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Data Representation
Prepared by:
Dr. Anju Sharma
SMCA
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CONTENTS
Data Types
Complements
Fixed Point Representation
Floating Point Representation
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Data Representation in Digital
System OR Data Types Numbers used in arithmetic computations
Letters of the alphabet used in data
processing (ASCII Code) Other symbols used for specific purpose
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Number Systems
Base or R adix r system : uses distinctsymbols for r digi ts
Most common number system :Decimal,Binary, Octal, Hexadecimal
Positional-value(weight) System : r 2 r 1r 0.r -1
r -2 r -3
Multiply each digit by an integer power of r and then form the sum of all weighteddigits
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Number Systems ± Decimal
Base 10
± Ten digits, 0-9
± Columns represent (from right to left) units, tens,
hundreds etc.
123
1v102 + 2v101 + 3v100
or
1 hundred, 2 tens and 3 units
Each position is a power of 10
3052 = 3 x 103
+ 0 x 102
+ 5 x 101
+ 2 x 100
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Bases
The base of a number is often indicated by asubscript. E.g. (123)10 indicates the base-10number 123.
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Binary
Base 2
± Two digits, 0 & 1
± Columns represent (from right to left) units,twos, fours, eights etc.
1111011
1v26 + 1v25 + 1v24 + 1v23 + 0v22 + 1v21 + 1v20
= 1v64 + 1v32 + 1v16 + 1v8 + 0v4 + 1v2 + 1v1
= 123
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Decimal to Binary Conversion
123 z 2 = 61 remainder 1
61 z 2 = 30 remainder 130 z 2 = 15 remainder 015 z 2 = 7 remainder 17 z 2 = 3 remainder 1
3z
2 = 1 remainder 11 z 2 = 0 remainder 1
Least significant bit (rightmost)
Most significant bit (leftmost)
(123)10 = (1111011)2
Example Converting (123)10 into binary
Read the result upward to give an answer of
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Hexadecimal
Base 16
± Sixteen digits, 0-9 and A-F (ten to fifteen)
± Columns represent (from right to left) units,16s, 256s, 4096s etc.
7B
7v161 + 11v160 = 123
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Decimal to Hex Conversion
123z
16 = 7 remainder 11 (or B)7 z 16 = 0 remainder 7
Answer : (123)10 = (7B)16
Converting (123)10 into hex
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Binary to Hex / Hex to Binary
Even very long numbers can be convertedeasily, treating each hex digit independently.
0111 1011
7 B
1011 1001 0110 1111 1010
B 9 6 F A
E.g.
Each group of four binary bits maps on to a single hexdigit.
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COMPLEMENTS
Adding to 1 to the r¶-1 complement
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Finding Two¶s Complement
Step 1: First complement all the bits
(that is find one¶s complement)
± Make all 1s as 0s and all 0s as 1s
Step 2:Then perform increment by 1
± Add 0001b
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Two¶s Complement as -ve
Number Two¶s complement is -ve number because
binary addition of a n-bit number with it¶s
complement gives nbit result with all bits = 0s
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Highest Two¶s Complement format +
ve Number A highest positive arithmetic number is
when at msb there is 0 and all
remaining bits are 1s
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Lowest Two¶s Complement format
í
ve Number A lowest negative arithmetic number is
when at msb there is 1 and all
remaining bits are 0s
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Arithmetic Numbers
Two¶s complement format arithmetic number
Maximum 8-bit number = 0111 1111( +127)
Minimum 8-bit number = 1000 0000 (í
128)
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Arithmetic Numbers
Two¶s complement format arithmetic
number
Maximum 16-bit number = 0111 1111 1111 1111( +32767)
Minimum 16-bit number
= 1000 0000 0000 0000 ( í
32768)
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EXAMPLE
Number +16392 0100 0000 0000 1000
One¶s complement 1011 1111 1111 0111
+ 0000 0000 0000 0001í163921011 1111 1111 1000
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Binary subtraction A í B
Add A with two¶s complement of B to find
A í B, provided we use two¶s
complementation for representation í ve
numbers
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Example: Find 129 í 128
0000 0000 1000 0001 [= +129d]
1111 1111 1000 0000 [= í 128d]
0000 0000 0000 0001 [= + 1d]
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SUMMARY
Two¶s complement is found by first
finding 1¶s complement and then adding
0001b.
Two¶s complement gives negative of a
given number
Adding a number with it¶s two¶s
complement gives all bits = 0s
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Binary Arithmetic - Addition
Binary long addition works just like
decimal long addition.
1 0 0 1 1 1
0 0 1 1 1 0
010
11
01
11
00
1
+
Carried digits
Result
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OVERFLOW
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Floating Point Representation