Data Representation (1) (1)

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Data Representation

Prepared by:

Dr. Anju Sharma

SMCA



CONTENTS

Data Types

Complements

Fixed Point Representation

Floating Point Representation



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



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



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



Bases

The base of a number is often indicated by asubscript. E.g. (123)10 indicates the base-10number 123.



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



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



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



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



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.





COMPLEMENTS

Adding to 1 to the r¶-1 complement



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



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



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



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



Arithmetic Numbers

Two¶s complement format arithmetic number

Maximum 8-bit number = 0111 1111( +127)

Minimum 8-bit number = 1000 0000 (í

128)



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)



EXAMPLE

Number +16392 0100 0000 0000 1000

One¶s complement 1011 1111 1111 0111

+ 0000 0000 0000 0001í163921011 1111 1111 1000



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



Example: Find 129 í 128

0000 0000 1000 0001 [= +129d]

1111 1111 1000 0000 [= í 128d]

0000 0000 0000 0001 [= + 1d]



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



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



OVERFLOW



Floating Point Representation



example

± 0.12 × 0.12 = 0.0144

would be expressed as

± (1.2 × 10í

1) × (1.2 × 10í

1) = (1.44 × 10í

2). In a fixed-point system with the decimal

point at the left, it would be

± 0.120 × 0.120 = 0.014.

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Data Representation (1) (1)