Number Systems & Conversions

Number Systems & Conversions

NUMBER SYSTEMS

WHAT IS NUMBER SYSTEM?

Like many words and phrases, the phrase “number system” has more than one meaning.

  1. A number system is nothing but a way to represent numbers.
  1. A collection of things (usually called numbers) together with operations on those numbers and the properties that the operations satisfy.
  2. A system for representing (that is expressing or writing) numbers of a certain type.
  3. a collection of numbers together with operations, properties of the operations, and a system of representing these numbers.

TYPES OF NUMBER SYSTEMS

AS PER REQUIREMENTS THERE ARE VARIOUS TYPES OF NUMBER SYSTEMS.

  1. NON POSITIONAL NUMBER SYSTEM
  2. POSITIONAL NUMBER SYSTEM
  3. DECIMAL NUMBER SYSTEM
  4. BINARY NUMBER SYSTEM
  5. OCTAL NUMBER SYSTEM
  6. HEXADECIMAL NUMBER SYSTEM

Non POSITIONAL NUMBER SYSTEMS

  • Characteristics :
  • Use only a few symbols called digits
  • These symbols represent different values depending on the position occupy in the number

POSITIONAL NUMBER SYSTEMS

  • Characteristics :
  • Use only a few symbols called digits
  • These symbols represent different values depending on the position occupy in the number
  • The value of each digit is determined by :
  • The digit itself
  • The position of the digit in the number
  • The base of the number system

( base = total number of digits in the number system )

  • The maximum value of a single digit is always equal to one less the value of the base

Example of DECIMAL NUMBER SYSTEMS

236810 = ( 2 x 103 ) + ( 3 x 102 ) + ( 6 x 101 ) + ( 8 x 100 )

= 2000 + 300 + 60 + 8

BINARY NUMBER SYSTEMS

  • Characteristics:
  • A positional number system
  • Has only 2 symbols or digits ( 0, 1 ). Hence, its base = 2
  • The maximum value of a single digit is 1 ( one less than the value of the base)
  • Each position of a digit represents a specific power of the base (2)
  • We use this number system in our computers

Example of BINARY NUMBER SYSTEMS

101012 = (1 x 24 ) + ( 0 x 23 ) + ( 1 x 22 ) + ( 0 x 21 ) + ( 1x 20 )

= 16 + 0 + 4 + 0 + 1

= 2110

BIT

  • Bit stands for binary digit
  • A bit in computer terminology means either a 0 or a 1
  • A binary number consisting of n bits is called an n-bit number

OCTAL NUMBER SYSTEMS

  • Characteristics:
  • A positional number system
  • Has only 8 symbols or digits ( 0, 1, 2, 3, 4, 5, 6, 7 ). Hence, its base = 2
  • The maximum value of a single digit is 7 ( one less than the value of the base)
  • Each position of a digit represents a specific power of the base (8)

Example of OCTAL NUMBER SYSTEMS

Since there are only 8 digits, 3 bits ( 23 = 8 ) are sufficient represent any octal number in binary

Example

20578 = ( 2 x 83 ) + ( 0 x 82 ) + ( 5 x 81 ) + ( 7 x 80 )

= 1024 + 0 + 40 + 7

= 107110

HEXADECIMAL NUMBER SYSTEMS

  • Characteristics:
  • A positional number system
  • Has total 16 symbols or digits ( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F ). Hence, its base = 16
  • The symbols A, B, C, D, E and F represent the decimal values 10, 11, 12, 13, 14 and 15 respectively
  • The maximum value of a single digit 15 ( one less than the value of the base )

HEXADECIMAL NUMBER SYSTEMS

  • Characteristics:
  • Each position of a digit represents a specific power of the base ( 16 )
  • Since there are only 16 digits , 4 bits ( 24 = 16 ) are sufficient to represent any hexadecimal number in binary

Example

1AF16 = ( 1 x 162 ) + ( A x 161 ) + ( F x 160 )

= 1 x 256 + 10 x 16 + 15 x 1

= 256 + 160 + 15

= 43110

Converting a number of another base a decimal number

  • Method:
  • Step 1 : Determine the column ( positional ) value of each digit
  • Multiply the obtained column values by the digits in the corresponding columns
  • Calculate the sum of these products

Converting a number of another base a decimal number

Example:

40678 = ?10

40678 = 4 x 83 + 0 x 82 + 6 x 81 + 7 x 80

= 4 x 512 + 0 + 48 + 7

= 2048 + 0 + 48 + 7

= 210310