1. Data Representation

2. Data Representation Input data transformed into output.
A computer is a device that:
Accepts input
Processes data
Stores data
Produces output
Input data transformed into output.
Data can be stored for repeated output.

3. Data Representation
Spreadsheet data  graphs 3D models  animation Vocals and MIDI  Song Bar code  Price of item Card and Pin #  Money from ATM

4. Data Representation
How can we represent information in a way that can be stored and manipulated by a computer?

5. Data Representation and Storage
External representation: computers use decimal digits (base ten), 26-character alphabet for easier human interaction via keyboard, terminal, printer
Internal representation: computers use binary system for numbers, letters, graphics, etc.

6. Data Representation
Internally, computers represent information as patterns of bits
A bit (binary digit) is either 0 or 1; these are symbols and have no numeric meaning
Storing a bit requires that a device can be in one (and only one) of just two states; analogous to true and false

7. Data Representation
Binary Numbers!!! Sound  pitch  number  binary number Letter  number  binary number Image  color at each pixel  number  binary number

9. Decimal Number Systems
Base 10
Digits - 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
e.g =
= 3 x x x100
= 3 x x x 1 =

10. Binary Number System Base 2 Digits 0, 1 e.g. 1102 =
= 1 x x x 20
= 1 x x x 1 =
= 6

11. Data Representation Solution: use a fixed number of digits.
But how many bits do we need?
1 binary digit  0 or 1  2 possible chars
2 binary digits  00, 01, 10, 11  4 chars
3 binary digits  000, 001, 010, 011, 100, 101, 110, 111  8 chars
Notice a pattern? 12, 24, 38, …
the total number of character that can be represented by n bits is 2n

13. Data Representation log2n
But how many bits are needed to store n symbols?
Or, how many bits are needed to represent n numbers?
log2n
How many bits do we need to represent 16 states, 63 states?

14. Let’s count 1 bit 2 bits 3 bits 000 ;0 001 ;1 010 ;2 011 ;3 100 ;4
; 0
; 1
2 bits
; 0
01 ; 1
10 ; 2 ;3
3 bits ;0 ;1 ;2 ;3 ;4 ;5 ;6 ;7

15. Binary Numerals: Convert to Decimal
Bits are numbered from the right
b7 b6b5b4b3b2b1b0
Subscripts represent the place value
bi has place value 2i
Convert to decimal
b7 * 27+b6*26 + b5*25 +b4*24 +b3*23 +b2*22 +b1*21 + b0*20

16. Binary to Decimal: Example
100 = 1 * * = 1 *

17. Data Representation
Binary to Decimal = 1 * * * * * 20

18. Data Representation
Binary to Decimal = 1 * * * * * 20 = = 19

19. Converting Decimal to Binary
Repeatedly divide by 2, recording remainders in reverse order
e.g. 53 / 2 = 26 R 1
26 / 2 = 13 R 0
13 / 2 = 6 R 1
6 / 2 = 3 R 0
3 / 2 = 1 R 1
1 / 2 = 0 R 1
giving

20. Excercise
56d = ?b
1011b = ?d

21. Addition & Subtraction=? =
011 – 010 =

22. 4 bits and Hex 0000 ;0 1000 ;8 0001 ;1 1001 ;9 0010 ;2 1010 ;10 (Ah);0 ;1 ;2 ;3 ;4 ;5 ;6 ;7 ;8 ;9
;10 (Ah)
;11 (Bh)
;12 (Ch)
;13 (Dh)
;14 (Eh)
;15 (Fh)

23. Storing Negative Values Two’s complement!

24. Storing Negative Values
Sign Magnitude
use first bit as sign bit, 0 = positive 1 = negative
e.g. 8-bits
= = −0
= = −1
… …
= = −127
Problem: has two 0s.
Using 2’s Complement instead

25. Two’s Complement Two’s complement if positive, use binary
if negative, complement bits and add one
e.g. −53
magnitude (binary rep for 53)
complement (flip each bit)
add (add 1, resulting
2’s complement rep for -53)

27. Examples: 2’s Complement Rep
3 bit pattern
000 ; 0
001 ; 1
010 ; 2
011 ; 3
100 ; -4
101 ; -3
110 ; -2
111 ; -1
We can use the 2’s complement code of 3 code to find that of -3
And vice versa

28. Exercise
Practice: 2 – 3 using 2’s complement representation as done by a computer
Using 3 bit pattern
(2)
(-3)
= (-1)
Using 8 bit pattern
(2)
(-3)
= (-1)

29. 8-bit Two’s Complement
= = = − = = −2 … … = = − = −128

30. 16-bit Two’s Complement
8-bit two’s complement range is − 27 to 27 − 1 − to 127
16-bit two’s complement range is − to 215 − 1 − 32,768 to 32,767

31. Overflow When a number is too big for the range, overflow will occur
Example: with 3 bit pattern
3 + 2 using 2’s complement
Two positive number add up to a negative number, overflow
Similarly, -4-3 results in a positive number, also overflow

32. Representing Real Numbers

33. Representing Real Numbers
A number with a whole part and a fractional part
, , 37.0, and
Positions to the right of the decimal point are the
tenths position: 10-1, 10-2 ,
Same rules apply in binary as in decimal
Decimal point is actually the radix point
Positions to the right of the radix point in binary are
2-1 (one half), 2-2 (one quarter),
2-3 (one eighth)

34. Representing Real number
A real value in base 10 can be defined by the following formula
The representation is called floating point because the number of digits is fixed but the radix point floats
A binary floating-point value is defined by the formula

36. Memory Sizes A byte is 8 bits. Kilobyte (K) = 210 = 1,024 bytes
Megabyte (Mb) = 220 = 1,048,576 bytes
Gigabyte (Gb) = 230 = 1,073,741,824 bytes

37. Character Representation

38. Character Representation
Assign a code to each character
ASCII
American Standard Code for Information Interchange
8 bits per character
256 possible codes with 8 bits
Unicode, 16 bits per character
International language coding standard
Superset of ASCII

39. ASCII
American Standard Code for Information Interchange (ASCII ) defines 256 symbols that can be stored in a byte. Each symbol corresponds to a number from
Symbol
Decimal
Binary @ 64 A 65 B 66 C 67 D 68 E 69 F 70 G 71 H 72

40. ASCII Code
Code Value Letter 0 Null character Special Control Characters 10 \n = New line 32 Space 33-47, 58-64, Punctuation A - Z a - z

41. Digits 0 through 9 in ASCII
Digit Dec Hex
… … …

42. Unicode Character Examples