data representation in computer

7/31/2019 Data Representation in Computer

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Data Representation in

Computer

CSC2101 COMPUTERORGANISATION


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Storing Data in Computer

Computer hardware can only store and process inbinary.

Data (text, number, audio, image, video, etc) has to be

converted to some form of binary representation

understand by the computer. For example, text isnormally represented in ACSII or Unicode, integer

numbers are represented in twos complement binary.

A 01000001 (ASCII code)

65 01000001 (8-bit twos complement)Both data was represented in the same binary code, so

it is important for computer to know the data type

(remember the variable declaration in C++ ?)


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Storing Data in Computer

And also due to computer storage (memory) isorganized in terms of 8-bit per unit (called 1 byte), data

must be represented in terms of 1 byte, 2 bytes, 4

bytes, and etc.

So an integer of 5 is represented as 00000101 in 8-bitsystem or 00000000 00000101 in 16-bit system rather

than 101.


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Signed Number Representation

The sign of a number is specified by the sign bit. Positive number is specified by a 0 bit and negative

number is specified by a 1 bit.

Two representations

Sign Magnitude easy for human but cannot be computed efficiently in

computer hardware.

Twos Complement Use by all computer systems nowadays


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Twos Complement (1)

Represents negative signed binary numbers in a formthat can be processed by a computer

The twos complement form is given by the definition

[N]2 = 2n

(N)2

where n is the number of bits in (N)2


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Twos Complement (2)

Example 1: Determine twos complement of (01100101)2

[N]2 = 28 (01100101)2

= (10000000)2 (01100101)2

= (10011011)2

Another method :( 01100101 )2 + 1 = 100110102 + 1

= 100110112


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Twos Complement (3)

Example 2: Determine signed binary representation of610in twos complement

Representing 610 with a byte 0000 0110

Twos complement 0000 0110 + 1= 1111 1001 + 1

= 1111 1010

To show that (1111 1010) is the negative of (0000 0110)

1111 1010 + 0000 0110 = 1 0000 0000

carry bit


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Twos Complement (4)

Example 3: Determine the decimal value of the following

(0101 0000)2

Sign bit is 0, therefore, it is a positive number

= 2

4

+ 2

6

= 8010

(1000 1111 0101 1101)2

Sign bit is 1, therefore, it is a negative number

= - (1000 1111 0101 1101 + 1)

= - (0111 0000 1010 0011)

= - (20 + 21 + 25 + 27 + 212 + 213 + 214)= - 2883510


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Twos Complement Arithmetic

Example : 1410 2010 = ?Can be rewritten as 1410 + (-2010)

1410 = (0000 1110)2

2010 = (0001 0100)2

2010 = 0001 0100 + 1 = 1110 1100

0000 1110

+1110 1100

1111 1010

to verify :

1111 1010 + 1 = - (0000 0110) = 610


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Overflow(1)

Data/numbers are stored in fixed size insidecomputers memory. For example, integer numbers is

usually stored as 8-bit or 16-bit (2 byte).

Overflow occurs when the result of the calculation

can not be represented by the fixed number of bitsused.

Example : Suppose we try to add the following two 8-

bit numbers 0110 11002 + 0101 11102 in an 8-bit

system. (continue next page)


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Overflow(2)

0110 1100 108+ 0101 1110 + 94-------------------------- ----------

1100 1010 -54

127-128

-ve +ve

0

21

126-127

-1-2

108

-54

Overflow !!

This is due to the sum is toobig to

be fitted (represented) into 8-bit

twos complement binary system.


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Overflow(3)

Overflow may happen when

Arithmetic Operation Overflow

positive + positive integer Possible

positive + negative integer No

negative + negative integer Possible

negative + positive integer No

positive - positive integer No

positive - negative integer Possible

negative - negative integer No

negative positive integer Posibble


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Carry

When the result of the calculation produces an extrabit out from the most significant bit position, we call

this bit as carry bit.

Example :

1110 1001

+ 0110 1100-----------------

1 0101 0101

Carry bit

msb lsb

msb = most significant bit

lsb = least significant bit


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8-bit Unsigned Number

Binary Hexadecimal Decimal

0000 0000 00 0

0000 0001 01 1

0000 0010 02 2

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0111 1111 7F 127

1000 0000 80 128

1000 0001 81 129

1000 0010 82 130

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1111 1110 FE 254

1111 1111 FF 255


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8-bit Signed Number (2s complement)

Binary Hexadecimal Decimal

0000 0000 00 0

0000 0001 01 1

0000 0010 02 2

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.

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0111 1111 7F 127

1000 0000 80 -128

1000 0001 81 -127

1000 0010 82 -126

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1111 1110 FE -2

1111 1111 FF -1


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Binary Coded Decimal (1)

Binary Coded Decimal (BCD) is used for representing

decimal digits 0 through 9

Each bit position in the code has a fixed numerical value

or weight associated with it

Digits represented by a given word can be found by

summing up the weighted bits

BCD uses 4 bits, with weights chosen to be the same as

those of a 4-bit binary integer


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Decimal Binary Hexadecimal BCD

0 0000 0 0000

1 0001 1 0001

2 0010 2 0010

3 0011 3 0011

4 0100 4 0100

5 0101 5 0101

6 0110 6 0110

7 0111 7 0111

8 1000 8 1000

9 1001 9 1001

10 1010 A 0001 0000

11 1011 B 0001 0001

12 1100 C 0001 0010

13 1101 D 0001 0011

14 1110 E 0001 0100

15 1111 F 0001 0101


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Example : Encode the decimal number 9750 in BCD

Individual digits are encoded91001, 70111, 50101, 00000

Individual codes are concatenated to1001 0111 0101 0000BCD


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BCD is used to encode numbers for output tonumerical displays, numerical inputs and for

representing numbers in processors that perform

decimal arithmetic

Applications Digital voltmeters, digital clocks (output) Electronic calculators (input)


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ASCII Code

ASCII stands for American Standard Code forInformation Interchange.

Computers can only understand numbers (binary), so an

ASCII code is the numerical representation of Englishalphabets and characters such as +' or'@.


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ASCII Table

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ASCII Code Example

If you store a string (such as How are you ?) into atext file, and then use a hexadecimal code editor to

open the text file, you will see the ASCII code (in

hexadecimal) like below

48 6F 77 20 61 72 65 20 79 6F 75 3F

H o w a r e y o u ?

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(space)(space)

data representation in computer

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