CS 2130
Lecture 23
Data Types
Scalar Types
•
•
•
•
•
•
•
int
float
char
long
short
boolean
enums*
Scalar Types
•
•
•
•
•
•
•
•
int
float
char
long
short
double
boolean
enums*
Integer Numbers
• *Some languages implement enumerated types in a
more restrictive fashion than C
Some languages implement enumerated
types in a more restrictive fashion than C
enum {FOO, BAR, BAZ, BLARG};
0
1
2
3
These become only allowed values
C Types
• C "thinks" of 3 basic types: char, int and float
Typical
Size*
Type
Comments
1
char
Really just a 1 byte int
2
short
4
int
at least as long as a short
8
long
at least as long as an int
4
float
8
double
*in bytes which are normally but not necessarily 8 bits
DON'T FORGET!
•signed
•unsigned
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?
Depends!
• Unsigned
• Signed
• 2N
• 2N-1 or 2N
• Numbers range from
0 to 2N-1
• Depends on representation
–
–
–
–
Signed Magnitude
One's Complement
Two's Complement
Biased/Excess Notation
Ranges
Unsigned
Signed Magnitude
One's Complement
Two's Complement
Biased/
Excess Notation
(Bias = B)
From
0
To
2N - 1
-(2N-1 - 1) +(2N-1 - 1)
-(2N-1 - 1) +(2N-1 - 1)
-(2N-1)
+(2N-1 - 1)
-B
2N - 1 - B
What?
N = 3
Binary Unsigned Signed One's Two's Excess Excess
Mag Comp Comp
3
4
000
001
010
011
100
101
110
111
Number
Stored
0
1
2
3
4
5
6
7
0
1
2
3
-0
-1
-2
-3
0
1
2
3
-3
-2
-1
-0
Number
Represented
0
1
2
3
-4
-3
-2
-1
-3
-2
-1
0
1
2
3
4
-4
-3
-2
-1
0
1
2
3
N = 4
Number Represented
Binary Unsigned Signed One's Two's Excess Excess
Mag Comp Comp
7
8
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
1
2
3
4
5
6
7
-0
-1
-2
-3
-4
-5
-6
-7
0
1
2
3
4
5
6
7
-7
-6
-5
-4
-3
-2
-1
-0
0
1
2
3
4
5
6
7
-8
-7
-6
-5
-4
-3
-2
-1
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
N = 8
Binary
00000000
00000001
00000010
...
01111111
10000000
10000001
...
11111110
11111111
Unsigned Signed One's Two's Excess Excess
Mag Comp Comp
127
128
0
1
2
...
127
128
129
...
254
255
0
1
2
...
127
-0
-1
...
-126
-127
0
1
2
...
127
-127
-126
...
-1
-0
0
1
2
...
127
-128
-127
...
-2
-1
-127
-126
-125
...
0
1
2
...
127
128
-128
-127
-126
...
-1
0
1
...
126
127
N = 32
Binary
00000000000000000000000000000000
00000000000000000000000000000001
00000000000000000000000000000010
...
01111111111111111111111111111111
10000000000000000000000000000000
10000000000000000000000000000001
...
11111111111111111111111111111110
11111111111111111111111111111111
Hex
0
1
2
...
7FFFFFFF
80000000
80000001
...
FFFFFFFE
FFFFFFFF
Unsigned
0
1
2
...
2147483647
2147483648
2147483649
...
4294967294
4294967295
N = 32
Hex
0
1
2
...
7FFFFFFF
80000000
80000001
...
FFFFFFFE
FFFFFFFF
Unsigned
Signed
Mag
One's
Comp
Two's
Comp
0
0
0
0
1
1
1
1
2
2
2
2
...
...
...
...
2147483647
2147483647 2147483647 2147483647
2147483648
-0 -2147483647 -2147483648
2147483649
-1 -2147483646 -2147483647
...
...
...
...
4294967294 -2147483646
-1
-2
4294967295 -2147483647
-0
-1
Where do bits come from?
Numeric Bits
14,000'
Bitaba Tree
Andes Mountains
Character
Bits
NOT
So what's the difference...
