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Lecture 2 The information layer Data and

Number Representations

Outline

- Data Representation
- Data and Information
- Data Types
- Digital Data Representation
- How To Represent Different Types of Data
- Number Representation
- Number Systems
- Conversion Between Number Systems
- Integer Representation
- Floating-point Representation
- Operations on Bits
- Arithmetic Operations
- Logical Operations

Objectives

- Differentiate between data and information.
- Explain how text, images, audio and video are

represented in computers. - Explain decimal notation, binary notation,

hexadecimal notation, and octal notation. - Apply conversions from one number system to

another.

Objectives

- Explain how integers are stored in computers

(sign-and-magnitude, ones complement, twos

complement). - Explain how the Excess system works.
- Explain how to represent a floating-point number

in computers. - Apply bit operations such as arithmetic

operations, logical operations, and shift

operations.

Part 1

- Data Representation

Overview

- Whats data?
- Data refers to the symbols that a computer

uses to represent facts (such as people, events

and things) and ideas. - Whats information?
- The words, numbers, and graphics used as the

basis for human actions and decisions. - The difference between data and information
- Data becomes information when it is presented

in a format that people can understand and use.

Graphics

Data

Information

Table

Overview (con.)

- Data entered into a computer is called input. The

processed information are called output. The

cycle of input, process, output, and storage is

called the information processing cycle.

information

Data

Overview (con.)

- There are several types of data.

The computer industry uses the termmultimedia

to define information that contains numbers,

text, images, audio, and video.

Note

Question How do you handle all these data types

(text, number, image, audio and video)?

Text

1234890.

Solution

- The most efficient solution is to use a uniform

representation of data. - All data types from outside a computer are

transformed into this uniform representation when

stored in a computer and then transformed back

when leaving the computer. - This universal format is called a bit pattern.

Digital Data Representation

- Digital device works with discrete, distinct data

or digits, such as 1 and 0. - Analog device works with continuous data.

- Most computers are digital computers because the

digital is a relatively simple, dependable, and

adaptable technology.

Some important concepts

- Bit short for binary digit, is the smallest unit

of data that can be stored in a computer. - In the binary system, each 0 and 1 is called a

bit. - In a two-state on/off arrangement (such as a

switch), one state can represent a 1 digit (on),

the other represents a 0 digit (off). - Computers use sequences of bits (bit pattern) to

represent all kinds of data.

Some important concepts (con.)

- Bit pattern is a sequence or a string of bits. It

is the responsibility of I/O devices or programs

to interpret a bit pattern as a number, text, or

some type of data.

1 0 0 0 1 0 1 0 1 0 1 1 1 1 1 1

- Byte is a kind of bit pattern. Its length is 8

bits.

1 byte 8 bits

Data Representation (con.)

- How to represent different types of data
- Number Representation
- Text Representation
- Image Representation
- Audio Representation

- In a computer, numbers are represented using the

binary system.

Note

Text Representation

- A piece of text in any language is a sequence of

symbols. - Symbol examples
- 26 uppercase letters (A, B, C Z)
- 26 lowercase letters (a, b, c z)
- 10 numeric characters (0, 1, 2 9)
- Others (? blank, newline, and tab.)
- In a bit pattern, the number of bits to represent

a symbol depends on how many symbols are in the

set. - --- More symbols mean a longer bit pattern.

Text Representation (con.)

Text Representation (con.)

- The relationship between the length of the bit

pattern and the number of symbols is logarithmic. - For example
- If you need 4 symbols, the length is 2 bits.
- log24 2 The forms are 00, 01, 10 and 11

Number of symbols Bit pattern length

2 1

16 4

128 7

. .

65,536 16

(No Transcript)

Text Representation (con.)

- Different sets of bit patterns have been designed

to represent text symbols. Each set is called a

code, or code scheme. - There are some common codes
- ASCII
- Extended ASCII
- EBCDIC
- Unicode
- ISO

Popular Code Schemes

- ASCII Code
- Stands for American Standard Code for Information

Interchange. It was developed by ANSI (American

National Standards Institute). - It is used on most microcomputers, many

minicomputers, and some mainframe computers. - This code uses 7 bits for each symbol, which

means 128 (27) different symbols can be defined.

