Computer Numerals for Efficient Numeracy for Bringing up Geniuses and Eradicating Poverty, Preventing Spread of AIDS(HIV),of Tuberculosis, and of Malaria, and Population Crisis
Computer numerals were developed using the concept of the binary system to form decimal numerals. This would allow addition, subtraction, multiplication, and division to be performed based solely on the knowledge of the 14 forms of the numerals and a few simple rules. These numerals were taught to several developmentally challenged individuals. After 4 days of instruction, they were able to add, subtract, and multiply with little assistance. A Protected Computer Numerals are proposed against forgery. An International Numeration System is proposed based on the form of computer numerals to facilitate international communication. A new type of abacus is proposed.
Basic education (Literacy and Numeracy) has been implicated by several researchers in the prevention of the spread of infectious diseases, as well as population control, based on the strong association between these factors and poverty and ignorance (1). The present paper proposes the use of the abacus numeral system to improve the efficiency of learning arithmetic. By memorizing the form of the numerals and applying a few simple rules, the ability to perform addition, subtraction, multiplication, and division appears to be attainable within a few months. The form of the decimal numerals is based on the concept of the binary system. However, knowledge of the binary system is not necessary in order to learn these numerals. A medieval Chinese numeral system (Table 1) by which addition and subtraction can be achieved by memorizing the form of the numerals has been reported previously (2).
Table 1. Medieval Chinese numerals used for calculation
However, because the Chinese numerals were developed on the quasi-quinary system, multiplication and division can only be performed by memorizing the multiplication table. A kind of decimal abacus (Papy's minicomputer) based on the binary system has been reported to facilitate addition and subtraction (3). Calculations of multiplication and division by Papy's minicomputer are possible without memorizing the multiplication table. They are, however, difficult and necessitate much time except for some simple cases. The abacus numeral system provides an improvement over the Chinese numeral system and Papy's minicomputer because multiplication and division can be performed easily without memorizing the multiplication table. The rules guiding sformation of Computer numerals are shown in Fig. 1.
Fig.1 Rules governing formation of computer numerals. A Order of height of ranks of numerals from the lowest are referred to as "ma", "mu", "mi", "ta", "tu", "ti", "sa". Two circles at a certain rank correspond with a circle one rank higher. Computer numerals for a given number are formed by putting open circles of that number at the lowest rank "ma" and then shifting the location of the rank of the circles by the above rule to minimize the number of circles. B Example of creating abacus numeral for number 5. C A rule of decimal system. One circle at rank "ta" and another circle at rank "mu" are transformed to one circle at rank "ma" of one higher order.
The actual forms of the numerals are shown in Table 2 .
Table 2.Table of Computer Numerals and International Numeration System.
Examples of addition by Computer Numerals are shown in Fig.2. The numerals to be added are joined to yield the resultant summed numeral.
Fig.2 Examples of addition: Resultant Computer Numerals are obtained by joining two Computer Numerals. Actual calculations can be simplified by moving coins or stones instead of circles (Fig.6), as is the case of a traditional abacus.
Examples of subtraction using these numerals are shown in Fig.3.
Fig.3 Examples of subtraction: Numerals to be subtracted are joined with filled circles or crosses. A filled circle and an open circle at the same rank cancel each other out. Actual calculations can be simplified by moving coins or stones instead of circles (Fig.6) , as is the case of a traditional abacus.
In the case of subtraction, the numerals to be subtracted are joined by filled circles or crosses to the subtracted numerals with open circles to yield the resultant numerals. When a filled circle or a cross and an open circle appear at the same rank, they cancel each other out. Multiplication is performed by reforming the numerals as follows: Multiplication by 1 entails the addition of a given number to zero once, resulting in the formation of the original numeral. Multiplication by 2 entails the addition of a given number to zero twice, resulting in the formation of a numeral with two open circles in the original ranks, which results in one open circle shifted one rank higher from the original rank [The abacus numeral for 2 consists of single circle in rank "mu" (one rank higher than rank "ma") and is equivalent to 2 circles in rank "ma"](Fig.4A).
Fig.4 Examples of multiplication： A 7x2 and 7x4 B 29x17 , 17 circles in rank "ma" become 17 in Abacus Numerals and 17 circles in A rank and B order higher from "ma" become Computer Numerals of 17 lifted by A ranks and B orders. The number 17 in triangle, diamond, and circle on the left hand side means 17 circles at the same rank and same order.