• Between a char 1 and an int 1
– about 48?
• Between the ip address 128.43.56.102 and
0x80263866 and -2,144,978,842 and 2,149,988,454
• The letter A and an int 65
What about these?
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
Don't forget...
ASCII Table (7-bit)
Decimal
Octal
----------000
000
001
001
002
002
003
003
004
004
005
005
006
006
007
007
008
010
009
011
010
012
011
013
012
014
013
015
014
016
015
017
016
020
017
021
018
022
019
023
020
024
Hex
--000
001
002
003
004
005
006
007
008
009
00A
00B
00C
00D
00E
00F
010
011
012
013
014
Binary
-----00000000
00000001
00000010
00000011
00000100
00000101
00000110
00000111
00001000
00001001
00001010
00001011
00001100
00001101
00001110
00001111
00010000
00010001
00010010
00010011
00010100
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)
LF
(Line Feed)
VT
(Vertical Tab)
FF
(Form Feed)
CR
(Carriage Return)
SO
(Serial In)(Shift Out)
SI
(Serial Out)(Shift Out)
DLE
(Data Link Escape)
DC1 (XON) (Device Control 1)
DC2
(Device Control 2)
DC3 (XOFF)(Device Control 3)
DC4
(Device Control 4)
Why
two
codes?
Don't forget...
ASCII Table (7-bit)
Decimal
Octal
----------021
025
022
026
023
027
024
030
025
031
026
032
027
033
028
034
029
035
030
036
031
037
032
040
033
041
034
042
035
043
036
044
037
045
038
046
039
047
040
050
Hex
--015
016
017
018
019
01A
01B
01C
01D
01E
01F
020
021
022
023
024
025
026
027
028
Binary
-----00010101
00010110
00010111
00011000
00011001
00011010
00011011
00011100
00011101
00011110
00011111
00100000
00100001
00100010
00100011
00100100
00100101
00100110
00100111
00101000
Value
----NAK
SYN
ETB
CAN
EM
SUB
ESC
FS
GS
RS
US
SP
!
"
#
$
%
&
'
(
(Negative Acknowledgement)
(Synchronous Idle)
(End of Trans. Block)
(Cancel)
(End of Medium)
(Substitute)
(Escape)
(File Separator)
(Group Separator)
(Request to Send)(Record Separator)
(Unit Separator)
(Space)
Don't forget...
ASCII Table (7-bit)
Decimal
Octal
----------041
051
042
052
043
053
044
054
045
055
046
056
047
057
048
060
049
061
050
062
051
063
052
064
053
065
054
066
055
067
056
070
057
071
058
072
059
073
060
074
Hex
--029
02A
02B
02C
02D
02E
02F
030
031
032
033
034
035
036
037
038
039
03A
03B
03C
Binary
-----00101001
00101010
00101011
00101100
00101101
00101110
00101111
00110000
00110001
00110010
00110011
00110100
00110101
00110110
00110111
00111000
00111001
00111010
00111011
00111100
Value
----)
*
+
,
.
/
0
1
2
3
4
5
6
7
8
9
:
;
<
Don't forget...
ASCII Table (7-bit)
Decimal
Octal
----------061
075
062
076
063
077
064
100
065
101
066
102
067
103
068
104
069
105
070
106
071
107
072
110
073
111
074
112
075
113
076
114
077
115
078
116
079
117
080
120
Hex
--03D
03E
03F
040
041
042
043
044
045
046
047
048
049
04A
04B
04C
04D
04E
04F
050
Binary
-----00111101
00111110
00111111
01000000
01000001
01000010
01000011
01000100
01000101
01000110
01000111
01001000
01001001
01001010
01001011
01001100
01001101
01001110
01001111
01010000
Value
----=
>
?
@
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Don't forget...
