1. CS 2130
Lecture 23
Data Types

2. Scalar Types
int
float
char
long
short
boolean
enums*

3. Scalar Types int float char long short double boolean Integer Numbers
enums*
*Some languages implement enumerated types in a more restrictive fashion than C
Integer Numbers

4. Some languages implement enumerated types in a more restrictive fashion than C
enum {FOO, BAR, BAZ, BLARG};
These become only allowed values

5. C Types C "thinks" of 3 basic types: char, int and float Typical Size*
Comments 1
char
Really just a 1 byte int 2 4
int
at least as long as a short 8
at least as long as an int
float
double
long
short
*in bytes which are normally but not necessarily 8 bits

6. DON'T FORGET!
signed
unsigned

7. More Bits, More Precision
int (4)...long (8)
Twice as long, twice as precise
Given an N bit number, how many different numbers can be represented?
Your final answer?

8. Depends! Unsigned 2N Numbers range from 0 to 2N-1 Signed 2N-1 or 2N
Depends on representation
Signed Magnitude
One's Complement
Two's Complement
Biased/Excess Notation

9. Ranges Unsigned Signed Magnitude One's Complement Two's Complement
Biased/
Excess Notation
(Bias = B)
From
-(2N-1 - 1)
-(2N-1) -B To
2N - 1
+(2N-1 - 1)
2N B

10. What? N = 3 Binary 000 001 010 011 100 101 110 111 Unsigned 1 2 3 4 51 2 3 4 5 6 7
Signed
Mag 1 2 3 -0 -1 -2 -3
One's
Comp 1 2 3 -3 -2 -1 -0
Two's
Comp 1 2 3 -4 -3 -2 -1
Excess 3 -3 -2 -1 1 2 4
Excess 4 -4 -3 -2 -1 1 2 3
Number
Stored
Number
Represented

11. N = 4 Number Represented Unsigned 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Binary
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Signed
Mag -0 -1 -2 -3 -4 -5 -6 -7
One's
Comp
Two's -8
Excess
N = 4
Number Represented

12. N = 8
Binary
...
Unsigned 1 2
...
127
128
129
254
255
Signed
Mag 1 2
...
127 -0 -1
-126
-127
One's
Comp 1 2
...
127
-127
-126 -1 -0
Two's
Comp 1 2
...
127
-128
-127 -2 -1
Excess
127
-127
-126
-125
... 1 2
128
Excess
128
-128
-127
-126
... -1 1
126
127

13. N = 32
Binary
...
Hex 1 2
...
7FFFFFFF
FFFFFFFE
FFFFFFFF
Unsigned 1 2
...

14. N = 32 Hex 1 2 ... 7FFFFFFF 80000000 80000001 FFFFFFFE FFFFFFFF1 2
...
7FFFFFFF
FFFFFFFE
FFFFFFFF
Unsigned 1 2
...
Signed
Mag 1 2
... -0 -1
One's
Comp 1 2
... -1 -0
Two's
Comp 1 2
... -2 -1

15. Where do bits come from? Bitaba Tree Andes Mountains Numeric Bits
14,000'
Numeric Bits
Character
Bits

16. NOT

17. So what's the difference...
Between a char 1 and an int 1
about 48?
Between the ip address and 0x and -2,144,978,842 and 2,149,988,454
The letter A and an int 65

18. What about these?

19. Key Point
Data is stored internally as binary numbers with no meaning to the computer.
All meaning is assigned by programmer using programming languages and other tools.
Example:
char c = 65;
printf(" %d \n", c);
printf(" %c \n", c);
Output: 65 A

20. Don't forget... Why two codes? ASCII Table (7-bit)
Decimal Octal Hex Binary Value
NUL (Null char.)
SOH (Start of Header)
STX (Start of Text)
ETX (End of Text)
EOT (End of Transmission)
ENQ (Enquiry)
ACK (Acknowledgment)
BEL (Bell)
BS (Backspace)
HT (Horizontal Tab)
A LF (Line Feed)
B VT (Vertical Tab)
C FF (Form Feed)
D CR (Carriage Return)
E SO (Serial In)(Shift Out)
F SI (Serial Out)(Shift Out)
DLE (Data Link Escape)
DC1 (XON) (Device Control 1)
DC (Device Control 2)
DC3 (XOFF)(Device Control 3)
DC (Device Control 4)
Why
two
codes?

