NUMBER SYSTEMS AND CODES

1
NUMBER SYSTEMS AND CODES
2
Outline

• Number systems
• Number notations
• Arithmetic
• Base conversions
• Signed number representation
• Codes
• Decimal codes
• Gray code
• Error detection code
• ASCII code

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Number Systems

• The decimal (real), binary, octal, hexadecimal
  number systems are used to represent information
  in digital systems. Any number system consists
  of a set of digits and a set of operators (, ?,
  ?, ?).

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Radix or Base
 
 
The radix or base of the number system denotes
the number of digits used in the system.
Decimal (base 10) 0 1 2 3 4 5 6 7 8 9
Binary (base 2) 0 1
Octal (base 8) 0 1 2 3 4 5 6 7
Hexadecimal (base 16) 0 1 2 3 4 5 6 7 8 9 A B C D E F
 
 
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Decimal Binary Octal Hexadecimal
00 0000 00 0
01 0001 01 1
02 0010 02 2
03 0011 03 3
04 0100 04 4
05 0101 05 5
06 0110 06 6
07 0111 07 7
08 1000 10 8
09 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F
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Positional Notation

• It is convenient to represent a number using
  positional notation. A positional notation is
  written as a sequence of digits with a radix
  point separating the integer and fractional part.
•  
•  
• where r is the radix, n is the number of
  digits of the integer part, and m is the number
  digits of the fractional part.

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Polynomial Notation

• A number can be explicitly represented in
  polynomial notation.
•  
•  
• where rp is a weighted position and p is the
  position of a digit.

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Examples

• In binary number system
•  
• In octal number system
• In hexadecimal number system

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Arithmetic
Addition In binary number system,
(101101)2 (11101)2 1111 1
  101101
  11101
  1001010
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Addition
In octal number system,
(6254)8(5173)8 1 1
  6254
  5173
  13447
In hexadecimal number system,
(9F1B)16 (4A36)16 1 1
  9F1B
  4A36
  D951
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Subtraction
In binary number system,
(101101)2 -(11011)2 10 10
 - 101101
 - 11011
  10010
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Subtraction
In octal number system,
(6254)8 -(5173)8 8
 - 6254
 - 5173
  1061
In hexadecimal number system,
(9F1B)16 -(4A36)16 16
 - 9F1B
 - 4A36
  54E5
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Multiplication
In binary number system,
(1101)2 ? (1001)2
? 1101
? 1001
1101
0000
  0000
  1101
  1110101
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Division
In binary number system,
(1110111)2 ?(1001)2 1101
1001 1110111
1001
  1011
1001
    1011
  1001
  10
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Base Conversions

• Convert (100111010)2 to base 8

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Base Conversion

• Convert (100111010)2 to base 10

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Base Conversion

• Convert (100111010)2 to base 16

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Base Conversion from base 8

• Convert (372)8 to base 2
• Convert (372)8 to base 10
• Convert (372)8 to base 16

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Base Conversion from base 16

• Convert (9F2)16 to base 2
• Convert (9F2)16 to base 8
• Convert (9F2)16 to base 10

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Binomial expansion (series substitution)

• To convert a number in base r to base p.
• Represent the number in base p in binomial
  series.
• Change the radix or base of each term to base p.
• Simplify.

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Convert Base 10 to Base r

• Convert (174)10 to base 8
• Therefore (174)10 (256)8

8 1 7 4 6 LSB
  8 2 1 5  
    8 2 2 MSB
      0 0  
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Convert Base 10 to Base r

• Convert (0.275)10 to base 8
• Therefore (0.275)10 (0.21463?)8

8 ? 0.275 ? 2.200 MSD
8 ? 0.200 ? 1.600  
8 ? 0.600 ? 4.800  
8 ? 0.800 ? 6.400  
8 ? 0.400 ? 3.200 LSD
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Convert Base 10 to Base r

• Convert (0.68475)10 to base 2
• Therefore (0.68475)10 (0.10101?)2

2 ? 0.68475 ? 1. 3695 MSD
2 ? 0.3695 ? 0.7390  
2 ? 0.7390 ? 1.4780  
2 ? 0.4780 ? 0.9560  
2 ? 0.9560 ? 1.9120 LSD
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Signed Number Representation

• There are 3 systems to represent signed numbers
  in binary number system
•  Signed-magnitude
• 1's complement
• 2's complement

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Signed-magnitude system

• In signed-magnitude systems, the most significant
  bit represents the number's sign, while the
  remaining bits represent its absolute value as an
  unsigned binary magnitude.
• If the sign bit is a 0, the number is positive.
• If the sign bit is a 1, the number is negative.

