Introduction to Number Systems

An Indian mathematician Aryabhatta is the father of the number system.  The technique to represent numbers in the computer system architectures known as number systems.

• There are four types of number systems: binary, octal, decimal and hexadecimal.
• The base of number system is called radix or base.
• Number systems help to represent numbers in a small symbol set.
• There are four number properties: commutative property, associative property, distributive property and identity property.
• Number properties are only associated with algebraic operations that are addition, subtraction, multiplication and division.
• The number system is simply a system to represent or express numbers. 
• It is a way of writing any number. 

Types of Number Systems

1. Binary Number (Base-2 or Radix-2) System 

          A number system that has only two digits 0 and 1 is called binary number system. The base/radix of the binary number system is 2.

• • The position of each number is the power of the base.
  • The English mathematician George Boole discovered binary number system.
  • 0 and 1 are Boolean values or Boolean digits.
  • Zero means FALSE, LOW or OFF.  
  • One means TRUE, HIGH or ON.
  • Binary number system takes the following format: (BN)₂         Where BN is binary number and base is 2.
  • In binary number system, binary number is made up of two symbols (or digits) 0 and 1.
  • Binary number system is said to Base-2 system.
  • The sequence of binary numbers goes as 00, 01, 10, 11, 100, 101, 110, 111, 1000 and so on.

       With representation, some examples are:

1. 1. (0)₂= 0 x 2⁰                  
   2. (1)₂ = 1 x 2⁰                 
   3. (10)₂ = 1 x 2¹ + 0 x 2⁰                           
   4. (11)₂ = 1 x 2¹ + 1 x 2⁰
   5. (100)₂ = 1 x 2² + 0 x 2¹ + 0 x 2⁰                                           
   6. (101)₂ = 1 x 2² + 0 x 2¹ + 1 x 2⁰
   7. (1000)₂ = 1 x 2³+ 0 x 2² + 0 x 2¹ + 0 x 2⁰                     
   8. (1001)₂ = 1 x 2³+ 0 x 2² + 0 x 2¹ + 1 x 2⁰ 
   9. (1000.0101)₂ = 1 x 2³+ 0 x 2² + 0 x 2¹ + 0 x 2⁰ + 0 x 2^-1+ 1 x 2^-2 + 0 x 2^-3 + 1 x 2^-4  

          The term binary digit is often abbreviated to the term bit. The term nibble is a group of 4 bits and byte is a group of 8 bits.

• • The Kilobyte (KB) unit of computer memory is a group of 1024 bytes.
  • A group of 1024 KB is called Megabytes (MB).
  • The collection of 1024 MB is said to be Gigabytes (GB).
  • One Terabytes (TB) unit of binary number is equal to 1024 GB.
  • The next higher data unit is Petabytes (PB).  

          2. Octal Number (Base-8 or Radix-8) System

          An Octal refers to the base-8 numbering system. A number system which has base-eight is called an Octal number system. It uses numbers from 0 to 7.

• • In Octal number system, an Octal number is made up of eight symbols (or digits) 0, 1, 2, 3, 4, 5, 6 and 7.
  • An Octal number system is said to Base-8 system.
  • An Octal number system takes the following format: (ON)₈                  Where ON is an Octal number and base is 8.
  • The sequence of an Octal numbers goes as 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 30 and so on.

           With representation, some examples are:

1. 1. (5)₈= 5 x 8⁰                 
   2. (10)₈ = 1 x 8¹ + 0 x 8⁰               
   3. (23)₈ = 2 x 8¹ + 3 x 8⁰     
   4. (80)₈ = 8 x 8¹ + 0 x 8⁰       
   5. (98)₈ = 9 x 8¹ + 8 x 8⁰                           
   6. (107)₈ = 1 x 8² + 0 x 8¹ + 7 x 8⁰
   7. (1000)₈ = 1 x 8³+ 0 x 8² + 0 x 8¹ + 0 x 8⁰ 

          3. Decimal Number (Base-10 or Radix-10) System

          In a decimal number system, each digit is denoted by integers from 0 to 9. Decimal number system is a base-10 system having 10 digits from 0 to 9. 

• • The decimal system consists of 10 single-digit numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. 
  • The decimal number system takes the following format: (DN)₁₀                  Where DN is a decimal number and base is 10.
  • The sequence of decimal numbers goes as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ……. and so on.

         With representation, some examples are:

1. 1. (9)₁₀= 9 x 10⁰               
   2. (13)₁₀ = 1 x 10¹ + 3 x 10⁰                      
   3. (128)₁₀ = 1 x 10² + 2 x 10¹ + 3 x 10⁰
   4. (1701)₁₀ = 1 x 10³ + 7 x 10² + 0 x 10¹ + 1 x 10⁰       = 1000 + 700 + 0 + 1   

         4. Hexadecimal Number (Base-16 or Radix-16) System

           In a hexadecimal number system, each digit is denoted by integers from 0 to 9 and each character is denoted by A to F.

• • Hexadecimal number system is a base-16 system having 10 digits from 0 to 9 and letters from A to F. 

• • The hexadecimal system consists of 10 single-digit numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
  •  It also consists of 6 letters: A, B, C, D, E and F.
  • The hexadecimal number system takes the following format: (HN)₁₆    Where HN is a hexadecimal number and base is 16.
  • The sequence of decimal numbers goes as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, ……. and so on.

        With representation, some examples are:

1. 1. (9)₁₆= 9 x 16⁰               
   2. (13A)₁₆ = 1 x 16² + 3 x 16¹ + A x 16⁰                 
   3. (1D7F)₁₆ = 1 x 16³ + D x 16² + 7 x 16¹ + F x 16⁰    = 4096 + 3328 + 112 + 15 = (7551)₁₆                            

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