Assembly Language for Intel-Based Computers DATA REPRESENTATION Part 1

Data Formats Computers Human communication Data formats: 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

Sources of Data Binary input Analog 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 Figure 3.1 with this color scheme Computer 1101000101010101… Input device A 1+2=3 math

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

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

Internal Data Representation Reflects 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)

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

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"

Data Types: 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

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

Data Representation Binary Numbers Translating between binary and decimal Binary Addition Integer Storage Sizes Hexadecimal Integers Translating between decimal and hexadecimal Hexadecimal subtraction Signed Integers Binary subtraction Character Storage

Binary, Octal, Decimal and Hexadecimal Digits Table 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

Binary Numbers Digits are 1 and 0 1 = true 0 = false MSB – most significant bit LSB – least significant bit Bit numbering:

Binary Numbers Binary integers can be signed or unsigned. Signed integer = either positive or negative Unsigned integer = only positive, including 0.

Unsigned Binary Integers Each digit (bit) is either 1 or 0 Each bit represents a power of 2 Every binary number is a sum of powers of 2

Translating Unsigned Binary to Decimal Weighted 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  9

Translating Binary to Decimal Try this 010011002 100000012 101010102 110011002

Translating Unsigned Decimal to Binary Repeatedly divide the decimal integer by 2. Each remainder is a binary digit in the translated value: 37 = 1001012

Translating Decimal to Binary Try this 55 16 77 111

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

Integer Storage Sizes Standard sizes: What is the largest unsigned integer that may be stored in 20 bits?

Hexadecimal Integers Binary values are represented in hexadecimal.

Translating Binary to Hexadecimal Each hexadecimal digit corresponds to 4 binary bits. Example: Translate the binary integer 000101101010011110010100 to hexadecimal:

Powers of 16 From 160 to 167 Used when calculating hexadecimal values up to 8 digits long:

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.

Converting Hexadecimal to Decimal (cont) Try this: 1C59 ABCD Hex 3BA4 equals 3 * 163 = 12 288 + 11 * 162 = 2 816 + 10 * 161 = 160 + 4 * 160 = 4 15,268

Converting Decimal to Hexadecimal Try this: 100 2108 decimal 422 = 1A6 hexadecimal

Hexadecimal Addition eg: 1 6 A  6 | 10 + 4 B  4 | 11 11 | 21 B 5 Divide the sum of two digits by the number base (16). The quotient becomes the carry value, and the remainder is the sum digit. eg: 1 6 A  6 | 10 + 4 B  4 | 11 11 | 21 B 5 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.

Solution A2 : A3 : 2 | 8 2 | 8 + 4 | 5 + B | 8 6 | 13 6 D 14 | 16 E 0 16 /16 = 1 remain 0 1/16 = 0 remain 1 14 = E

Hexadecimal Subtraction When a borrow is required from the digit to the left, add 16 (decimal) to the current digit's value: 16 2 8 16 3 9 5 3 9 5 - D A - 13 10 2 11 11 2 B B C 6 12 6 - A 2 -10 2 2 4 (16+5)-10 =11 (16+8)-13 =11

Hexadecimal Subtraction 6 16 7 5 4 7 2 E 16 + 5 = 21 21 – 7 = 14 14 = E F = 15 15 – 2 = 13 13 = D E =14 B = 11 14 – 11 = 3 E F B 2 3 D

Try this Hexadecimal addition Hexadecimal subtraction D3A + 19 6B2 + FF Hexadecimal subtraction D1A - 339 A11 - 99 74B – 8F

Signed Integers The highest bit indicates the sign. 1 = negative, 0 = positive If the highest digit of a hexadecimal integer is > 7, the value is negative. Examples: 8A, C5, A2, 9D

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)

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)

1’s vs. 2’s Complements Choice made by computer designer 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

Forming the Two's Complement Negative numbers are stored in two's complement notation Represents the additive Inverse Note that 00000001 + 11111111 = 00000000

Forming Two’s Complement of Hexadecimal Step 1: reverse the bits = 95C2 Step 2: Add 1 = 95C3

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

Guidelines Convert signed binary to decimal If number is negative, form two complement notation Convert to decimal Convert signed decimal to binary Convert to binary Convert signed decimal to hexadecimal Convert to hexadecimal Convert signed hexadecimal to decimal

**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

Ranges of Signed Integers 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?

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" next: Boolean Operations

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

Learn How To Do the Following: Convert signed decimal to binary - 43 =? 43= 00101011 Two’s Complement: 0010 1011 1101 0100 + 1 1101 0101 Convert signed decimal to hexadecimal -18 =? * Twist 18=12 hexadecimal Two’s complement: 15 15 - 1 2 E D + 1 E E

Learn How To Do the Following: Convert signed hexadecimal to decimal - 8C =? 8= -ve integer hexadecimal Two complements: 15 15 - 8 C 7 3 + 1 7 4 =(7 X 16 1) + (4 X 16 0) =112+4=116 8C = -116