Numeral System

List of Numeral System - Unary, Binary, octal and Decimal Number System

A numeral system (or a system of numeration) is a writing representation of numbers in a given set, using digits or other symbols in a sorted way. The same set of symbols can be used in a different numeral system to define different number like “10” represents the number ten in the decimal numeral system and the number two in the binary number system. The numeral system is also known as number systems. The number which is represented by the numeral is called its value.
Ideally, a numeral system will:

  • • Represent a useful set of numbers.

  • • Give every number represented in the numeral system a unique representation.

  • • Clearly display the arithmetic and the algebraic construction of the number.

  • For example, the usual decimal numeration gives every non-zero number a unique representation, as a finite number of sets. However when decimal representation is used for a rational number, then such numbers contains an infinite number of representation for example 3.36 is also written as 3.35999, 3.36000, 3.360 etc., all of which meant the same except for some scientific and another context where a greater precision is needed.

    Why numeration systems exist

    The need of the numeration system can be described by the help of an example by illustrating the three following primary reasons: 

    First, it is necessary to tell the number of items contained in a collection or set of those items. To do that, you must have some method for counting the items. The total number of items is described as the cardinal number. For example, if a basket contains 30 oranges, then 30 would be a cardinal number since it tells how many of an item there are.

    Second, Numbers can also be used to represent the sequence or order of items. For example, the individual oranges in the basket could be sequenced according to a way in which they were picked.

    Finally, numbers can be used for the purpose of identification of a particular item. Some method must be devised to keep the record of savings accounts, credit card accounts, drivers' licenses, and other kinds of records for different people separated from each other.

    History

    The most commonly used numeral system is the Hindu-Arabic numeral system. Two Indian mathematicians are credited to developing it. Aryabhata who developed the place value notation in the 5th century and Brahmagupta who introduced the symbol for zero a century later. The numeral system which is developed by the Indians slowly spread to other surrounding countries due to their commercial and military activities. Even today, the Arabs call their numeral system “Raqam Al-Hind”. The Arabs translated the Indian numeral system and spread them to the world due to their trade links. Hence the current western numeral system is the modification of the Hindu numeral system which was developed in India, which exhibits a great similarity with Sanskrit- Devanagari notation. 

    The bases of numeration systems

    All the numeration systems have been created by taking certain numbers as the base. The base of a numeration system is the highest number that can be counted without the repetition of any previously counted number. In the decimal system used in most parts of the modern world, the base is 10. 

    The base chosen for a numeration system often reflects actual methods of counting used by humans. For example, the decimal system may have developed because most humans have ten fingers because it an easy way to create numbers to count off one's ten fingers at a moment.

    Position in Numeral System

    In a positional base b numeral system, b basic symbols (or digits) corresponding to the first b natural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the given formula is defined. The symbol in the last position has its own value, and as it moves to the left its value is multiplied by the base.

    Number b= [dN…d2d1d0]b = \[\sum_{n-0}^{N}dn\:b^{n}=\] d0b0 + d1b1 + d2b2 + … + dNbN


    b - numeral system base
    dn - the n-th digit
    n -starts from negative number if the number has a fraction part.
    N+1 - the number of digits

    List of numeral systems

    Unary numeral System

    One of the simplest numeral systems is the unary numeral system, in which every natural number is represented by a corresponding number of symbols. If a symbol “+” is chosen, for example, then the number four is represented as “++++”. The unary system is limited for the small numbers, although it plays an important role in data compression in computer science. 
    The unary numerals are also used in the neural circuits.

    Binary Numeral System

    In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or the binary numeral system, which uses only two symbols: typically "0" (zero) and "1" (one). The base-2 numeral system is a positional notation with a base of 2. Each digit is referred to as a bit. The application of binary systems is in digital computing, imperial, and customary volume.

    Here is an example:

    101012 = 1×24+0×23+1×22+0×21+1×20 = 16+4+1= 21
    101112 = 1×24+0×23+1×22+1×21+1×20 = 16+4+2+1= 23

    The octal numeral system

    The octal numeral system is a base 8 numeral system which consists of any of the following eight (8) symbols or digits namely 0, 1, 2, 3, 4, 5, 6, and 7. Octal numerals can be made from binary numerals by grouping binary digits following each other continuously into a group of three. Octal are used for compact notation of binary numbers.

    Here is an example:

    308 = 3×81+0×80 = 24
    43078 = 4×83+3×82+0×81+7×80= 2247

    Decimal Numeral System

    A decimal (dec) number system is a base 10 numeral system which consists of any of the following ten (10) symbols or digits namely 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. A decimal point positioned at any place within a decimal number is used to generate fractional numbers. Decimal numeral system is generally used by modern civilization.

    To represent the value 10 we have to write it by using two digits. To represent the value 100 we have to write it by using three digits, hence the use of multiple digits to represents the higher values. Each of these upcoming digits is associated with place value. Each place value is associated with a power of ten.

    105104103102101100
    Hundreds of ThousandsTens of ThousandsThousandsHundredsTensOnes
    100000100001000100101


    Here is an example:
    253810 = 2×103+5×102+3×101+8×100

    Hexadecimal Numeral System

    The Hexadecimal Numeral System is a base-16 numeral system. Therefore, the 10 digits 0 to 9 and the 6 letters A to F are used in a hexadecimal numeral system. The Hexadecimal digit A is represented by the decimal number 10 and the Hexadecimal digit F is represented the decimal number 15.

    Here is an example:
    2816 = 2×161+8×160 = 40
    2F16 = 2×161+15×160 = 47

    DecimalBinaryOctalHexadecimal
    0000000
    1000111
    2001022
    3001133
    4010044
    5010155
    6011066
    7011177
    81000108
    91001119
    10101012A
    11101113B
    12110014C
    13110115D
    14111016E
    15111117F