Number System:

· Every computer stores number, letters and other special character in a coded form.

· Number system are basically of two types:

Ø  Non-positional number system

Ø  Positional number system

· Non-positional number system:

o   We have symbol such as I for 1, II for 2, III for 3, IV for 4, and V for 5 etc.

o  It is very difficult to perform arithmetic with such as a number system; positional number systems were developed as the centuries passed.

· Positional number system:

o  In positional number system there are only a few symbols called digits, and these symbols represent different values depending on the position they occupy in the number.

o   The value of the each digit in positional number is determined by

§ The digit itself

§ The position of the digit in the number

§ The Base or Radix of the number system (where BASE is defined as the total number of digits available in the number system)

· There are only four types, which are including in the number system.

     (1) Decimal number system (Base – 10)

            Symbols: 0,1,2,3,4,5,6,7,8,9

     (2) Binary number system (Base – 2)

            Symbols: 0, 1

     (3) Octal number system (Base – 8)

            Symbols: 0, 1,2,3,4,5,6,7

     (4) Hexadecimal number system (Base - 16)

            Symbols: 0,1,2,3,4,5,6,7,8,9, A,B,C,D,E,F

Conversion of number system

Ø  A one of the number system that is converting into another number system for representation of data is called Conversion of number system. 

  1. Decimal to Binary
  2. Decimal to Octal
  3. Decimal to Hexadecimal
  4. Binary to Decimal
  5. Binary to Octal
  6. Binary to Hexadecimal
  7. Octal to Decimal
  8. Octal to Binary
  9. Octal to Hexadecimal
  10. Hexadecimal to Decimal
  11. Hexadecimal to Binary
  12. Hexadecimal to Octal

Decimal number system.

Ø  A number system that has 0 to 9 digits number is called Decimal number system.

Ø  The Decimal number system contains 10 unique symbols: 0,1,2,3,4,5,6,7,8,9.

Ø  In Decimal numbers which are before the decimal places is counting with following formula: 100, 101, 102, 103… 10n-1.

Ø  In Decimal numbers which are after the decimal places is counting with following formula: 10-1, 10-2, 10-3… and so on.

Ø  Any number of any magnitudes can be expressed by using of positional weighting.

                        Example: 6484


                        4 lots of         1(100) = 4 *       1   =       4

                        8 lots of       10(101) = 8 *     10  =      80

                        4 lots of     100(102) = 4 *    100 =    400

                        6 lots of   1000(103) = 6 *  1000 =  6000

                                                                         TOTAL       6484

          So, the Decimal number system Base or Radix is 10.

Binary number system.

Ø  A number system that has 0 and 1 digits number is called Binary number system.

Ø  0 and 1 is also called Bits i.e. Binary digits.

Ø  The Binary number system contains 2 unique symbols: 0, 1.

Ø  In Binary numbers which are before the decimal places is counting with following formula: 20, 21, 22, 23… 2n-1.

Ø  In Binary numbers which are after the decimal places is counting with following formula: 2-1, 2-2, 2-3… and so on.

            Example 1: 1010

1 lots of   2(21) = 1 *    2  =    2

0 lots of   1(20) = 0 *    1  =    0

0 lots of   4(22) = 0 *    4  =    0

1 lots of   8(23) = 1 *    8  =    8 

                                                      TOTAL   10

So, the Binary number system Base or Radix is 2.

Octal number system.

Ø  A number system that has 0 to 7 digits number is called Octal number system.

Ø  The Octal number system contains 8 unique symbols: 0, 1, 2, 3, 4, 5, 6, 7.

Ø  In Octal numbers which are before the decimal places is counting with following formula: 80, 81, 82, 83 … 8n-1.

Ø  In Octal numbers which are after the decimal places is counting with following formula: 8-1, 8-2, 8-3… and So on.

Binary

Octal

Binary

Octal

000

0

100

4

001

1

101

5

010

2

110

6

011

3

111

7

 

 

 

 

Example: 96


6 lots of 6(80) = 6 * 1 =  6

9 lots of 9(81) = 9 * 8 = 72

 TOTAL            78

So, the Octal number system Base or Radix is 8

Hexadecimal Number System.

Ø  A number system that has 0 to 9 digits and A to F Alpha is called Hexadecimal number system because of 16 digits.

Ø The Hexadecimal number system contains 16 unique symbols: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F.

Ø  In Hexadecimal numbers which are before the decimal places is counting with following formula: 160, 161, 162, 163, …, 16n-1.

Ø  In Hexadecimal numbers which are after the decimal places is counting with following formula: 16-1, 16-2, 16-3,…, and So on.

Example: 9E


E lots of 14(160) = 14*1 =  14

9 lots of 9(161)   = 9*16 = 144

                                                TOTAL     158

So, the Hexadecimal number system Base or Radix is 16.

Data Representation and Number System


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