nummòlt help files: For UNO magazine: unpublished. - nummòlt archivos de ayuda: Para la revista UNO: No publicado. 

nummòlt.Org is an endeavour to explain mathematical by means of graphical computer science applications. One is not simple static or preprogrammed graphs like a film. They are small computer science toys that behave under rules inspired by the physics, and therefore in the mathematics. The work of nummòlt.Org is more general, but in this one article we treat of the basic and original piece: nummòlt: The manipulation of whole numbers with its three defined operations: Addition, Subtraction and Multiplication. The program allows to make divisions, only in the sense of inverse operation of the multiplication. And never like resolution of a created problem. In order to solve divisions, it is necessary to introduce the concept of rational number. And the analogy of nummòlt no longer is useful. Our applications try to create clear images that they can constitute a correct idea of the operation of the numbers, and allow to reason, using mentally what it has been seen in the computer screen. 
In 1993, a little worried about the use of the calculator like habitual method of calculation, I began an investigation on the visual numerical calculation with manipulation of blocks.
The elementary operations in the school, always have had a magical aspect, (based on rules and tables) and in a single step always in the sense of the resolution.
The calculator, represents the upper expression of the magic: The fact to tighten a key, gives the result us.
I decided to go in opposite sense:
To try to look about what it happens within the computer (or of the head) when a calculation is made. And to avoid the use of rules, to be able to get to make reasonings.
The result, in 1998 after three attempts and three years of work, was nummòlt.
Nummòlt is a calculator with small manipulable blocks that are due to be modifying to obtain the resolution of an initial expression.
I have persecuted to construct a set of computer science facts coherent, and what here I explain, it is the application, result of my reflections.
Throughout this explanation, I am going to avoid many aspects manipulative, very difficult to explain in a text, and send to the real use of the program for the interested people.
Nummòlt is a program of public, gratuitous and accessible use: they can find and to unload at:
First of all , nummòlt consists of an editor of numerical expressions and elementary operations. Once written the expression, a button is due to press, and appears the expression in graphical form. The amounts that appear, cannot be modified graphically as far as the resulting amount. What one considers graphically always represents the same amount. What we modified graphically is the form to represent it.
Let us begin by the representation way:
The numbers are visualized by small cards. The units by squares of value 1, the tens, by rectangles of value 10, and so on.
The small cards, stick to each other, are ordered and they are piled up of an automatic form, following the principle of which never a great card hides a small card.
Of this form, the cards unit, always are above of everything in the visual representation (that is to say, technically, they are the last ones that are drawn), because they are smallest.
Let us see for example the 123: (fig 1)
The 100 remain to the bottom, above has two cards of 10 and finally, the 3 of 1.
Figura 1.

As it is possible to be seen, there is a small horizontal and vertical phase angle to visualize better the depth of the expression.
The stages single take place when there are cards in an order. That is to say, when there is no amount in an order, the corresponding stage does not take place (Figure 2) can be verified in number 1002:

Figura 2.

In case the reading of these cards caused some difficulty, there is an automatic device of reading, that is put into operation when mouse upon a card set leaves itself (Figure 3):

Figura 3.

The positive numbers have a green-blue aspect (Figure 3).
Negative numbers are reddish (Figure 4):
The labels that indicate the amounts are yellow.

Figura 4.

In order to transfer the small piles of cards (the numbers) it is necessary to drag with the left button of the mouse tightened on the deepest card. The drag on one upper level, that is, of lower order, or lower in its own order, causes the decomposition of the number in two, based on the selected card.

In figure 5 we see the turn out to drag the -246 by the last ten card:

Figura 5.

The background of the screen, has become darker. This happens when the number which it are expressed in the board is not simplest possible.
Therefore, already we have expressed most elementary of nummòlt: the sum. If the sum of two numbers considers, appear two small piles of cards on a dark gray background. If they are transferred and they are superposed, the result is the sum of the two numbers.
Unless in some of the orders they appear more than 9 cards. Then, of 10 in 10, they are due to turn the excess of cards others of the upper order:
Look please at the sum of 251 + 382 (Figure 6 and 7):

Figura 6.
Figura 7.

In figure 6 is observed the result of the simple superposition of both numbers .