21. Don't forget... ASCII Table (7-bit) Decimal Octal Hex Binary Value
NAK (Negative Acknowledgement)
SYN (Synchronous Idle)
ETB (End of Trans. Block)
CAN (Cancel)
EM (End of Medium)
A SUB (Substitute)
B ESC (Escape)
C FS (File Separator)
D GS (Group Separator)
E RS (Request to Send)(Record Separator)
F US (Unit Separator)
SP (Space) !
" # $ %
& ' (

22. Don't forget... ASCII Table (7-bit) Decimal Octal Hex Binary Value )
A * B
C , D E
F /
A :
B ;
C <

23. Don't forget... ASCII Table (7-bit) Decimal Octal Hex Binary Value
D =
E >
F ? @ A B C D E F G H I
A J
B K
C L
D M
E N
F O P

24. Don't forget... ASCII Table (7-bit) Decimal Octal Hex Binary Value Q R S T U V W X Y
A Z
B [
C \
D ]
E ^
F _ ` a b c d

25. Don't forget... ASCII Table (7-bit) Decimal Octal Hex Binary Value e f g h i
A j
B k
C l
D m
E n
F o p q r s t u v w x

26. Don't forget... Why is this at the end? ASCII Table (7-bit)
Decimal Octal Hex Binary Value y
A z
B {
C |
D }
E ~
F DEL
Why is this at the end?

27. Unsigned
Add 89 and 205
Step 1: Convert 89 and 205 to binary

28. Converting to Binary Use dc Use some other calculator
See man dc
Use some other calculator
Learn how to do it by hand !!!

29. Old Stuff

30. Converting to Binary 89 odd 1 44 even 0 22 even 0 11 odd 1 5 odd 1

31. Converting to Binary 205 odd 1 102 even 0 51 odd 1 25 odd 1 12 even 0

32. Why does this work? 47 odd 1 23 odd 1 11 odd 1 5 odd 1 2 even 0
48 even 0
24 even 0
12 even 0
6 even 0
3 odd 1
1 odd 1
0 even 0

33. Unsigned Add 89 and 205 01011001 11001101 100100110 Result = ??? + 1 11 1 1 1 10
Significance of this?

34. What happened?
Overflow (carry out from high order bit) means that the result is bigger than the allowed number of bits
= 294
Assumption for this discussion: Two N bit numbers may be added together and overflow can be detected
BTW: When adding two N bit numbers together...what is the maximum number of bits the result can have?
Can we do subtraction?

35. Signed Magnitude Both positive and negative zero
Equal number of positives and negatives
Easy to interpret
First bit is the sign
Remaining bits are number
Sounds ideal?

36. Signed Magnitude? Add these two numbers 01011001 11001101
What does this represent?

37. Signed Magnitude? What does this represent? 010110012 = 8910
= 8910
= -7710
How do we do the problem in decimal arithmetic?
Note: This is not subtraction. Addition of signed numbers
Recall 3rd Grade

38. Signed Magnitude? What does this represent? 010110012 = 8910
= 8910
= -7710
If signs are different sign of result will be sign of larger operand

39. Signed Magnitude? What does this represent? 010110012 = 8910
= 8910
= -7710
= 1210
What are the implications of this operation?

40. One's Complement
Easy
Create binary representation of negative number, ignoring sign
Flip all bits
Example -94:
Flip Bits

41. One's Complement?
What does this represent?

42. One's Complement? What does this represent? 010110012 = 8910
= 8910
= -5010

43. One's Complement? What does this represent? 010110012 = 8910
= 8910
= -5010
3910
Suppose we just add?

44. One's Complement? What does this represent? 010110012 = 8910
= 8910
= -5010
= 3810
So we ignore the overflow and have to add 1 to get the right answer?
Is this useful?