Popular Code Schemes

- Features of ASCII code
- Range of 7-bit pattern is from 0000000 to

1111111. - The first pattern (000 0000) represents the null

character. - The last pattern (111 1111) represents the delete

character. - There are 31 control characters.

(No Transcript)

Popular Code Schemes

- Extended ASCII Code
- To make the size of each pattern 1 byte (8 bits)

, the ASCII bit patterns are augmented with an

extra 0 at the left. - The code is from 00000000 to 01111111. The left

bit (0) is extended bit.

- EBCDIC Code
- Stands for Extended Binary Coded Decimal

Interchange Code. It was commonly used in IBM

mainframes. - It uses 8-bits pattern to represent 256

characters.

Popular Code Schemes

- Unicode
- Although previous codes can handle English and

European languages well, it cannot handle all the

characters of some other languages, such as

Chinese and Japanese. Unicode, which was

developed to deal with such languages, uses 2

bytes (16 bits) to handle 65 536 characters.

- ISO Code
- It is developed by ISO (International

Organization for Standardization), which uses 4

bytes (32 bits) to handle 4 294 967 296 (232)

characters.

Image Representation

- Bitmap
- --- an image is divided into a matrix of pixels.
- --- each pixel is assigned a bit pattern.
- --- to represent a black and white image, 1

represents white pixel and 0 represents black

one. - --- to represent a color image, a pixel is

decomposed into three colors (RGB).

Image Representation (con.)

Image Representation (con.)

Image Representation (con.)

For a color image, each pixel has three bit

patterns one to represent the intensity of the

red color, one to represent the intensity of the

green color, and one to represent the intensity

of the blue color.

Image Representation (con.)

- Vector
- ---an image is decomposed into a combination of

curves and lines. - ---each curve or line is represented by a

mathematical formula. - ---For example, a line may be described by the

coordinates of its endpoints, and a circle may be

described by the coordinates of its center and

the length of its radius. - ---they use much less storage space than bitmap

images, but do not look as realistic as bitmap

images.

Image Representation (con.)

Image Representation (con.)

- Video
- ---is composed of a series of frames. A movie is

a series of frames shown one after another to

create the illusion of motion. - ---each frame could be stored as a bitmap.
- ---a digital video requires tremendous storage

capacity. - ---Refer to chapter 15 for details about video

compression.

Audio Representation

- ---Audio is a representation of sound or music.
- ---music, voice and sound effects can all be

recorded as waveform, which is by nature analog

data. - ---samples of the sound are collected as periodic

intervals and stored as numeric data.

Audio Representation (con.)

- The steps to change audio data to bit patterns
- The analog data is sampled. Sampling means

measuring the value of the signal at equal

intervals. - The samples are quantized. Quantization means

assigning a value to a sample. - The quantized values are changed to binary

patterns. - The binary patterns are stored.

Audio Representation (con.)

Audio Representation (con.)

Part 2

- Number Representation

- Some essential concepts

- Number system any system of naming or

representing numbers, also called number

representation system or numeration system. - Carry happens when the sum or product of two or

more digits equals or exceeds the base of the

number system.

- Some essential concepts

- Base the number of digits in a number system.
- Position value the value associated with each

digit place, also called radix, weight or

positional value. - --For example for the decimal number 125, the

position values associated with the character

1,2,5 are respectively 102,101,100, therefore,

(125)10110221015100.

Question how do we compare these two numbers,

25.6 and 52.6, in the decimal system?

- (25.6)10 2 101 5 100 6 10-1
- (52.6)10 5 101 2 100 6 10-1
- (N)R ? Ki ? Ri Ki?0,1,,R-1

n

i-m

- n number of digits of the integer - 1
- m number of digits of the decimal fraction
- R base
- Ri Position values

- Different Number Systems

- Binary System (Base 2) consists of 2 digits

(symbols). - Octal System (Base 8) consists of 8 digits.
- Decimal System (Base 10) consists of 10 digits.
- Hexadecimal System (Base 16) consists of 16

digits.

