1. Assembly Language for Intel-Based Computers
DATA REPRESENTATION
Part 1

2. 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

3. 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 …
Input device
A 1+2=3 math

4. 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

5. 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

6. 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)

7. 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

8. 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: " "
ASCII decimal
string of digits: "65"
ASCII hexadecimal
string of digits: "9C"

9. 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

10. 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

11. 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

12. Binary, Octal, Decimal and Hexadecimal Digits Table
System
Base
Possible Digits
Binary 2
0 1
Octal 8
Decimal 10
Hexadecimal 16
A B C D E F

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

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

15. 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

16. 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 = decimal 9:
How to calculate?
( 0 27)+(0  26)+(0  25)+(0  24)+(1  23)+(0  22)+ (0  21)+ (1  20)
 9

17. Translating Binary to Decimal
Try this

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

19. Translating Decimal to Binary
Try this 55 16 77
111

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

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

22. Hexadecimal Integers
Binary values are represented in hexadecimal.

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

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

25. 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.

26. Converting Hexadecimal to Decimal (cont)
Try this:
1C59
ABCD
Hex 3BA4 equals
3 * =
+ 11 * 162 =
+ 10 * 161 =
* 160 =
15,268

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

28. 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 B8
21/16 = 1, remain 5
1/16 = 0, remain 1
Important skill: Programmers frequently add and subtract the addresses of variables and instructions.

29. 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

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

31. Hexadecimal Subtraction
7 5
4 7
2 E
= 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

32. Try this Hexadecimal addition Hexadecimal subtraction D3A + 19
6B2 + FF
Hexadecimal subtraction
D1A - 339 A
74B – 8F

33. 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

34. 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)

35. 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: = +255
Signed: = = -127
Note: 2 values for 0: +0 ( ) and -0 ( )

36. 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

37. Forming the Two's Complement
Negative numbers are stored in two's complement notation Represents the additive Inverse
Note that =

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

39. 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

40. 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

41. **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

42. 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?

43. Numeric Data Representation
pure binary
can be calculated directly
ASCII binary
string of digits: " "
ASCII decimal
string of digits: "65"
ASCII hexadecimal
string of digits: "9C"
next: Boolean Operations

44. 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
(First complement)
(Two’s Complement)
=( 1 X 2 ^ 5) =16
=16

45. Learn How To Do the Following:
Convert signed decimal to binary
- 43 =?
43=
Two’s Complement:
Convert signed decimal to hexadecimal
-18 =?
* Twist
18=12 hexadecimal
Two’s complement:
15 15
E D
E E

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