The conversion of 10 tens in 1 hundred is made by means of a double click on the column of the tens. The program turns 10 cards of ten 10 cards 00, and creates a new one of 100. Cards 0 or 00 are erased little by little, by a permanent system, that cleans the board of this type of expressions. Figure 7 shows the final result: 633.
As we have seen before, there are positive cards, and negative.
They are annulled among them, if they are of the same order.
This is the base of subtraction on nummòlt.
Let us see an example that requires a subtraction with borrowing 25-18:

Figura 8.
Figura 9.
Figura 10.
Figura 11.

In figure 8 the operation considers. By the properties of cards, 5 negative cards of unit are annulled, with the 5 positive units of the 25 (figure 9). A ten is decomposed in its units (Figure 10) and are annulled the positive and negative tens, and the 3 corresponding negative units with the positive ones. Result: 7.

One of the intentions of this program, is to deal with naturalness the negative numbers: The negative result of a subtraction, does not need a complementary explanation, (speaking of thermometers and  temperatures below zero or other slippery analogies).
Simply, there are positive and negative cards, like the matter and the antimatter are annulled among them. If there are more negative cards in the end, the result is negative, and if it happens the opposite, the result is positive.
In order to be able to continue, it is necessary to introduce the parenthesis concept now. Although for the sum, it is not necessary, it agrees to show now how the interpreter of nummòlt visualizes the following expression: ((2+4)+(1+3)):

Figura 12.

Nummòlt turns the parentheses windows of Windows ®, and places in its interior the operations or numbers that were within the respective parentheses.

The resolution process, already is evident. First, to solve the inner operations of the parentheses, when being solved the operations (of sum in this case) the bottom of each one of the windows becomes in a ligther color and in this point, the window can be eliminated (the parenthesis) clicking double on a zone without cards of the own clarified window.
Also it is possible to create new parentheses. In this one point, only by diversion. If we mark a window (of selection) dragging the mouse with the left button tightened, appears a rectangle formed by a black line. The rectangle must include the cards that we want to introduce in the parenthesis (a new window) (Figure 13).

Figura 13.

When surrounding the cards and loosen the button, appear a dialog box. We choose the option "new parenthesis" (figure 14):

Figura 14.
Figura 15.

This operation has been made to show the mechanism of parenthesis creation, and for to see the result (figure 15), and as a introductory step to the way to express the multiplication.
The parentheses have the sign property. There are positive parentheses, and negatives. Here are two different ways to express the same amount. (- (- 2)) (figure 16) and ((2)) (Figure 17):

Figura 16.
Figura 17.


When we double_click on the red parenthesis, the window disappears, and 2 number interior (negative) becomes 2 positive. When we double_click on the positive parenthesis, simply, disappears the window, and the 2 interior falls on the back window.

The parentheses that are created during the resolution are always positive.
In this point, and with the knowledge of the parentheses in nummòlt, we can see how the multiplications are expressed:
I must clarify, that this is not, nor I try that it is, the best way to explain the multiplication. But given the analogy that raises nummòlt, és the most coherent way.
A multiplication of type 3 * 7, considers in the form ([ + 3 ] * [ + 7 ] + 0).
Figure 18 shows it:

Figura 18.

The windows lined to 45º (parenthesis or hooks in the expression) are parenthesis with special properties for the multiplication. They only exist when they contain something. (the multiplication by zero, does not have sense here).
The parenthesis piling up, expresses the priority of the multiplication with respect to the rest of operations. If we raised 2 + 3 * 4,  occurs by understood that first there must solve 3 * 4 and soon to add 2. There are more reasons. The multiplication is considered  like a reversible operation, and by steps. Therefore, in the process of multiplication, it is isolated of the rest of operations, for greater clarity and accuracy.
In order to carry out the multiplication, a unique procedure in the program is due to follow: To drag cards from a window to the other, of one in one.
Whenever this becomes, the transferred card, becomes zero, and the multiplication of the card transferred by the totality of cards takes place in the back window . The result, appears in the back window (between the two windows of multiplication). The procedure must be repeated while there are cards. We follow the previous operation  (Figure 18) and solve:

Figura 19.
Figura 20.
Figura 21.

For greater clarity and simplicity, we have operated dragging 3 cards of the left, adding groups of 7 units to the back window. Is not iprescindible to act thus. Any card displacement gives a correct result. In order to understand this, he is better to resort to the analogy of the multiplication like expression of an area. There is a good first subject of reflection with the students here: Why is indifferent the card that we drag?