0 1

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9 A B C D E F

Question how do we tell which number system we

are using?

- Subscript method

In C 0101 101 0x101

(101)2 (101) 8 (101) 10 (101) 16

Binary Octal Decimal Hexadecimal

101B 101O 101D 101H

- Prefix method

- Postfix method

Binary System

- Its base is 2. It has only two digits,1 and 0.
- There will be a carry when the result equals 2.

Position Value Table

243

Binary Position Value Table

Position Power Decimal Value Binary Value

216 65536 10000 0000 0000 0000

210 1024 100 0000 0000

29 512 10 0000 0000

28 256 1 0000 0000

27 128 1000 0000

26 64 100 0000

25 32 10 0000

24 16 1 0000

23 8 1000

22 4 100

21 2 10

20 1 1

Decimal Binary Binary (16-bit system)

0 0 0000 0000 0000 0000

1 1 0000 0000 0000 0001

2 10 0000 0000 0000 0010

3 11 0000 0000 0000 0011

4 100 0000 0000 0000 0100

5 101 0000 0000 0000 0101

6 110 0000 0000 0000 0110

7 111 0000 0000 0000 0111

8 1000 0000 0000 0000 1000

9 1001 0000 0000 0000 1001

10 1010 0000 0000 0000 1010

11 1011 0000 0000 0000 1011

12 1100 0000 0000 0000 1100

13 1101 0000 0000 0000 1101

14 1110 0000 0000 0000 1110

15 1111 0000 0000 0000 1111

How to represent (16)10 in the 16-bit system??

How computer capacity is expressed bit by bit

- The following terms are used to denote capacity
- Bit In the binary system, the binary digit (bit)

0 or 1is the smallest unit of measurement. - Byte A group of 8 bits is called a byte, and a

byte represents one character, digit, or other

value. - Kilobyte A kilobyte(K, KB) is about 1000 bytes.

(Actually, it's precisely 1024, that is 210bytes,

but the figure is commonly rounded.)

How computer capacity is expressed bit by bit

- Megabyte A megabyte (M, MB) is about 1 million

bytes (220). - Gigabyte A gigabyte (G, GB) is about 1 billion

bytes (230). - Terabyte A terabyte (T, TB) represents about 1

trillion bytes (240). - Petabyte(250) A new measurement accommodates the

huge storage capacities of modern databasesa

petabyte represents about 1 million gigabytes!

Octal System

- Its base is 8. It has eight digits 0, 1, 2, 3,

4, 5, 6 and 7. - There will be a carry when the result equals 8.
- A 3-bit pattern can be represented by an octal

digit, and vice versa.

Bit Pattern Oct Digit Bit Pattern Oct Digit

000 0 100 4

001 1 101 5

010 2 110 6

011 3 111 7

Decimal System

- Nowadays, there are two dominant number systems

in the world decimal system and binary system. - It has ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and

9.

Hexadecimal System

- Its base is 16. It has sixteen digits 0, 1, 2,

3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F. - A 4-bit pattern can be represented by a

hexadecimal digit, and vice versa.

Bit Pattern Hex Digit Bit Pattern Hex Digit

0000 0 1000 8

0001 1 1001 9

0010 2 1010 10

0011 3 1011 11

0100 4 1100 12

0101 5 1101 13

0110 6 1110 14

0111 7 1111 15

Decimal Binary Octal Hexadecimal

0 0 0 0

1 1 1 1

2 10 2 2

3 11 3 3

4 100 4 4

5 101 5 5

6 110 6 6

7 111 7 7

8 1000 10 8

9 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

- Conversion between number systems

Binary to Decimal

Decimal to Binary

Binary to Octal

Octal to Binary

Binary to Hexadecimal

Hexadecimal to Binary

Binary to Decimal Conversion

- Solution
- Step 1 multiply each binary digit by its

corresponding position value. - Step 2 add all multiplication results together

to get the decimal number.

