Number system

 Topic Outline

  • Number system
  • Decimal number system
  • Binary number system
  • Octal number system
  • Hexadecimal number system

Number system

Set of values used to represent quantities is known as the number system. A number system can be used to represent the number of people in an organization or the number of students in a class. Digital computers represent all data in binary numbers. The total number of digits used in the number system is called it's base. The base is written after the number as subscript as 102. The number system is as follow:

(1) Decimal number system
(2) Binary number system
(3) Octal number system
(4) Hexadecimal number system

The decimal number system is used in general. The computer uses a binary number system. Octal and Hexadecimal number systems are also used in computers.

(1) Decimal Number System

It consists of ten digits from 0 to 9. These digits can be used to represent numeric values. The base of a decimal number system is 10. The value represented by the individual digit depends on the weight and position of the digit. Each number in the decimal number system consists of digits. These digits are located at different positions. 

  • Position of the first digit towards the right side of the decimal point is -1. 
  • Similarly the position of the first digit towards the left side of the decimal point is 0.
  • Value of the number is determined by multiplying the digits with the weight of their position and adding the results. This method is an expansion method.

Most Significant Digit

The leftmost digit of a number has the highest weight, so it is called MSD. Digit 2 in the number 2258 is the most significant digit.

Least Significant Digit

The rightmost digit of a number has the lowest weight, so it is called LSD. The digit 8 in the number 2258 is the least significant digit.

Example(1) 

The weight and positions of each digit of the number 2258 are as follow:


Position 

3 

2 

1 

0 

Weight 

103 

102 

101 

100 

Face Value 

2 

2 

5 

8 


Above Table indicates that:

The value of digit 2 = 2*103 = 2*1000 =2000 

The value of digit 2 = 2*102 = 2*100 =200 

The value of digit 5 = 5*101 = 5*10 =50 

The value of digit 8 = 8*100 = 8*1 =8 

The actual number can be found by adding values obtained by the digits as follow:

2000+200+50+8= 225810 

Example(2) 

The weight and positions of each digit of the number 234.67 are as follow:


Position 

2 

1 

0 

 

-1 

-2 

Weight 

102 

101 

100 

 

 10-1 

 10-2

Face Value 

2 

3 

4 

. 

6 

7 


Above Table indicates that:

The value of digit 2 =2*102 = 200 

The value of digit3 = 3*101= 30  

The value of digit4 = 4*100 = 4 

The value of digit6 = 6*10-1 = 0.6

The value of digit7 = 7*10-2 = 0.07 

The actual number can be found by adding values obtained by the digits as follow:

200+30+4+0.6+0.07= 234.67

(2) Binary Number System

Digital computers represent data and information in the binary number system. It consists of two digits 0 and 1. The base of a binary number system is 2. Each digit or bit can be 0 or 1. The positional value of each digit in the binary number system is twice that Place value or Face value of the digit of its right side. The weight of each position is a power of 2.

 Place value of the digits according to position and weight is as follow:


Position 

2 

1 

0 

Weight 

22 

21 

20 


Example(1)  Convert 111012  into a decimal number.

Position 

4 

3 

2 

1 

0 

Weight 

24 

23 

22 

21 

20 

Face Value 

1 

1 

1 

0 

1 


11101 = 1*24+1*23+1*22+0*21+1*20 

            = 1*16+1*8+1*4+0*2+1*1 

            = 16+8+4+0+1 

            = 2910 

Example(2)  Convert 111.1012  into a decimal number.


Position 

2 

1 

0 

 

-1 

-2 

-3 

Weight 

22 

21 

20 

 

2-1 

2-2 

2-3 

Face Value 

1 

1 

1 

. 

1 

0 

1 


111.101 = 1*22+1*21+1*20+1*2-1 +0*2-2+1*2-3 

               = 1*4+1*2+1*1+1/2+0/4+1/8 

               = 4+2+1+0.5+0+0.125 

               = 7.62510 

(3) Octal Number System

It consists of eight digits from 0 to 7 . Base of the octal number system is 8. Each digit position in this number system represents a power of 8. Any digit in this number system is always less than 8. The number 4618   is valid in this number system. The number  8538  is not valid in this number system because 8 is not a valid digit.

The place value of each digit according to position and weight is as follow:


Position 

3 

2 

1 

0 

Weight 

83 

82 

81 

80 


Example(1)  Convert 588 into a decimal number.


Position 

1 

0 

Weight 

81 

80 

FaceValue 

5 

8 


588 = 5*81+8*80 

       = 5*8+8*1 

       = 40+8 

       = 4810 

Example(2)  Convert 223.738 into decimal number.


Position 

2 

1 

0 

 

-1 

-2 

Weight 

82 

81 

80 

 

8-1 

8-2 

Face value 

2 

2 

3 

. 

6 

3 


223.638 = 2*82+2*81+3*80+6*8-1+3*8-2 

               = 2*64+2*8+3*1+6/8+3/64 

               = 128+16+3+0.75+0.046875 

               = 147.796875 => 147.79710 


(4) Hexadecimal Number System

It consists of 16 digits from 0 to 9 and A to F. Decimal numbers from 10 to 15 represent alphabet A to F. Base of the hexadecimal number system is 16. Each digit position in this number system represents a power of 16. The number 67416  is a valid hexadecimal number.

The value of A is 10, B is 11, C is 12, D is 13, E is 14, F is 15. 

The place value of each digit according to position and weight is as follow:


Position 

3 

2 

1 

0 

Weight 

163 

162 

161 

160 


Example(1)  Convert  3E16  into a decimal number.

3E16   = 3*161+E*160                                                         • E=14          

         = 3*16+14*1 

         = 48+14 

         = 6210 

Example(2)  Convert  1C3F.1416  into decimal number.

Position 

3 

2 

1 

0 

 

-1 

-2 

Weight 

163 

162 

161 

160 

 

16-1 

16-2 

Face Value 

1 

C 

3 

F 

. 

1 

4 



1C3F.1416 = 1*163+C*162+3*161+F*160+1*16-1+4*16-2       • C=12 , F=15 

                 = 1*4096+12*256+3*16+15*1+1/16+4/256 

                 = 4096+3072+48+15+0.0625+0.015625 

                 = 7231.078125 => 7231.078110 


Comments

  1. Mahrukh NaeemNovember 30, 2021 at 8:50 AM

    Love the way you make every difficult thing so simple and easy 😍👏

    ReplyDelete

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