ASSEMBLY LANGUAGE FOR INTEL-BASED COMPUTERS

DATA REPRESENTATIONPART 1

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DATA FORMATS

Computers

• Process and store all forms of data in binary format

Human communication

• Includes language, images and sounds

Data formats:

• Specifications for converting data into computer-usable form

• Define the different ways human data may be represented, stored and processed by a computer

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SOURCES OF DATABinary input

◦ Begins as discrete input ◦ Example: keyboard input such as A 1+2=3 math◦ Keyboard generates a binary number code for each key

Analog◦ Continuous data such as sound or images◦ Requires hardware to convert data into binary numbers

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Computer

1101000101010101…

Input device

A 1+2=3 math

Figure 3.1 with this color scheme

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COMMON DATA REPRESENTATIONS

Type of Data Standard(s)

Alphanumeric Unicode, ASCII, EDCDIC

Image (bitmapped) GIF (graphical image format)TIF (tagged image file format)PNG (portable network graphics)

Image (object) PostScript, JPEG, SWF (Macromedia Flash), SVG

Outline graphics and fonts

PostScript, TrueType

Sound WAV, AVI, MP3, MIDI, WMA

Page description PDF (Adobe Portable Document Format), HTML, XML

Video Quicktime, MPEG-2, RealVideo, WMV 4

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INTERNAL COMPUTER DATA FORMAT

All data stored as binary numbers

Interpreted based on

• Operations computer can perform• Data types supported by programming language

used to create application

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INTERNAL DATA REPRESENTATIONReflects the

◦ Complexity of input source◦ Type of processing required

Trade-offs◦ Accuracy and resolution

Simple photo vs. painting in an art book◦ Compactness (storage and transmission)

More data required for improved accuracy and resolution Compression represents data in a more compact form Metadata: data that describes or interprets the meaning of data

◦ Ease of manipulation: Processing simple audio vs. high-fidelity sound

◦ Standardization Proprietary formats for storing and processing data (WordPerfect vs.

Word) De facto standards: proprietary standards based on general user

acceptance (PostScript)

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5 SIMPLE DATA TYPES

Boolean: 2-valued variables or constants with values of true or false

Char: Variable or constant that holds alphanumeric character

Enumerated◦User-defined data types with possible values listed

in definition Type DayOfWeek = Mon, Tues, Wed, Thurs, Fri, Sat, Sun

Integer: positive or negative whole numbers Real

◦Numbers with a decimal point◦Numbers whose magnitude, large or small, exceeds

computer’s capability to store as an integer

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DATA TYPES: NUMERIC

Used for mathematical manipulation◦ Add, subtract, multiply, divide

Types◦ Integer (whole number)◦ Real (contains a decimal point)

pure binary◦ can be calculated directly

ASCII binary◦ string of digits: "01010101"

ASCII decimal◦ string of digits: "65"

ASCII hexadecimal◦ string of digits: "9C"

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DATA TYPES: ALPHANUMERIC

Alphanumeric: • Characters: b T• Number digits: 7 9• Punctuation marks: ! ; • Special-purpose characters: $ &

Numeric characters vs. numbers• Both entered as ordinary characters• Computer converts into numbers for calculation

• Examples: Variables declared as numbers by the programmer (Salary$ in BASIC)

• Treated as characters if processed as text• Examples: Phone numbers, ZIP codes

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ALPHANUMERIC CODES

Arbitrary choice of bits to represent characters

• Consistency: input and output device must recognize same code

• Value of binary number representing character corresponds to placement in the alphabet

• Facilitates sorting and searching

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DATA REPRESENTATION

Binary Numbers◦ Translating between binary and decimal

Binary AdditionInteger Storage SizesHexadecimal Integers

◦ Translating between decimal and hexadecimal◦ Hexadecimal subtraction

Signed Integers◦ Binary subtraction

Character Storage

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BINARY, OCTAL, DECIMAL AND HEXADECIMAL DIGITS TABLE

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System Base Possible Digits

Binary 2 0 1

Octal 8 0 1 2 3 4 5 6 7

Decimal 10 0 1 2 3 4 5 6 7 8 9

Hexadecimal

16 0 1 2 3 4 5 6 7 8 9 A B C D E F

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BINARY NUMBERS

Digits are 1 and 0

• 1 = true• 0 = false

MSB – most significant bit

LSB – least significant bit

Bit numbering:

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015

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

MSB LSB

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BINARY NUMBERS

Binary integers can be signed or unsigned.