Start from 0

0 1 0 1 1 0 1

Binary Number

6 5 4 3 2 1 0

Position

26

24

22

25

23

21

20

Position Value

026 125 024 123 122 021 120

0 32 0 8 4 0 1 ( 45 )10

Result

Decimal to Binary Conversion

- Solution
- Step 1 divide the number by 2 and write the

quotient and remainder. - Step 2 use the remainder (from step 1) as the

corresponding binary digit (from right to left),

1 or 0 . - Step 3 check whether the quotient is 0 or not.

If it is zero, skip to the Step 4 otherwise, use

the quotient as the number and go back to Step 1.

- Step 4 stop and put all remainders together to

get the binary number.

Example Convert the decimal number 35 to binary.

Stop when the quotient is 0

Quotient

Decimal Number

17

8

2

1

4

0

35

1

0

0

1

1

0

Remainder

Binary Number 100011

Questions

- What if the number has a fraction part?
- How do we convert octal to decimal, and decimal

to octal? - How do we convert hexadecimal to decimal, and

decimal to hexadecimal?

Binary to Octal Conversion

- Solution
- Step 1 Organize the pattern into groups of 3

(from right to left). - Step 2 Transform each group into an octal digit.

1 1 1 1 1 1 0 0 1 1 1 0 0 1 0 0

1

7

6

3

4

4

If the leftmost bit pattern does not contain 3

digits, add extra 0s to the left.

Octal to Binary Conversion

- Solution
- Transform each octal digit into a 3-bit binary

pattern.

Octal Number

Octal Number

1 4 2

5 6 2

001

100

010

101

110

010

Binary Number

Binary Number

Binary to Hexadecimal

- Solution
- Step 1 Organize the bit pattern into groups of 4

(from right to left). - Step 2 Transform each group into a hexadecimal

digit.

1 1 1 1 1 1 0 0 1 1 1 0 0 1 0 0

F

C

E

4

If the leftmost bit pattern does not contain 3

digits, add extra 0s to the left.

Hexadecimal to Binary

- Solution
- Transform each hexadecimal digit into a 4-bit

binary pattern.

Hexadecimal Number

2 4 C

0010

0100

1100

Binary Number

Exercises

- (10 111 011.110 1)2 (?)8
- (273.64)8
- (6754.32)8 (?)2
- (110 111 101 100. 011 010)2
- (1011 1110 0110. 1101 1)2 (?)16
- (BE6.D8)16
- (A7B8.C9)16 (?)2
- (1010 0111 1011 1000. 1100 1001)2

- Integer Representation

- Integers are the whole numbers, which include

positive integers and negative integers. - The figure below shows different integers.

Integers

Unsigned

Signed

Sign-and- Magnitude

Ones Complement

Twos Complement

Unsigned Integers

- Range of unsigned integers 0 ( 2N-1 )
- ---N is the number of bits the computer

allocates to store an unsigned integer. - ---For example
- 8 -bit computer 0 255
- 16-bit computer 0 65,535
- How to store an unsigned integer
- Step 1 Convert the number to binary.
- Step 2 If the number of bits is less than N, 0s

are added to the left of the binary number so

that there is a total of N bits.

Unsigned integers in two different computers

Add zeros to the left when needed

Decimal 8-bit Allocation 16-bit Allocation

7

234

258

24,760

1,245,678

0000 0111

0000 0000 0000 0111

0000 0000 1110 1010

1110 1010

overflow

0000 0001 0000 0010

0110 0000 1011 1000

overflow

overflow

overflow

Exceeds the capacity of the storage

Signed Integers

- Sign-and-Magnitude Representation

- The leftmost bit defines the sign of the number

0 for positive, and 1 for negative. - ---For example in an 8-bit allocation, the

leftmost bit shows the sign and the other seven

bits represent the absolute value. - Range of signed integers -(2N-1-1) ( 2N-1-1)
- ---N is the number of bits allocated to

represent one sign-and-magnitude integer.

