Calculator on Java Using the Method of Raoelina Andriambololona

: The method of Raoelina Andriambololona consists to apply tools from matricial calculation and linear algebra in Arithmetics. This method permits to simplify arithmetic operations and to suppress many existing inconsistencies in writing, in enunciating


Introduction
RAOELINA ANDRIAMBOLOLONA has done research to improve the Malagasy scientific language instead of doing a translation from French language. He forged a scientific approach, starting from the concept itself.
The column -matrix layout of the basic vectors, which results from a convention, reveals the only four possible logical and coherent writings of integer and decimal numbers. The choice adopted, and therefore the convention chosen, must be consistent with the way of writing: line writing (LR) from the left-hand side (L) to the right-hand side (R) or line writing (RL) from the right-hand side (R) to the left-hand side (L). Thus, the arrangement of numbers is by construction consistent with writing; it must be the same for the reading, which must obviously follow the writing [1] [2] [4] In Malagasy language, we write numbers from the lefthand side (L) to the right-hand side (R), and we read it from the right-hand side (R) to the left-hand side (L). This inconsistency between writing and reading in malagasy language has stimulated RAOELINA ANDRIAMBOLOLONA to write several articles on this subject. He designed a rational logical matrix operation technique consistent with the Malagasy spoken numeration of numbers (malagasy people write the number of LRi (increasing order) and read LRd (decreasing order).
In this work, we design a calculator, to understand better the use of this calculation technique, and to become familiar with the notation LRi i.e left hand side (L) to right hande side (R) and increasing (i) order. This calculator does not only display the LRi numbers, but is designed with the calculation basis (illustrated in examples of this article). Raoelina Andriambololona and Hanitriarivo Rakotoson have programed it in javaScript [9]. Nirina Gilbert Rasolofoson and Raoelina Andriambololona have designed an ALU and code converter using matrix calculation [7]. In this paper, we apply this method in framework of computer science using java programming language.

Recall of the Method of Raoelina
Andriambololona in Arithmetics [1 -7] A number is meaningless unless we give the basis. For example 1101 makes no sense if we do not specify the number basis, for instance binary or decimal basis.

Basis, Systems, Writing of Numbers
Let a fixed natural integer numbers called "basis", and the sequence formed by the powers ( > 0, = 0, < 0) . is called the "basis vector" using the language of linear algebra.
The intrinsic (i.e. independent on the basis) number may be written on the basis vector as = where the , not all of which are zero, are natural integers strictly less than . is called the "component" of on the basis vector .
The uniqueness of is deduced from the linear independence of the basis vectors.
We have put the basic vectors in column -matrix or row -matrix form by following a well -defined order (either in decreasing (d) order or in increasing (i) order). The rowmatrix or column -matrix formed by the components ranked in a well-defined order represents the intrinsic number in the basis We have four choices corresponding to the conventions taken (column -matrix or row -matrix, increasing or decreasing order). We fix our choice, so that the writing of the number is compatible with the writing used (writing on row from left-hand side to right-hand side or from right-hand side to left-hand side).
We mark by a sign (by a comma in French, German, Malagasy, or by a dot in English), put after the component corresponding to , thus separating the components of with the positive and negative powers .
The values of , strictly less than the number , are the system digits.

Basis Column -Matrix
We introduce the columns -matrix basis vector (1) maybe written matricially as where the row -matrix , -( ) is the writing in increasing order of the number in the numerical basis ) +.

Remarks
a. The choice of the decomposition has been fixed so as to have the writing of numbers from left-hand side to the right-hand side in increasing order (LRi). b. Let us note that this arrangement (LRi) may be made to be consistent with the rules of the four operations (addition, subtraction, multiplication and division). c. Theorem The writing , -( ) is unique. d. Theorem All the of (1) and (2) are strictly less than , ( ∈ )0, 1, … , − 1+).
the numbers = (0, 1, 2, … . , − 1) are called the digits of the number in the numerical basis . e. We do not need to call upon the Euclidean division of by and its properties (in particular, the property of uniqueness and condition < ) which is already contained in the decomposition. f. Theorem The column -matrix basis defined in the paragraph 2.2.2 is a bijection of the row -matrix on (2).

The Rules of Addition
Let = ∑ # and 5 = ∑ 5 6 # , (2.3.1). we can assume 7 5 > 7 without losing in generality with 5 ≠ 0 is chosen in such a way that + 5 < for all and in particular for = 7′.
All the elements of all the columns of the row -matrix , -( + ′) are zero or strictly less than .
The addition is thus carried out from left-hand side to right-hand side in increasing order (LRi).
Example 1 Using the Method of Raoelina Andriambololona Add in the basis 10 the following intrinsic numbers and ′

Example 2
We suppose that we are in basis 10. The reader will notice the positions of the carry over that is not indicated.

Rules of Multiplication
Let and ′ be two integers, defined by the decompositions (1) We are getting In calculating the components of product × 5 that must be less than , we must carry over the deductions in the subtotals We will illustrate the calculation rules by explaining the products, subtotals and deductions in Example 3.

Calculator on Java Using the Method of Raoelina Andriambololona
In this third part, we will use Malagasy words like marika, isa, fanampiana, fangalana, fampitomboana, fizarana.... to name the classes of digits, numbers, addition, subtraction, multiplication and division.
Same overview of the program excerpt is given

Class Marika
This is the class for the object Marika. Marika is an object of the form .

Class Isa
This is the class for the object Isa. Isa = . + … . + . Code

Conclusion
Let us point out that the malagasy people read the numbers in increasing order LRi even if they write them in decreasing order LRd. The investigation to look for to improve this inconsistency has led Raoelina Andriambololona to use matrix calculation in arithmetics. The latter makes clear the writing and arrangement of operations on numbers. The LRi disposition is consistent with the rules of operations (addition, multiplication, subtraction, division), which cannot be the case of the international LRd writing of numbers with the usual rules of operations. Enunciating (reading of numbers) must be consistent with the writing of numbers too. This fact has encouraged us to write down a program on java, which is a free software, object-oriented, easy to learn, and platform-independent. It can be moved easily from one computer to another and it has the ability to run the same program on many different systems.