45. Two's Complement Calculate One's Complement Add 1 Example -94:
Produce One's Complement
Why do we do this?
Hardware Reasons

46. Why does it work? Consider decimal numbers.
Suppose we have 3 digits available
We can use 10's complement!
Example -94
First form 9's complement of 094
905
Add 1
906
But what is this?
= ?
So what?

47. Add 107 and -94 Decimal Form 107 -94 Rewrite as 906 - 1000 Add 13

48. Two's Complement?
What does this represent?

49. Two's Complement? What's the decimal answer? 010110012 = 8910
= 8910
= -5110

50. Two's Complement? What happens if we add the binary numbers?
= 8910
= -5110
3810

51. Two's Complement? The eight bit result is correct 010110012 = 8910
= 8910
= -5110
3810

52. Two's Complement? Overflow? 010110012 = 8910 110011012 = -5110
= 8910
= -5110
What's the rule?
Significance of this?

53. Consider 1 digit decimal numbers 2 2 -2 2 -2 -2 2 4 4 -4 -4 -2
Positive numbers:
Negative numbers: ( )
Bad Ok Ok Bad

54. Recall N = 3 Binary 000 001 010 011 100 101 110 111 Unsigned 1 2 3 4 51 2 3 4 5 6 7
Signed
Mag 1 2 3 -0 -1 -2 -3
One's
Comp 1 2 3 -3 -2 -1 -0
Two's
Comp 1 2 3 -4 -3 -2 -1
Excess 3 -3 -2 -1 1 2 4
Excess 4 -4 -3 -2 -1 1 2 3
Number
Stored
Number
Represented

55. Consider 3 bit numbers: 2 -2 2 -2 3 3 -3 -3 5 1 -1 -5 010 110 010 110
Max allowed 3 (011)
Min allowed -4 (100)
Bad Ok Ok Bad
If carry in to sign bit != carry out of sign bit: Overflow

56. But why? Back to decimal land...
Using complement notation to represent negative numbers looks like this:
Assume 3 decimal digits available (no sign)
Want -42
Subtract 42 from 1000 = 958 (Ten's Complement)
So we get
If we want to add 98 and -42 (Should get 56)
= 56

57. Think about forming the complement
100000
42653
57347
Another way:
Write down 9's complement
57346
Then add 1

58. Binary works the same way!
What's the 2's complement of 56?
Assume we have an 8 bit machine
56 =
Easier way
Start with:
Take 1's complement:
Add 1:

59. Binary works the same way!
Bad Ok Ok Bad

60. Excess Notation Need to define bias
Add bias to number to be represented
Example:
Bias = 127
Number to be represented: 42
Calculate: = 169
Convert to binary unsigned magnitude
To decode, subtract bias
Calculate = 42
We will see Excess Notation again with floating point

61. Binary Coded Decimal
For certain applications such as representing currency amounts, numbers must be exact.
The problems of representing decimal floating point numbers with binary representations cannot be tolerated
Solution: BCD or Binary Coded Decimal

62. BCD Encode each decimal digit as a sequence of binary digits 0: 0000
1: 0001
2: 0010
3: 0011
4: 0100
5: 0101
6: 0110
7: 0111
8: 1000
9: 1001
Note: 1010, 1011, 1100, 1101, 1110 and 1111 not used.

63. BCD Example: Decimal 4045 Binary: 1111 1100 1101 (12 digits)
BCD: (16 digits)
BCD is less efficient from the standpoint of number of bits required to store a number.

64. BCD Special routines are required to do math Example: 0011 3 0100 4
(less 10...okay)
(>9 add 6)
0110
(with carry)

65. BCD
Typical approaches to negative numbers use 9 and 10's complement arithmetic
BCD numbers are implemented in COBOL

66. Questions?

67. Key Ideas The computer just manipulates bits.
The same bit pattern can mean different things based on programmer and software
Different representations are used internally based on considerations such as
Cost
Speed
Typical time vs space tradeoffs.
YOU REALLY NEED TO UNDERSTAND THIS STUFF

68. Questions?