Note

- There are two 0s in sign-and-magnitude

representation positive zero (0) and negative

zero (-0). - ---For example In an 8-bit allocation,
- 0 ? 00000000 -0 ? 10000000

- How to store a sign-and-magnitude integer
- Step 1 Convert the number to binary, the sign is

ignored. - Step 2 If the number of bits is less than N-1,

0s are added to the left of the number so that

there is a total of N-1 bits. - Step 3 Add sign-bit to the left ( to make it N

bits ) if its positive, add 0 if negative,

add 1.

Example Store 7 in an 8-bit memory location,

and store 258 in a 16-bit memory location using

sign-and-magnitude representation .

7

Question how about 7 in a 16-bit allocation,

and -258 in an 8-bit allocation?

1

1

1

1

1

1

0

0

0

0

0

0

0

0

0

1

1

1

-258

1

0

0

0

0

0

0

1

0

1

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

1

0

1

Signed Integers

- Ones Complement Representation

- To represent a positive number, use the

convention adopted for an unsigned integer. To

represent a negative number, complement the

positive number. - In ones complement, the complement of a number

is obtained by changing all 0s to 1s and all 1s

to 0s. - The leftmost bit defines the sign of the number

0 for positive, and 1 denotes negative. - Range of signed integers -(2N-1-1) ( 2N-1-1)
- ---N is the number of bits allocated to

represent a ones complement integer.

Note

- There are two 0s in ones complement

representation, too positive zero (0) and

negative zero (-0). - ---For example In an 8-bit allocation,
- 0 ? 00000000 -0 ? 11111111

- How to store a ones complement integer
- Step 1 Convert the number to binary, the sign is

ignored. - Step 2 0s are added to the left of the number to

make a total of N bits. - Step 3 If the sign is positive, no action is

needed. If negative, every bit is complemented.

Example Store 7 and -7 in an 8-bit memory

location, and store 258 in a 16-bit memory

location using ones complement representation .

7

-7

1

1

1

1

1

1

1

1

1

0

0

0

0

0

1

1

1

0

0

0

0

0

1

1

1

1

0

0

0

1

1

0

0

0

0

0

0

1

0

-258

1

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

1

1

1

1

1

1

0

1

1

1

1

1

1

0

1

1

Note

- Ones complementing means reversing all bits.
- If you ones complement a positive number, you

get the corresponding negative number. - If you ones complement a negative number, you

get the corresponding positive number. - If you ones complement a number twice, you get

the original number.

Signed Integers

- Twos Complement

- Range of signed integers -(2N-1) ( 2N-1-1)
- --- N is the number of bits allocated to

represent a twos complement integer. - How to store a twos complement integer
- Step 1 Convert the number to binary, ignore the

sign. - Step 2 0s are added to the left of the number to

make a total of N bits. - Step 3 If the sign is positive, no action is

needed. If negative, leave all the rightmost 0s

and the first 1 unchanged. Complement the rest of

the bits.

Twos complement is the most common, the most

important, and the most widely used

representation of integers today.

- For twos complement one can also use the

following formula to compute the negative

representation - Negative(I)2K-I, where k is the number of

bits - Example Assume four bits are used for twos

complement representation, for the number 0010,

we can derive its negative representation as

follows - Negative(0010) 24-00101110

Example Store 7 in an 8-bit memory location,

and store 40 in a 16-bit memory location using

twos complement representation .

7

1

1

1

1

1

1

0

0

0

0

0

1

0

1

0

0

0

-40

0

0

0

0

0

0

1

0

1

0

0

0

0

0

0

0

1

1

1

1

1

1

1

1

1

0

1

1

0

0

0

1

Note

For negative numbers Twos complement ones

complement 1

Note

- In twos complement representation, the leftmost

bit defines the sign of the number. - If it is 0, the number is positive.
- If it is 1, the number is negative.
- There is only one 0 in twos complement
- For example in an 8-bit allocation,
- 0 ? 00000000
- If you twos complement a positive number, you

get the corresponding negative number. - If you twos complement a negative number, you

get the corresponding positive number. - If you twos complement a number twice, you get

the original number.