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Signed-magnitude system
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1's Complement system

• A 1's complement system represents the positive
  numbers the same way as in the signed-magnitude
  system. The only difference is negative number
  representations.
• Let be N any positive integer number and be a
  negative 1's complement integer of N. If the
  number length is n bits, then

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Example of 1's Complement

• For example in a 4-bit system, 0101 represents 5
  and
• 1010 represents ?5

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1's Complement system
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2's Complement system

• A 2's complement system is similar to 1's
  complement system, except that there is only one
  representation for zero.
• Let be N any positive integer number and
• be a negative 2's complement integer of
  N. If the number length is n bits, then

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Example of 2's Complement

• For example in a 4-bit system, 0101 represents 5
  and
• 1011 represents ?5

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2's Complement system
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Addition and Subtraction in Signed and Magnitude
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Addition and Subtraction in 1s Complement
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Addition and Subtraction in2s Complement
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Overflow Conditions

• Carry-in ? carry-out
• 0111 1000
• 5 0101 -5 1011
• 3 0011 -4 1100
• -8 1000 7 10111
• Carry-in carry-out
• 0000 1110
• 5 0101 -2 1110
• 2 0010 -6 1010
• 7 0111 -8 11000

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Addition and Subtraction inHexadecimal System
Addition
Subtraction
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Codes

• Decimal codes
• Gray code
• Error detection code
• ASCII code

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Decimal codes
Decimal Digit BCD Excess-3 2421
8421
0 0000 0011 0000
1 0001 0100 0001
2 0010 0101 0010
3 0011 0110 0011
4 0100 0111 0100
5 0101 1000 1011
6 0110 1001 1100
7 0111 1010 1101
8 1000 1011 1110
9 1001 1100 1111
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Gray Code
Decimal Equivalent Binary Code Gray Code
0 0000 0000
1 0001 0001
2 0010 0011
3 0011 0010
4 0100 0110
5 0101 0111
6 0110 0101
7 0111 0100
8 1000 1100
9 1001 1101
10 1010 1111
11 1011 1110
12 1100 1010
13 1101 1011
14 1110 1001
15 1111 1000
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Error detection code
Parity Bit (odd) Message
1 0000
0 0001
0 0010
1 0011
0 0100
1 0101
1 0110
0 0111
0 1000
1 1001
1 1010
0 1011
1 1100
0 1101
0 1110
1 1111
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Error detection code
Parity Bit (even) Message
0 0000
1 0001
1 0010
0 0011
1 0100
0 0101
0 0110
1 0111
1 1000
0 1001
0 1010
1 1011
0 1100
1 1101
1 1110
0 1111
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ASCII Code

• ASCII American Standard Code for Information
  Interchange.
• Used to represent characters and textual
  information
• Each character is represented with 1 byte
• upper and lower case letters a..z and A..Z
• decimal digits -- 0,1,,9
• punctuation characters -- , .
• special characters -- _at_ /
• control characters -- carriage return (CR) , line
  feed (LF), beep

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Assignment 1

• Page 74
• 1.1 Only AB and A?B (a), (c), (f), and (g)
• 1.2 Only AB and A?B (a), (c)
• 1.3 Only AB and A?B (a), (c)
• 1.4 (a), (c), (e)
• 1.5 (a), (c), (e)
• 1.6 (a), (e)
• 1.7 (a), (b)
• 1.8 (a), (b)
• 1.10 (a), (c)
• 1.11 (a), (c)
• 1.12 (a), (c)
• 1.13 (a), (b)