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Figura 1.
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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.
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Figura 1.
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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:
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Figura 2.
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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):
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Figura 3.
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The positive numbers have
a green-blue aspect (Figure 3).
Negative numbers are reddish
(Figure 4):
The labels that indicate
the amounts are yellow.
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Figura 4.
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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.
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Figura 5.
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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):
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Figura 6.
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Figura 7.
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In figure 6 is observed the result of the simple superposition of both numbers .
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Figura 8.
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Figura 9.
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Figura 10.
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Figura 11.
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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.
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Figura 12.
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Nummòlt turns the parentheses windows of Windows ®, and places in its interior the operations or numbers that were within the respective parentheses.
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Figura 13.
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When surrounding the cards
and loosen the button, appear a dialog box. We choose the option "new parenthesis"
(figure 14):
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Figura 14.
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Figura 15.
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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):
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Figura 16.
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Figura 17.
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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.
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Figura 18.
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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:
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Figura 19.
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Figura 20.
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Figura 21.
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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?
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Figura 22.
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Figura 23.
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Figura 24.
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Figura 25.
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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:
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Figura 26.
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Figura 27.
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Figura 26.
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Figura 29.
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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)):
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Figura 30.
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Figura 31.
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Finally,
we are going to solve an operation based on a geometric reasoning 99*99+99*2+1:
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Figura 32.
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We simplified the inner expressions
of the multiplications adding pairs of +1-1:
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Figura 33.
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We multiply:
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Figura 34.
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We solve the inner expressions
of the parentheses:
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Figura 35.
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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:
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Figura 36.
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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)
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Figura 37
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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.