Summary of integer representation

Unsigned ------------ 0 1 2 3 4 5 6 7 8 9 10 11 1

2 13 14 15

Sign-and-Magnitude --------- 0 1 2 3 4 5 6

7 -0 -1 -2 -3 -4 -5 -6 -7

OnesComplement --------- 0 1 2 3 4 5 6 7

-7 -6 -5 -4 -3 -2 -1 -0

TwosComplement -------- 0 1 2 3 4 5 6 7

-8 -7 -6 -5 -4 -3 -2 -1

Contents of Memory------------ 0000 0001 0010 001

1 0100 0101 0110 0111 1000 1001 1010 1011 1100 110

1 1110 1111

Excess system

- In an excess conversion, a positive number,

called the magic number, is used in the

conversion process. - The magic number is normally (2N-1) or ( 2N-1-1),

where N is the bit allocation. For example, if N

is 8, the magic number is either 128 (Excess_128)

or 127(Excess_127). - To represent a number in Excess, use the

following procedure - Step 1 Add the magic number to the integer.
- Step 2 Change the result to binary and add 0s to

make a total of N bits.

Example Represent 25 in Excess 127 using an

8-bit allocation.

- Step 1 Add 127 to 25 and get 102.
- Step 2 Change the result to binary, which is

1100110. - Step 3 Add one 0 to make a total of 8 bits. The

representation is 01100110

- Floating-point Representation

Fraction Part

1 4 . 2 3 4

Integer Part

1 4

2 3 4

See Decimal to binary Conversion

- To convert a floating-point number to binary
- Step 1 Convert the integer part to binary.
- Step 2 Convert the fraction part to binary.
- Step 3 Put decimal point between the two parts.

Repetitive Multiplication Method

Changing fractions to binary

Multiply 2 to gain integer part

Multiply by 2 to get integer part

Stop when the result is 0

0.250

0.500

1.000

0.000

0.125

0 .

0

0

1

Transform the fraction 0.4 to a binary of 6 bits?

0.4

0.8

1.6

1.2

0.4

0.8

1.6

0 .

0

1

1

0

0

1

Example transform the number 49.58 to a binary (

3 bits after the decimal point)?

( 49 . 58 )10 ( . )2

110001

100

Integer Part

Fraction Part

- Floating-point Representation

- Normalization the moving of the decimal point so

that there is only one 1 to the left of the

decimal point (1. XXXXXXXXX) .

6 digits

261.01000111001

1010001 . 11001

3 digits

- 2-31.110011

- 0 . 001110011

2e 1.XXXXXXX

Sign (1 bit)

Mantissa (unsigned integer)

Exponent (Excess representation)

- Floating-point Representation

- IEEE Standard

Single-precision format 4 bytes 32 bits

sign

1

8

23

exponent

mantissa

1

10

52

Double-precision format 8 bytes 64 bits

- Floating-point Representation

- IEEE Standard

Example Show the representation of the

normalized number 26 1.01000111001

- The sign is positive. The Excess_127

representation of the exponent is 133. You add

extra 0s on the right to make it 23 bits. The

number in memory is stored as 0 10000101

01000111001000000000000

Example Interpret the following 32-bit

floating-point number 1 01111100

11001100000000000000000

- The sign is negative. The exponent is 3 (124

127). The number after normalization is

-2-3 1.110011

Quiz

- The decimal equivalent of the binary number

11.11 is . - Show the following number in 32-bit IEEE format

. - -2-51.01101001
- Suppose the following bit pattern represents a

value stored in twos complement notation. Find

the twos complement representation of the

negative of the value. - 01010101

Part 3

- Operations on Bits

Bit Operations

Arithmetic

Logical

Shift

Arithmetic operations

- Arithmetic operations involve adding,

subtracting, multiplying, dividing, and so on. - You can apply these operations to integers and

floating-point numbers. - The multiplication operation can be implemented

in software using repeated addition or in

hardware using other techniques. - The division operation can also be implemented in

software using repeated subtraction or in

hardware using other techniques.

