Number-Systems Notes | EduRev

: Number-Systems Notes | EduRev

 Page 1


 Number Systems 
• Stone Age: knots, some stone marks 
• Roman Empire: more systematic notation  I, 
II, III, IV, V, VI, VII.VIII, IX, X, C=100, 
D=500, M=1000, L=50 
• Concept of zero by  
– Maya- I century, Hindu-V century 
• Positional-value systems: decimal, binary, 
octal, etc..  
Page 2


 Number Systems 
• Stone Age: knots, some stone marks 
• Roman Empire: more systematic notation  I, 
II, III, IV, V, VI, VII.VIII, IX, X, C=100, 
D=500, M=1000, L=50 
• Concept of zero by  
– Maya- I century, Hindu-V century 
• Positional-value systems: decimal, binary, 
octal, etc..  
Positional-Value System 
• The value of a digit (“digit” from Latin word 
for finger) depends on its position  
5  6  7  .  9  1  4 
MSD      Decimal     LSD 
                point    
Positional values      2    1    0        -1    -2    -3 
(weights)               10 10 10     10  10 10 
We will write ( 5 6 7. 9 1 4) 
10 
Page 3


 Number Systems 
• Stone Age: knots, some stone marks 
• Roman Empire: more systematic notation  I, 
II, III, IV, V, VI, VII.VIII, IX, X, C=100, 
D=500, M=1000, L=50 
• Concept of zero by  
– Maya- I century, Hindu-V century 
• Positional-value systems: decimal, binary, 
octal, etc..  
Positional-Value System 
• The value of a digit (“digit” from Latin word 
for finger) depends on its position  
5  6  7  .  9  1  4 
MSD      Decimal     LSD 
                point    
Positional values      2    1    0        -1    -2    -3 
(weights)               10 10 10     10  10 10 
We will write ( 5 6 7. 9 1 4) 
10 
Binary: 
Base-2 Number System 
1   0   1   1   1   1  .   0   0   1  
2   2    2    2    2    2       2    2    2 
  5   4   3    2    1    0       -1   -2   -3 
We write:  ( 1 0 1 1 1 1 . 0 0 1 )
2 
base point or radix 
Digits are called bits 
Page 4


 Number Systems 
• Stone Age: knots, some stone marks 
• Roman Empire: more systematic notation  I, 
II, III, IV, V, VI, VII.VIII, IX, X, C=100, 
D=500, M=1000, L=50 
• Concept of zero by  
– Maya- I century, Hindu-V century 
• Positional-value systems: decimal, binary, 
octal, etc..  
Positional-Value System 
• The value of a digit (“digit” from Latin word 
for finger) depends on its position  
5  6  7  .  9  1  4 
MSD      Decimal     LSD 
                point    
Positional values      2    1    0        -1    -2    -3 
(weights)               10 10 10     10  10 10 
We will write ( 5 6 7. 9 1 4) 
10 
Binary: 
Base-2 Number System 
1   0   1   1   1   1  .   0   0   1  
2   2    2    2    2    2       2    2    2 
  5   4   3    2    1    0       -1   -2   -3 
We write:  ( 1 0 1 1 1 1 . 0 0 1 )
2 
base point or radix 
Digits are called bits 
Binary Representation 
• The basis of all digital data is binary representation. 
• Binary  -  means ‘two’ 
– 1, 0 
– True, False 
– Hot, Cold 
– On, Off 
• We must be able to handle more than just values for 
real world problems 
– 1, 0,  56 
– True, False, Maybe 
– Hot, Cold, LukeWarm, Cool 
– On, Off, Leaky 
Page 5


 Number Systems 
• Stone Age: knots, some stone marks 
• Roman Empire: more systematic notation  I, 
II, III, IV, V, VI, VII.VIII, IX, X, C=100, 
D=500, M=1000, L=50 
• Concept of zero by  
– Maya- I century, Hindu-V century 
• Positional-value systems: decimal, binary, 
octal, etc..  
Positional-Value System 
• The value of a digit (“digit” from Latin word 
for finger) depends on its position  
5  6  7  .  9  1  4 
MSD      Decimal     LSD 
                point    
Positional values      2    1    0        -1    -2    -3 
(weights)               10 10 10     10  10 10 
We will write ( 5 6 7. 9 1 4) 
10 
Binary: 
Base-2 Number System 
1   0   1   1   1   1  .   0   0   1  
2   2    2    2    2    2       2    2    2 
  5   4   3    2    1    0       -1   -2   -3 
We write:  ( 1 0 1 1 1 1 . 0 0 1 )
2 
base point or radix 
Digits are called bits 
Binary Representation 
• The basis of all digital data is binary representation. 
• Binary  -  means ‘two’ 
– 1, 0 
– True, False 
– Hot, Cold 
– On, Off 
• We must be able to handle more than just values for 
real world problems 
– 1, 0,  56 
– True, False, Maybe 
– Hot, Cold, LukeWarm, Cool 
– On, Off, Leaky 
Number Systems 
• To talk about binary data, we must first talk about 
number systems 
• The decimal number system (base 10) you should 
be familiar with! 
– A digit in base 10 ranges from 0 to 9. 
– A digit in base 2 ranges from 0 to 1 (binary number 
system).  A digit in base 2 is also called a ‘bit’. 
– A digit in base R can range from 0 to R-1 
– A digit in Base 16 can range from 0 to 16-1 
(0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F).  Use letters A-F to 
represent values 10 to 15.  Base 16 is also called 
Hexadecimal or just ‘Hex’. 
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