Signed integer = either positive or negative

Unsigned integer = only positive, including 0.

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UNSIGNED BINARY INTEGERS

Each digit (bit) is either 1 or 0

Each bit represents a power of 2

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Every binary number is a sum of powers of 2

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TRANSLATING UNSIGNED BINARY TO DECIMALWeighted positional notation shows how to calculate the decimal value of each binary bit:

dec = (Dn-1 2n-1) (Dn-2 2n-2) ... (D1 21) (D0 20)D = binary digit

binary 00001001 = decimal 9:

How to calculate?

( 0 27)+(0 26)+(0 25)+(0 24)+(1 23)+(0 22)+ (0 21)+ (1 20)

0 + 0 + 0 + 0 + 8 + 0 + 0 + 1

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TRANSLATING BINARY TO DECIMAL

Try this

010011002

100000012

101010102

110011002

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TRANSLATING UNSIGNED DECIMAL TO BINARYRepeatedly divide the decimal integer by 2. Each remainder is a

binary digit in the translated value:

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37 = 1001012

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TRANSLATING DECIMAL TO BINARY

Try this

55

16

77

111

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BINARY ADDITION

Starting with the LSB, add each pair of digits, include the carry if present.

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INTEGER STORAGE SIZES

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What is the largest unsigned integer that may be stored in 20 bits?

Standard sizes:

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HEXADECIMAL INTEGERS

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Binary values are represented in hexadecimal.

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TRANSLATING BINARY TO HEXADECIMAL

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• Each hexadecimal digit corresponds to 4 binary bits.

• Example: Translate the binary integer 000101101010011110010100 to hexadecimal:

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POWERS OF 16FROM 160 TO 167

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Used when calculating hexadecimal values up to 8 digits long:

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CONVERTING HEXADECIMAL TO DECIMAL

Multiply each digit by its corresponding power of 16:

decimal = (D3 163) + (D2 162) + (D1 161) + (D0 160)

Hex 1234 equals (1 163) + (2 162) + (3 161) + (4 160), or decimal 4,660.

Hex 3BA4 equals (3 163) + (11 * 162) + (10 161) + (4 160), or decimal 15,268.

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CONVERTING HEXADECIMAL TO DECIMAL (CONT)

Hex 3BA4 equals

3 163 = 12 288

+ 11 * 162 = 2 816

+ 10 161 = 160

+ 4 160 = 4

15,268

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Try this:

1. 1C592. ABCD

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CONVERTING DECIMAL TO HEXADECIMAL

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decimal 422 = 1A6 hexadecimal

Try this:

1. 1002. 2108

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HEXADECIMAL ADDITION

Divide the sum of two digits by the number base (16). The quotient becomes the carry value, and the remainder is the sum digit.

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Try This 36 28 28 +42 + 45 + B8

21/16 = 1, remain 5

1/16 = 0, remain 1

Important skill: Programmers frequently add and subtract the addresses of variables and instructions.

eg: 1

6 A 6 | 10+ 4 B 4 | 11

11 | 21 B 5

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SOLUTION A1 :

36

+ 42

78

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A2 :

2 | 8

+ 4 | 5

6 | 13 6 D

13 = D

A3 : 1

2 | 8

+ B | 8

14 | 16 E 0

16 /16 = 1 remain 0

1/16 = 0 remain 114 = E

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HEXADECIMAL SUBTRACTIONWHEN A BORROW IS REQUIRED FROM THE DIGIT TO THE LEFT, ADD 16 (DECIMAL) TO THE CURRENT DIGIT'S VALUE:

C 6 12 6

- A 2 -10 2

2 4

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16

2 8 16

3 9 5 3 9 5

- D A - 13 10

2 11 11

2 B B (16+5)-10 =11

(16+8)-13 =11

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HEXADECIMAL SUBTRACTION

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1) 16 + 5 = 21

2) 21 – 7 = 14

3) 14 = E

6 16

7 5- 4 7 2 E

E F- B 2 3 D

1) F = 152) 15 – 2 = 133) 13 = D4) E =145) B = 116) 14 – 11 = 3

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TRY THIS

Hexadecimal addition

1. D3A + 19

2. 99+ A11

3. 6B2 + FF

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Hexadecimal subtraction

1. D1A - 339

2. A11 - 99

3. 74B – 8F

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SIGNED INTEGERS

The highest bit indicates the sign. 1 = negative, 0 = positive

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If the highest digit of a hexadecimal integer is > 7, the value is negative. Examples: 8A, C5, A2, 9D

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SIGNED-INTEGER REPRESENTATION

No obvious direct way to represent the sign in binary notation

Options:

• Sign-and-magnitude representation• 1’s complement• 2’s complement (most common)

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SIGN-AND-MAGNITUDE

Use left-most bit for sign

• 0 = plus; 1 = minus

Total range of integers the same

• Half of integers positive; half negative• Magnitude of largest integer half as large

Example using 8 bits:

• Unsigned: 1111 1111 = +255• Signed: 0111 1111 = +127

1111 1111 = -127• Note: 2 values for 0:

+0 (0000 0000) and -0 (1000 0000)

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1’S VS. 2’S COMPLEMENTS

Choice made by computer designer

1’s complement

• Reverse the bit• Easier to change sign• Addition requires extra end-around carry• Algorithm must test for and convert -0

2’s complement simpler

• Additional add operation required for sign change

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FORMING THE TWO'S COMPLEMENT

Negative numbers are stored in two's complement notation

Represents the additive Inverse

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Note that 00000001 + 11111111 = 00000000

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FORMING TWO’S COMPLEMENT OF HEXADECIMAL

6A3D

1. Step 1: reverse the bits = 95C2

2. Step 2: Add 1 = 95C3

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LEARN HOW TO DO THE FOLLOWING:

Form the two's complement of a hexadecimal integer

Convert signed binary to decimal

Convert signed decimal to binary

Convert signed decimal to hexadecimal

Convert signed hexadecimal to decimal

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GUIDELINES

Convert signed binary to decimal1. If number is negative, form two complement notation2. Convert to decimal

Convert signed decimal to binary1. Convert to binary 2. If number is negative, form two complement notation

Convert signed decimal to hexadecimal1. Convert to hexadecimal2. If number is negative, form two complement notation

Convert signed hexadecimal to decimal1. If number is negative, form two complement notation2. Convert to decimal

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**NOTES**

Any hexadecimal number

• If digit is >= 8, the number is negative• If digit is <=7, the number is positive

Eg :

• 8A20 = negative• 7FD9 = positive

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RANGES OF SIGNED INTEGERS

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The highest bit is reserved for the sign. This limits the range:

Practice: What is the largest positive value that may be stored in 20 bits?

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NUMERIC DATA REPRESENTATION

pure binary

• can be calculated directly

ASCII binary

• string of digits: "01010101"

ASCII decimal

• string of digits: "65"

ASCII hexadecimal

• string of digits: "9C"

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next: Boolean Operations

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LEARN HOW TO DO THE FOLLOWING:

Form the two's complement of a hexadecimal integer

• 3D C2 (First complement)

+ 1

C3 (Two’s Complement)

Convert signed binary to decimal

• 1111 0000 0000 1111 (First complement) + 1 0001 0000 (Two’s Complement) 0001 0000 =( 1 X 2 ^ 5) =16 =16

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LEARN HOW TO DO THE FOLLOWING:Convert signed decimal to binary

• - 43 =?

43= 00101011 Two’s Complement: 0010 1011 1101 0100 + 1 1101 0101

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•Convert signed decimal to hexadecimal -18 =?* Twist

18=12 hexadecimal

Two’s complement: 15 15 - 1 2 E D + 1 E E

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LEARN HOW TO DO THE FOLLOWING:

Convert signed hexadecimal to decimal- 8C =?8= -ve integer hexadecimalTwo complements: 15 15 - 8 C 7 3 + 1 7 4

=(7 X 16 1) + (4 X 16 0)=112+4=1168C = -116

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