ASCII Table (7-bit)
Decimal
Octal
----------081
121
082
122
083
123
084
124
085
125
086
126
087
127
088
130
089
131
090
132
091
133
092
134
093
135
094
136
095
137
096
140
097
141
098
142
099
143
100
144
Hex
--051
052
053
054
055
056
057
058
059
05A
05B
05C
05D
05E
05F
060
061
062
063
064
Binary
-----01010001
01010010
01010011
01010100
01010101
01010110
01011111
01011000
01011001
01011010
01011011
01011100
01011101
01011110
01011111
01100000
01100001
01100010
01100011
01100100
Value
----Q
R
S
T
U
V
W
X
Y
Z
[
\
]
^
_
`
a
b
c
d
Don't forget...
ASCII Table (7-bit)
Decimal
Octal
----------101
145
102
146
103
147
104
150
105
151
106
152
107
153
108
154
109
155
110
156
111
157
112
160
113
161
114
162
115
163
116
164
117
165
118
166
119
167
120
170
Hex
--065
066
067
068
069
06A
06B
06C
06D
06E
06F
070
071
072
073
074
075
076
077
078
Binary
-----01100101
01100110
01100111
01101000
01101001
01101010
01101011
01101100
01101101
01101110
01101111
01110000
01110001
01110010
01110011
01110100
01110101
01110110
01110111
01111000
Value
----e
f
g
h
i
j
k
l
m
n
o
p
q
r
s
t
u
v
w
x
Don't forget...
ASCII Table (7-bit)
Decimal
Octal
----------121
171
122
172
123
173
124
174
125
175
126
176
127
177
Hex
--079
07A
07B
07C
07D
07E
07F
Binary
-----01111001
01111010
01111011
01111100
01111101
01111110
01111111
Value
----y
z
{
|
}
~
DEL
Why is this at the end?
Unsigned
• Add 89 and 205
• Step 1: Convert 89 and 205 to binary
Converting to Binary
• Use dc
– See man dc
• Use some other calculator
• Learn how to do it by hand !!!
Old Stuff
http://www.pgh.net/~newcomer/hp16c.htm
Converting to Binary
89
44
22
11
5
2
1
0
odd
even
even
odd
odd
even
odd
even
1
0
0
1
1
0
1
0
01011001
Converting to Binary
205
102
51
25
12
6
3
1
odd
even
odd
odd
even
even
odd
odd
1
0
1
1
0
0
1
1
11001101
Why does this work?
47
23
11
5
2
1
0
0
odd
odd
odd
odd
even
odd
even
even
1
1
1
1
0
1
0
0
48
24
12
6
3
1
0
0
00101111
even
even
even
even
odd
odd
even
even
0
0
0
0
1
1
0
0
00110000
Unsigned
• Add 89 and 205
01011001
11001101
100100110
Significance of this?
Result = ???
+
0
1
0
0
1
1
1 10
What happened?
• Overflow (carry out from high order bit) means that
the result is bigger than the allowed number of bits
89 + 205 = 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?
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?
Signed Magnitude?
• Add these two numbers
01011001
11001101
• What does this represent?
Signed Magnitude?
• What does this represent?
010110012 = 8910
110011012 = -7710
• How do we do the problem in decimal arithmetic?
• Note: This is not subtraction. Addition of signed
numbers
– Recall 3rd Grade
Signed Magnitude?
• What does this represent?
010110012 = 8910
110011012 = -7710
0
If signs are different
sign of result will be
sign of larger operand
Signed Magnitude?
• What does this represent?
010110012 = 8910
110011012 = -7710
000011002 = 1210
• What are the implications of this operation?
One's Complement
• Easy
• Create binary representation of negative number,
ignoring sign
• Flip all bits
• Example -94:
01011110
• Flip Bits
10100001
One's Complement?
• What does this represent?
01011001
11001101
One's Complement?
• What does this represent?
010110012 = 8910
110011012 = -5010
One's Complement?
• What does this represent?
010110012 = 8910
110011012 = -5010
3910
• Suppose we just add?
One's Complement?
• What does this represent?
010110012 = 8910
110011012 = -5010
1001001102
3910
001001102 =
3810
• So we ignore the overflow and have to add 1 to get
the right answer?
• Is this useful?