Multiplication by 9 is performed by joining the numeral nine times, resulting in 9 open circles in the original rank. That results in a numeral with each circle shifted three ranks higher in addition to each circle in the original rank [the Computer Numerals for 9 is equivalent to 9 circles in rank "ma" and consists of one open circle in rank "ta" (three ranks higher than rank "ma") and another in rank "ma"]. Multiplication by number of higher order is shown in Fig.4B. Examples of division are shown in Fig. 5.
Fig.5 Examples of division: Actual calculation can be simplified by moving coins or stones instead of circles (Fig. 6), as is the case of a traditional abacus.
In the same way, abacus numerals can be used for calculation of square roots without remembering the multiplication table.
Mathematics instruction using abacus numerals
The author used Computer Numerals to instruct 5 mentally handicapped volunteers at the Niihari-En Institute with the cooperation of the president, Mr. Kyuugo Tsukada. The students consisted of one 23 year-old who had undergone brain surgery and had an IQ of 61, three subjects (aged 22, 20 and 26 years) who had Down's syndrome and had an IQ of 39, 32 and 32, respectively, and one subject (26 years old) who was mentally deficient and had an IQ of 35. They could understand Arabic numerals but could not perform addition with the carrying procedure nor subtraction with the borrowing procedure. They could not perform multiplication. All students except one of the two with an IQ of 32 showed interest in the activity and after 4 days of instruction, 2 and half hours per day, 4 of the 5 volunteers were able to add, subtract and multiply, although with frequent errors if not assisted.
Protected Computer Numerals
In order to prevent forgery, which can be performed easily by adding extra circles to Computer Numerals, vacant positions are occupied by a "+" symbol to form protected Computer Numerals (Table 3). The protected Computer Numerals can be employed for important documents such as bonds.
Table 3. Protected Computer Numerals against forgery.
International Numeration System
Using Computer Numerals, an International Numeration System can easily be developed a priori, as outlined briefly in Table 2. The numeration to show the order can be developed from Fig. 1 by changing the consonants, while vowels remain unchanged. In order to show the order of magnitude from 1 to 9 or from -1 to -9, the consonants are changed from "t" and "m" to "s" and "t". For example, 6 E9 is changed to "miu satan" and 8 E-9 as "ta satakk". To show the order of magnitude from 10 to 90 or from -10 to -90, the consonants are changed to "f "and "p" instead of "t" and "m". And to show the order of magnitude from 100 to 900 or from -100 to -900, the consonants are changed to "sh"and "k" instead of "t" and "m". For example, 9E977 is changed to "tama shakapiupatiutan" and 9.76 E-778 to "tama nikk miuma miu kiukapiupasakk". The number 1998 is numerated as "ma tuan tama tun tama tan ta (in)". Thus, the International Numeration System appears to provide a means of facilitating international communication (4). Learning calculations including zero and negative numbers can be represented concretely by abacus numerals, and thus become understandable by intuition (c.f. Fig.3 C and D).
Table 4. International Numeration System.
Abacus for Computer Numerals
A newly proposed Abacus System is shown in Fig. 6. This abacus can be constructed simply using an iron plate and magnets, cloth and coins, or lines on the ground and small stones, ceramic pieces, shells or grains (5).
Fig.6 Proposed Abacus System: Two left columns: 47+24 = 71, middle column: 8 - 4 = 4, two right columns: 8 x 6 = 48.
The use of Computer Numerals proposed in this work appears to be able to facilitate mathematics instruction. It seems to be beneficial not only in developing countries but also in developed countries. Since mathematics learning is basic to any science including higher mathematics and physics, the use of abacus numerals may facilitate gifted children in becoming better researchers early in their careers. The logical nature of calculation by Computer Numerals seems to stimulate logical understanding of matters in early lives. Further, as the sciences and technologies develop, more time seems to be required by conventional methods of instruction for acquiring only basic knowledge for developing research in the sciences. Therefore, for further advances in the sciences in t he near future, innovation in instruction seems be required especially in mathematics, reading and writing , and in international languages. The use of an octonal system instead of the decimal system in common practice also appears to facilitate the application of the abacus numerals due to the simplified carrying and borrowing procedures.
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July 30~August 6, 2000, Chiba, JAPAN
The author is indebted to Mr. Kyuugo Tsukada for his cooperation in teaching Computer (Abacus) Numerals to the mentally handicapped youths. The author is also thankful to Dr. Nobuhiko Nohda and Dr. Jerry P. Becker for their valuable comments.
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