Number Systems

Two types of number systems are:

•  Non-positional number systems
•  Positional number systems         

                               

•  Non-positional number systems

• Characteristics 

 Use symbols such as I for 1, II for 2, III for 3, IIII for 4, IIIII for 5, etc. 

Each symbol represents the same value regardless of its position in the number .

 The symbols are simply added to find out the value of a particular number.

•  Difficulty 

 It is difficult to perform arithmetic with such a number system 

• Positional Number Systems 

              Characteristics

 Use only a few symbols called digits

These symbols represent different values depending on the position they occupy in the number

 The value of each digit is determined by: 

1. The digit itself 

2. The position of the digit in the number 

3. 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 than the value of the base.

• Decimal Number System

              Characteristics 

1.  A positional number system 
2. Has 10 symbols or digits (0, 1, 2, 3, 4, 5, 6, 7, 8,  9).  Hence, its base = 10 
3. The maximum value of a single digit is 9 (one less than the value of the base) 
4. Each position of a digit represents a specific power of the base (10) 
5. We use this number system in our day-to-day life

Example

258610 = (2 x 103) + (5 x 102) + (8 x 101) + (6 x 100)

= 2000 + 500 + 80 + 6

• Binary Number System

 Characteristics 

1.  A positional number system 
2.  Has only 2 symbols or digits (0 and 1).  Hence its base = 2 
3.  The maximum value of a single digit is 1 (one less than the value of the base) 
4.  Each position of a digit represents a specific power of the base (2) 
5.  This number system is used in computers

Example

101012 = (1 x 24) + (0 x 23) + (1 x 22) + (0 x 21) x (1 x 20) = 16 + 0 + 4 + 0 + 1 =  2110

• Octal Number System 

Characteristics

1.  A positional number system 
2.  Has total 8 symbols or digits (0, 1, 2, 3, 4, 5, 6, 7).  Hence, its base = 8 
3.  The maximum value of a single digit is 7 (one less than the value of the base 
4.  Each position of a digit represents a specific power of the base (8)
5.  Since there are only 8 digits, 3 bits (23 = 8) are sufficient to 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 System 

Characteristics

1.  A positional number system 
2.  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
3.  The symbols A, B, C, D, E and F represent the decimal values 10, 11, 12, 13, 14 and 15 respectively 
4.  The maximum value of a single digit is 15 (one less than the value of the base)
5.  Each position of a digit represents a specific power of the base (16) 
6.  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