Download Negative Integers - Department of Computer and Information Science

Department of Computer and Information Science,
School of Science, IUPUI
CSCI N305
Information Representation:
Negative Integer Representation
Dale Roberts
Negative Numbers in Binary
Four different representation schemes are used
for negative numbers
1. Signed Magnitude
Left most bit (LMB) is the sign bit :
0  positive (+)
1  negative (-)
Remaining bits hold absolute magnitude
Example:
210  0000 0010b
Try, 1000 0100b = -410
-210  1000 0010b
Q: 0000 0000 = ?
1000 0000 = ?
Dale Roberts
1’s Complement
2. One’s Complement
– Left most bit is the sign bit :
•
•
0  positive (+)
1  negative (-)
– The magnitude is Complemented
Example:
210  0 000 0010b
-210  1 111 1101b
Exercise: try - 410 using 1’s Complement
Q: 0000 0000 = ?
1111 1111 = ?
Solution:
410 = 0 000 0100
-410 = 1 111 1011
b
b
Dale Roberts
2’s Complement
3. 2’s Complement
• Sign bit same as above
• Magnitude is Complemented first and a “1” is added to
the Complemented digits
Example:
210  0 000 0010b
1’s Complement  1 111 1101b
+
1
-210  1 111 1110b
Exercise: try -710 using 2’s Complement
710  0000 0111
b
1’s Complement 1111 1000b
+
1
-710  1111 1001b
Dale Roberts
2’s Complement
Example: 7+(-3)
[hint]: A – B = A + (~B) +1
710 = 0000 0111b
310 = 0000 0011b
1’s complement 1111 1100
b
2’s complement 1111 1101  -3
10
b
1 1111 111 carry
7+(-3) 
0000 0111
+1111 1101
ignore 1 0000 0100  0000 0100  410
Dale Roberts
Three Representation of Signed Integer
Representation
0 0000
0 0001
0 0010
0 0011
0 0100
0 0101
0 0110
0 0111
0 1000
0 1001
0 1010
0 1011
0 1100
0 1101
0 1110
0 1111
1 0000
1 0001
1 0010
1 0011
1 0100
1 0101
1 0110
1 0111
1 1000
1 1001
1 1010
1 1011
1 1100
1 1101
1 1110
1 1111
Value Representation
Signed Magnitude
1's Complement 2's Complement
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
-13
-14
-15
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-15
-14
-13
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
-0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-16
-15
-14
-13
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
Dale Roberts
Negative Numbers in Binary (cont.)
4. Excess Representation
– For a given fixed number of bits
the range is remapped such that
roughly half the numbers are
negative and half are positive.
Example: (as left)
Excess – 8 notation for 4 bit numbers
Binary value = 8 + excess-8
value
MSB can be used as a sign bit,
but
If MSB =1, positive number
If MSB =0, negative number
Excess Representation is also
called bias
Numbers
Binary
Value
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Notation
Excess – 8
Value
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Dale Roberts
Fundamental Data Types
• With vs. without using sign bit
For a 16 bit binary pattern:
2 byte unsigned (Default type is int)
0000 0000 0000 0000 (  0D)
0000 0000 0000 0001 (  1D )
0000 0000 0000 0010 (  2D )
….
0111 1111 1111 1111 (  32767D  215 -1)
1000 0000 0000 0000 (  32768D  215)
….
1111 1111 1111 1111 ( 216 –1)
2 byte
int
1000 0000 0000 0000 (  -32768D  - 215 )
1000 0000 0000 0001 (  -32767D  - 215 +1)
….
1111 1111 1111 1110 (  - 2D )
1111 1111 1111 1111 (  - 1D )
0000 0000 0000 0000 (  0D )
0000 0000 0000 0001 (  1D )
0000 0000 0000 0010 (  2D )
….
0111 1111 1111 1111 (  32767D  215 -1)
Dale Roberts
Fundamental Data Types
Four Data Types in C (assume 2’s complement, byte machine)
Data Type
char
Abbreviation
Size (byte)
Range
char
1
-128 ~ 127
unsigned char
1
0 ~ 255
2 or 4
-215 ~ 215-1 or -231 ~ 231-1
2 or 4
0 ~ 65535 or 0 ~ 232-1
int
int
unsigned int
unsigned
short int
short
2
-32768 ~ 32767
unsigned short
int
unsigned short
2
0 ~ 65535
long int
long
4
-231 ~ 231-1
unsigned long
int
unsigned long
4
0 ~ 232-1
float
4
double
8
Note:
27 = 128, 215 =32768, 215 = 2147483648
Complex and double complex are not available
Dale Roberts
Acknowledgements
These slides where originally prepared by Dr. Jeffrey Huang, updated by Dale
Roberts.
Dale Roberts