CSE 271- Signed/Unsigned Complements
K. Dudeck
Page 1 of 3
Comparison Between Unsigned and Signed 2's Complement
Binary Number Arithmetic (for n=5 bits).
BINARY
Range:
Example
Numbers:
00000 -> 01111
10000 -> 11111
N1=00100
N2=01001
N3=01110
N4=10010
N5=10100
N6=11101
DECIMAL EQUIVALENT
UNSIGNED
SIGNED 2's
0 -> 15
0 -> +15
16 -> 31
-16 -> -1
N1=4
N1=+4
N2=9
N2=+9
N3=14
N3=+14
N4=18
N4=-14
N5=20
N5=-12
N6=29
N6=-3
Addition
Examples
1)
N1:
00100
N2: + 01001
----0] 01101
4
+ 9
----13
+ 4
+ + 9
----+13
2)
N2:
01001
N3: + 01110
----0] 10111
9
+ 14
----23
+ 9
+ +14
----- 9 .
3)
N2:
01001
N4: + 10010
----0] 11011
9
+ 18
----27
+ 9
+ -14
----- 5
4)
N3:
01110
N5: + 10100
----1] 00010
14
+ 20
----2 .
+14
+ -12
----+ 2
5)
N4:
10010
N5: + 10100
----1] 00110
18
+ 20
----6 .
-14
+ -12
----+ 6 .
6)
N5:
10100
N6: + 11101
----1] 10001
20
+ 29
----17 .
-12
+ - 3
-----15
. - Overflow
CSE 271- Signed/Unsigned Complements
K. Dudeck
Page 2 of 3
DECIMAL EQUIVALENT
UNSIGNED
SIGNED 2's
BINARY
Subtract
Examples:
7)
N2:
01001
N1: - 00100 -> +
----1] 00101
1]
01001
11100
----00101
9
- 4
----5
+ 9
- + 4
----+ 5
8)
N1:
00100
N2: - 01001 -> +
----0] 11011
0]
00100
10111
----11011
4
- 9
----27 .
+ 4
- + 9
----- 5
9)
N4:
10010
N3: - 01110 -> +
----1] 00100
1]
10010
10010
----00100
18
- 14
----4
-14
- +14
----+ 4 .
10) N3:
01110
N4: - 10010 -> +
----0] 11100
0]
01110
01110
----11100
14
- 18
----28 .
+14
- -14
----- 4 .
11) N5:
10100
N6: - 11101 -> +
----0] 10111
0]
10100
00011
----10111
20
- 29
----23 .
-12
- - 3
----- 9
12) N6:
11101
N5: - 10100 -> +
----1] 01001
1]
11101
01100
----01001
29
- 20
----9
- 3
- -12
----+ 9
13) N6:
11101
N1: - 00100 -> +
----1] 11001
1]
11101
11100
----11001
29
- 4
----25
- 3
- + 4
----- 7
4
- 29
----7 .
+ 4
- - 3
----+ 7
13) N1:
00100
N6: - 11101
----0] 00111
00100
+ 00011
----0] 00111
. - Overflow
CSE 271- Signed/Unsigned Complements
K. Dudeck
Page 3 of 3
CONCLUSIONS:
When adding or subtracting binary numbers, there is no difference in the
binary result between unsigned and signed. The only distinction between the
two systems is the decimal interpretation of the result. Binary addition is
simply carried out as usual. Binary subtraction is accomplished by taking the
two's complement of the subtrahend and adding it to the minuend.
The
interpretation of the results are shown below:
Unsigned Binary Numbers.
1. Only positive numbers exist; therefore the most significant bit is
not an indication of sign.
2. When adding, the answer is correct if there is no carry bit generated
(ex. 1,2), if a carry is generated, there is an overflow and more
bits are needed to represent the result (ex. 4,5).
3. When subtracting, the answer is correct only when a carry bit is
generated (ex. 7,12). If the carry bit is cleared, the subtrahend is
greater than the minuend which generates a negative result, which is
invalid in an unsigned system (ex. 8,10).
Signed 2's Complement Binary Numbers.
1. Positive numbers are identified by a most significant bit equal to
0, while negative numbers have most significant bits (MSBs) equal
to 1 and with the negative quantity expressed in its 2's
complement. The carry bit has no meaning in a signed 2's
complement system.
2. When adding, the answer is always correct if the MSBs of the added
numbers are different (ex. 3,4). "Adding a positive and negative
number can never produce an overflow." If the MSBs are the same
and the MSB of the result is different, an overflow has occurred
(ex. 2,5). "Adding two positive numbers can not produce a negative
result or adding two negative numbers can not produce a positive."
3. When subtracting, the answer is always correct if the MSBs of the
subtracted numbers are the same (ex. 8,12). "Subtracting numbers
of the same sign can never produce an overflow." If the MSBs are
different, and the MSB of the result is different than the
minuend, an overflow has occurred (ex. 9,10).
"Subtracting a
number of different sign from the minuend is like adding a number
of the same sign to the minuend, the result must have the same
sign as the minuend."