Arithmetic Operations

Table adding bits

Number of 1s Result Carry

0 0

1 1

2 0 1

3 1 1

- Rule of Adding Integers in Twos Complement
- Add 2 bits and propagate the carry to the next

column. If there is a final carry after the

leftmost column addition, discard it.

Example add the following numbers in twos

complement representation(24) (-17) (127)

(3) .

?

(24) (-17)

(127) (3)

7

130

1 1 1 1 1

1 1 1 1 1 1 1

Carry

0 0 0 1 1 0 0 0

0 1 1 1 1 1 1 1

1 1 1 0 1 1 1 1

0 0 0 0 0 0 1 1

0 0 0 0 0 1 1 1

1 0 0 0 0 0 1 0

Result

discard

?

(-126)10

Whats wrong??? Data overflow !!!

Overflow

1 1 1 1 , 1 1 1 1

(-1) 0 -1

- Overflow will happen when the value exceeds the

range of the allocation. - Range of numbers in twos complementrepresentatio

n - - (2N-1) --- 0 --- (2N-1 1)

1273 -126

1271 -128

1 0 0 0 , 0 0 1 0

1 0 0 0 , 0 0 0 0

Note

When you do arithmetic operations on numbers

in a computer, remember that each number and the

result should be in the range defined by the bit

allocation.

- To subtract, negate (twos complement) the second

number and add. For example - (101) - (62) ?? (101) (-62)

Carry 1 1 0 1 1 0 0 1

0 1 1 1 0 0 0 0 1

0 ----------------------------------Result 0

0 1 0 0 1 1 1 ?

39The leftmost carry is discarded.

Addition and subtraction for floating-point

numbers

- Check the signs. If different, use the sign of

the number whose absolute value is larger. - Move the decimal point to make the exponents the

same. - Add or subtract the mantissas (including the

integer part and fraction part). - Normalize the result before storing in memory.
- Check for overflow.

Example

Add two floats0 10000100 1011000000000000000000

00 10000010 01100000000000000000000

Solution

The exponents are 5 and 3. The numbers are25

1.1011 and 23 1.011Make the exponents

the same.(25 1.1011) (25 0.01011) ? 25

10.00001After normalization 26 1.000001,

which is stored as0 10000101

000001000000000000000000

Logical Operations

- Some essential concepts

Its a cat , not a tiger !

True

Its a red tulip !

False

- Each logical variable can only be True or False.
- In the computer system, 1 means True, 0

means False.

Some essential concepts

- According to the number of operands they take,

operators can be categorized as unary, binary

and Ternary.

Some essential concepts

Some essential concepts

- A truth table shows all the possible input

combinations of input with the corresponding

output.

x y x AND y

0 0 0

0 1 0

1 0 0

1 1 1

x NOT x

0 1

1 0

x y x OR y

0 0 0

0 1 1

1 0 1

1 1 1

x y x XOR y

0 0 0

0 1 1

1 0 1

1 1 0

NOT operator

The NOT operator has one input. It inverts bits

that is, it changes 0 to 1 and 1 to 0.

Example

Use the NOT operator on the bit pattern 10011000

Solution

Target 1 0 0 1 1 0 0 0 NOT

------------------Result

0 1 1 0 0 1 1 1

AND operator

Example

Use the AND operator on bit patterns 10011000 and

00110101.

Solution

Target 1 0 0 1 1 0 0 0 AND

0 0 1 1 0 1 0 1

------------------Result 0

0 0 1 0 0 0 0

Inherent rule of the AND operator

If a bit in one input is 0, you can quickly

conclude that the result is 0.

OR operator

Example

Use the OR operator on bit patterns 10011000 and

00110101

Solution

Target 1 0 0 1 1 0 0 0 OR

0 0 1 1 0 1 0 1

------------------Result 1 0

1 1 1 1 0 1

Inherent rule of the OR operator

If a bit in one input is 1, you can quickly

conclude that the result is 1.

