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Ahmad Apriyanto
Graduate of Mathematics Education from Faculty of Education Universitas Siswa Bangsa Internasional (USBI), the transformation of Sampoerna School of Education (SSE) Jakarta.
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Saturday, September 28, 2013

Summary on Numbers and Numeration System

Created by Ahmad Apriyanto, Iis Rosita, and Novi Handayani

        Natural number idea comes after human’s culture, a long time ago human use notches to mark weather exchange because at that time weather exchange was very useful to their life like hunting, farming and so on. Therefore, the idea of natural number was not invented by human but it comes from nature, nature have some pattern, then human realize that, after that to represent it they use tally marks or any other way to count, that was the idea of natural number. It also showed that the beginning step of mathematics was counting because at that time human using mathematics just for marks nature pattern or solve their own problems like time to hold religion ceremony, in other words they count it. Because of that, all of mathematical concept are relating to counting because counting is the basic of all mathematical concept. As Morris Klein said that counting exist in almost all branch of mathematics like probability, statistic, analysis, function, and topology.
        Actually, different culture will develop different ways of counting. That happened because every culture has their own way of life, different need and different season and the important is they have their own problem to solve. For example, the dammara trib in Africa is trading tobacco stick for one sheep, that’s why they use only the multiplication of two like 2 sticks = 1 sheep. 4 sticks = 2 sheeps and so on. Every culture long time ago also use different symbol to count, for example Ishango use bone to count but Sumarian use stone to count.
        Besides counting, there are also numeral systems. There are two kinds of numeral arrangements; additional (unary) and positional. Unary numeral system (sign-value notation) is the simplest numeral system because every natural number is represented by a corresponding number of symbols. For example, if the symbol   is chosen, then the number 5 would be represented by /////. Moreover, the unary notation can be abbreviated by introducing different symbols for certain new values. For example, the symbol  / stands for one, the symbol * stands for ten, and the symbol # stands for a hundred, then the number 231 would be represented as ##***/. The benefit of this numeral arrangement doesn’t exist directly in the present numeral system, but it really plays an important role in theoretical computer science, such as Elias Gama Coding which commonly used in data compression, expresses arbitrary-sized numbers by using unary to indicate the length of a binary numeral.
Another numeral arrangement is positional numeral system (place-value notation), in this system, each symbol (called digit) conveys two things; face value and place value. The face value is the inherent value of the symbol, meanwhile place value is the power of the base which is associated with the position that the digit occupies in the numeral. For example is number 4564 in base ten, the symbol 4 has the same face value but different place value; the first 4 in the left denotes four 1000s and the second 4 in the right  denotes four 1s. The benefit of this numeral system is more efficient compared with the unary system.
There are some bases that used in ancient civilization and our present world as shown in the table below.

Name of System
Binary (dyadic)
Octal (octenary)
Decimal (decadic, denary)
Duodecimal (duodenary)
Hexadecimal (hexadecadic)
Vigesimal (vicenary)

There are some explanations about the reasons of choosing each number base. Babylonian constantly use of the number 60 as a base (known as Sexagesimal), which still exist in our division degrees, hours and minute, into sixty sub units. Babylonian were interested as they were in watching the stars, early came to believe that the circle of the year consist of 360 days. They knew that the side of regular in scribed hexagon is equal to the radius of the circle. This property suggesting the division of 360 into six equal parts and 60 being thus looked upon as a kind of mystic number, 60 was chosen because of its integral divisors 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 thus rendering work with its fractional parts very simple.
For base 20 (known as Vigesimal), numeration system that was used in Mayan numeration, was the usual mode of counting in many ancient cultures because humans have 10 fingers and 10 toes on which to count. In Duodecimal numeration, the number twelve has set its mark on many aspects of our environment. For example, there are twelve months in a year, a clock dial shows twelve hours, twelve items to make a dozen, twelve dozen to make a gross, and a foot is twelve inches.
Another base used is base 2 which was investigated and set into a serious numerical system by the eminent German mathematician Gottfried Wilhelm von Leibniz, to whom is attributed this very unmathematic thesis of abstract theology: God, represented by numeral 1, created the Universe out of nothing, represented by 0. As the time goes, in computer practice, octal and hexadecimal numbers are often used to represent large binary numbers which may be difficult to handle because of their length. The usefulness of these forms of notation arises from the ease of conversion between the number systems, as both eight (23) and sixteen (24) are powers of 2, the base of the binary system.
The numeral system used in our modern world is the base ten. By using the base ten, we can handle in symbolizing the very large numbers or the very small numbers (using scientific notation). For example, the diameter of the Milky Way, the galaxy in which our Earth is an insignificant speck of dust, is on the order of 100 trillion meters – a 1 followed by 20 zeros. Another example is the diameter of poliomyelitis virus is some 28 thousandths of a millionth part of a meter which is symbolized by 0.000 000 028 or 0.0728 or 2.8 x 10-8.
        Numerical systems, major scripts, and calendric system are developing. There are some similarities on the development of numerical systems, major scripts, and calendrical system such as if an area has numerical systems, the area also has major scripts and also calendrical system; the number systems, major scripts, and the calendric system depend on the area and needs of the areas; and number systems, major scripts, and calendrical system have spread to another area which close with the founder area. Each numerical system, major script, and calendrical system is different from one to another, each of them has uniqueness. Now, most of the areas in the worlds use same numerical systems, major scripts, and calendrical system based on mutual agreements to make communication easier.

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