigate across the equivalent fractions. (and select the range of available denominators)

In 2015 after the development of Touch Integers ℤ (+ - × ÷) (https://play.google.com/store/apps/details?id=com.nummolt.touch.integers)

This time, I was able to afford the next step:

Explain fractions and rational numbers as the prime factorization of its members: numerator and denominator.

Interact with the fraction, build fractions adding or removing prime factors in the numerator or in the denominator.

Simplify fractions dragging common prime fractions to the "common" zone, and invert fractions with the "^-1" button.

Touch Fraction ℚ is a complete tool to understand positive fractions, negative fractions, positive and negative numerators, positive and negative denominators, equivalent fractions, and inverted fraction.

Chandra Mukerji - Cambridge University Press,Sep 25, 1997-History-393 pages

There: "Chandra Mukerji highlights the connections between the seemingly disparate activities of engineering and garden design, showing how the gardens at Versailles showcased French skills in using nature and art to design a distinctively French landscape and create a naturalized political territoriality."

From my point of view is a suggestive research field. I created a group google plus to continue the search :

Touch Integers is the evolution of the Touch decimals Place value ±. (in the same blog) Touch decimals could not easily multiply or divide numbers:

I've started my reflections about this 20 years ago:

Is very easy add and subtract graphically. One can regroup the tokens of each order, regroup, carry or borrow tokens, and you can obtain the result in a simulation of abacus.

But not so easy to practice multiplication or division in this visual and interactive way

I looked the inside of the numbers:

Inside the numbers there are the components of the number: The prime factors.

To multiply two integers you must regroup the components of the two numbers.

To divide a integer, you must separate the components.

The program only works with integers. adds, subtract, multiplies and divides (but only exact division)

At left: two abacuses (two numbers stacked). At right two circles with the prime factors. (two circles with prime numbers stacked) At right edge: all the prime numbers available to the app. To create a number: Tap on the cells at left. The app shows the number To add: Drag the tokens from one abacus to the other. To subtract: Tap the sign key and drag from one abacus to the other. To multiply: (the numbers must be previously created with the earlier previous steps) Drag from one prime circle to the other prime circle. To Divide a number: Drag the prime numbers outside the prime circle: Release prime factors to the other prime circle (integer division and multiplication) Release prime factors between the prime circles (integer division) Release prime factors in the list of the right edge: (integer division and erase the prime factor) Scroll and pick a prime number from the list of the right edge:

And release it in the free zone, or in a prime circle (multiplication) Playing with 12*12:

Creating two prime numbers: Multiply them. Restoring to the original state. Throwing the prime numbers to the primes list.

Picking numbers from the big list of prime numbers: Playing with 2; 3; 5; 7; 11; 13. 1001; 30 and 30,030

(In the current version the top prime number available is 19,874,419)

Nearly 20 years of support and patience with me. And specially for the vision about the position of the places in "Touch decimals Place value ±" which allowed continuing the work well.