Two's Complement
• Calculate One's Complement
• Add 1
• Example -94:
01011110
• Produce One's Complement
10100001
• Add 1
10100010
• Why do we do this?
• Hardware Reasons
Why does it work?
• Consider decimal numbers.
• Suppose we have 3 digits available 000 - 999
• We can use 10's complement!
• Example -94
• First form 9's complement of 094
905
• Add 1
906
• But what is this?
906 - 1000 = ?
• So what?
Add 107 and -94
• Decimal Form
107
-94
• Rewrite as
107
906 - 1000
• Add
1013 - 1000
13
Two's Complement?
• What does this represent?
010110012
110011012
Two's Complement?
• What's the decimal answer?
010110012 = 8910
110011012 = -5110
Two's Complement?
• What happens if we add the binary numbers?
010110012 = 8910
110011012 = -5110
3810
Two's Complement?
• The eight bit result is correct
010110012 = 8910
110011012 = -5110
1001001102
3810
3810
Two's Complement?
• Overflow?
010110012 = 8910
110011012 = -5110
1001001102
3810
Significance of this?
• What's the rule?
Consider
• 1 digit decimal numbers
– Positive numbers: 0 1 2 3 4
– Negative numbers: 9 8 7 6 5 (-1 -2 -3 -4 -5)
2
2
4
2
4
6
-2
4
2
2
-4
-2
-2 -2
-4 -2
-6 -4
2
2
4
2
4
6
8
4
12
2
6
8
8 8
6 8
14 16
Bad
Ok
Ok
Bad
Recall
N = 3
Binary Unsigned Signed One's Two's Excess Excess
Mag Comp Comp
3
4
000
001
010
011
100
101
110
111
Number
Stored
0
1
2
3
4
5
6
7
0
1
2
3
-0
-1
-2
-3
0
1
2
3
-3
-2
-1
-0
Number
Represented
0
1
2
3
-4
-3
-2
-1
-3
-2
-1
0
1
2
3
4
-4
-3
-2
-1
0
1
2
3
Consider
• 3 bit numbers:
– Max allowed 3 (011)
– Min allowed -4 (100)
2
3
5
-2
3
1
2
-3
-1
-2
-3
-5
010
011
0101
Bad
110
011
1001
Ok
010
101
0111
Ok
110
101
1011
Bad
If carry in to sign bit != carry out of sign bit: Overflow
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 958 - 1000
• If we want to add 98 and -42 (Should get 56)
98 0
958 - 1000
1056 - 1000 = 56
Think about forming the complement
100000
42653
57347
Another way:
42653
Write down 9's complement
57346
Then add 1
57347
Binary works the same way!
What's the 2's complement of 56?
Assume we have an 8 bit machine
56 = 111000
100000000
-111000
11001000
Easier way
Start with:
Take 1's complement:
Add 1:
00111000
11000111
11001000
Binary works the same way!
2
3
5
010-0
011-0
101
-3
Bad
-2
3
1
110-1000
0110
1001-1000
1
Ok
2
-3
-1
0100
101-1000
111-1000
-1
Ok
-2
-3
-5
110 -1000
101 -1000
1011-10000
-5
Bad
Excess Notation
•
•
•
•
•
•
•
Need to define bias
Add bias to number to be represented
Example:
Bias = 127
Number to be represented: 42
Calculate: 42 + 127 = 169
Convert to binary unsigned magnitude 10101001
• To decode, subtract bias
• Calculate 169 - 127 = 42
We will see Excess Notation again with floating point
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
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.
BCD
• Example: Decimal 4045
• Binary:
1111 1100 1101 (12 digits)
• BCD: 0100 0000 0100 0101 (16 digits)
• BCD is less efficient from the
standpoint of number of bits required
to store a number.
BCD
• Special routines are required to do
math
Example: 0011
3
0100
4
0111
7 (less 10...okay)
0111
1000
1111
0110
1 0101
7
8
15 (>9 add 6)
5 (with carry)
BCD
• Typical approaches to negative numbers use 9 and
10's complement arithmetic
• BCD numbers are implemented in COBOL
Questions?
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
Questions?