As I have insinuated previously, the passages in the mechanism of the multiplication, are reversible. Let us imagine that we are in the step corresponding to figure 20. It is necessary to mark a parenthesis, as we have done in figure 13 on a group of 7 cards. Then it appears the picture of dialogue of figure 14. In this case the button of "inverse operación of multiplication" is due to tighten; (“/”) And the program asks if it is desired to obtain operation 7 * 1 or 1 * 7. (To add to the right 7 units or to add to the left 1 unit)
But, and if it is wished to start off of a state like the one of figure 21? Without no multiplication in march? And if it is wanted to raise a division with remainder?
We are going to experiment dividing 13 between 3:
nummòlt does not make divisions properly. We must start off of number 13, and manipulate it.

Figura 22.
Figura 23.
Figura 24.
Figura 25.

In this resolution, I have omitted many steps to be able to see the fast result. It has been grouped, and in the end it exceeds a 1. It is the remainder of the division.
Finally, it only is to explain a last resource of nummòlt: The neutral element of the sum: the 0 can be expressed here of another form. Adding positive and negative amounts at the same time, and in the same amount. This is obtained clicking with the right button of the mouse on any free space of any window. When doing this, appears a dialog box that asks what positive and negative amount at the same time is wanted to add.
Until now, we have seen that when we superpose equal amounts of different sign, everything becomes zeros, and by an automatic device of harvesting of zeros, everything disappears. This procedure is reversible: From anything, we can make appear in negative positive and the wished amount. It does not affect the result. Let us see an example of the utility of this mechanism: Let us express of a simpler graphical form number 99998:

Figura 26.
Figura 27.
Figura 26.
So that he appears +2 -2 is necessary, as is already said, to click with the right button of the mouse. Then it appears a dialog box that allows at the same time to choose the amount required in positive and negative cards. When adding the +2 to the 99998, through multiple doubles click on the columns, expression 100000 - 2 is obtained. A much more comfortable expression to manipulate with nummòlt, and to do a quick mental calculus.
Nummòlt can be useful simply to interpret correctly the written expressions.
Let us see as it interprets ambiguous expressions: For example 2 + 3 * 2:
Figura 29.

The priority of the multiplication is always considered on the sum or the subtraction, by principle, and the same structure of the program.
Expressions with parenthesis and without visualizing multiplication operators -(2(2+4)):

Figura 30.
Or expressions, shown here by the pleasure to see them ((((((((((1)))))))))):
Figura 31.

Finally, we are going to solve an operation based on a geometric reasoning 99*99+99*2+1:

Figura 32.

We simplified the inner expressions of the multiplications adding pairs of +1-1:

Figura 33.

We multiply:

Figura 34.

We solve the inner expressions of the parentheses:

Figura 35.

We suppress the parentheses and we carry out the sum, and we turn the groups of 10 cards of the same order in one of the upper order:

Figura 36.

Finally, I must clarify technical questions: Nummòlt, does not calculate anything. The program consists of the rules of behavior of the whole numbers, parenthesis, etc. The program does not verify operations. In an inferior window, read the board, and interpret what there is in him. The mechanisms of reading of the board, have as limit the Long Integer (2^30) Therefore greater numbers cannot be read nor be written. But the program in himself, does not have amount limit. (Only the dimension of the screen) You can test with a multiplication of type 100000000 * 100000000 * 100000000. (Not very often a quadrillion with all the zeros in a computer can be seen) (Figure 37)

Figura 37

How much greater it is the screen, better it works.
Nummòlt can be amusing to make calculations with really great numbers.
But the initial edition window only admits numbers of maximum 9 numbers.
Simultaneously, it is necessary to clarify that errors with the graphical manipulation cannot be committed. The rules are strict, and a number of a parenthesis cannot be transferred, to another one (except in the special case of the multiplication), or any manipulation that can alter the result. The result is always upon the board. Any modification of the total amount cannot be made that expresses the board. The total amount is only introduced by the initial edit box, before pushing the button that translates the expression written to the graphical and manipulable expression.
Nummòlt is a tool to raise numerical operations and to represent them graphically. It does not have any preprogrammed activity. It is thought like tool to help to visualize numbers. The final adressee of the program, is the student, but the support of the professor is necessary.
Our applications try to create clear images that they can constitute a correct idea of the operation of the numbers, and allow to reason, using mentally what it has been seen in the screen.

June 2002

© 2002: Maurici Carbó Jordi.

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