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2                                     NUMBER SYSTEMS AND

                                                                 ENCODING SCHEMES











          Chapter Outline


          2.1 Decimal Number System (Base-10)                2.2 Binary Number System (Base-2)
          2.3 Octal Number System (Base-8)                   2.4 Hexadecimal Number System (Base-16)
          2.5 Expressing Numbers in Different Number Systems   2.6 Encoding Schemes






        Introduction

        Humans have been counting for thousands of years. Initially, the fingers of hands and feet were used for counting.
        Indeed, in different civilisations, people organised the counting of objects in groups of five (number of fingers in a hand),
        ten (number of fingers in two hands), and twenty (number of fingers in the hand and the feet). Thus, several number
        systems were developed for expressing numbers. The number systems enable us to express physical entities such as
        length, distance, speed, time, and weight. While the decimal system has evolved as the most popular number system
        for expressing numbers by humans, the computer systems are based on the binary system that involves only binary
        digits (often called bits) 0 and 1. Although computers use binary representation of numbers, while entering data from a
        keyboard, it is often convenient to aggregate the bits in groups of three (octal system) bits or groups of four (hexadecimal
        system). In this chapter, we will discuss decimal, binary, octal, and hexadecimal systems of representing numbers.

        2.1 Decimal Number System (Base-10)

        The decimal number system uses ten symbols and is known as the base-10 number system. The decimal system's
        ten most popular symbols are Arabic numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. While dealing with different number
        systems, it is often convenient to enclose the numbers in parentheses and write the base of the number system as
        a subscript. Thus, we would express the decimal numbers 2390, 4920.67, and 1001 as (2390) , (4920.67) , and
                                                                                               10
                                                                                                          10
        (1001) , respectively.
              10
               While dealing with different number systems, it is a good practice to enclose the number in parenthesis followed by
               its base value as a subscript. For example, we write the decimal number 298 as (298) .
                                                                                   10

        2.1.1 Positional Number System

        We say that a number system is positional if the position of a symbol in the representation of the number impacts
        its value. In this chapter, we will study positional number systems, even though there are number systems, such as
        the Roman number system, which are not positional. In contrast, the decimal number system is a positional number

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