XOR operator

Example

Use the XOR operator on bit patterns 10011000 and

00110101.

Solution

Target 1 0 0 1 1 0 0 0 XOR

0 0 1 1 0 1 0 1

------------------Result 1

0 1 0 1 1 0 1

Inherent rule of the XOR operator

If a bit in one input is 1, the result is the

inverse of the corresponding bit in the other

input.

Mask

A mask is a special bit pattern used to modify

another bit pattern. We can use masks to unset,

set, or reverse specific bits.

Example of unsetting specific bits

Example

Use a mask to unset (clear) the 5 leftmost bits

of a pattern. Test the mask with the pattern

10100110.

Solution

The mask is 00000111. Target 1 0 1 0 0 1 1 0

ANDMask 0 0 0 0 0 1 1 1

------------------Result

0 0 0 0 0 1 1 0

Example

Imagine a power plant that pumps water to a city

using eight pumps. The state of the pumps (on or

off) can be represented by an 8-bit pattern. For

example, the pattern 11000111 shows that pumps 1

to 3 (from the right), 7 and 8 are on while pumps

4, 5, and 6 are off. Now assume pump 7 shuts

down. How can a mask show this situation?

Solution

Use the mask 10111111 to AND with the target

pattern. The only 0 bit (bit 7) in the mask turns

off the seventh bit in the target. Target 1 1

0 0 0 1 1 1 ANDMask 1 0

1 1 1 1 1 1

------------------Result 1 0 0 0

0 1 1 1

Example of setting specific bits

Example

Use a mask to set the 5 leftmost bits of a

pattern. Test the mask with the pattern 10100110.

Solution

The mask is 11111000. Target 1 0 1 0 0 1 1 0

ORMask 1 1 1 1 1 0 0 0

------------------Result

1 1 1 1 1 1 1 0

Example

Using the power plant example, how can you use a

mask to to show that pump 6 is now turned on?

Solution

Use the mask 00100000. Target 1 0 0 0 0 1 1 1

ORMask 0 0 1 0 0 0 0 0

------------------Result

1 0 1 0 0 1 1 1

Example of flipping specific bits

Example

Use a mask to flip the 5 leftmost bits of a

pattern. Test the mask with the pattern 10100110.

Solution

Target 1 0 1 0 0 1 1 0 XOR Mask

1 1 1 1 1 0 0 0

------------------Result 0 1 0

1 1 1 1 0

Shift operations

If a bit pattern can be shifted to the right or

to the left.

Example

Show how you can divide or multiply a number by2

using shift operations.

Solution

If a bit pattern represents an unsigned number, a

right-shift operation divides the number by two.

The pattern 00111011 represents 59. When you

shift the number to the right, you get 00011101,

which is 29. If you shift the original number to

the left, you get 01110110, which is 118.

Example

Use a combination of logical and shift

operations to find the value (0 or 1) of the

fourth bit (from the right).

Solution

Use the mask 00001000 to AND with the target to

keep the fourth bit and clear the rest of the

bits.

Solution (continued)

Target a b c d e f g h AND Mask

0 0 0 0 1 0 0 0

------------------Result

0 0 0 0 e 0 0 0 Shift the new pattern three

times to the right 0000e000 ? 00000e00 ?

000000e0 ? 0000000eNow it is easy to test the

value of the new pattern as an unsigned integer.

If the value is 1, the original bit was 1

otherwise the original bit was 0.

Objectives

- Differentiate between data and information.
- Explain how text, images, audio and video are

represented in computers. - Explain decimal notation, binary notation,

hexadecimal notation, and octal notation. - Apply conversions from one number system to

another.

Objectives

- Explain how integers are stored in computers

(sign-and-magnitude, ones complement, twos

complement). - Explain how the Excess system works.
- Explain how to represent a floating-point number

in computers. - Apply bit operations such as arithmetic

operations, logical operations, and shift

operations.

Thats all for this lecture!

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