01’s Complement Form
Definition:
A system used in some
computers to represent negative numbers. To negate a number, each bit of the
number is inverted (zeros are replaced with ones and vice versa). This has the consequence
that there are two representations for zero, either all zeros or all ones.
Number representation
Positive
numbers are the same simple, binary system used by two's complement and
sign-magnitude. Negative values are the bit complement of the corresponding
positive value. The largest positive value is characterized by the sign
(high-order) bit being off (0) and all other bits being on (1). The smallest
negative value is characterized by the sign bit being 1, and all other bits
being 0. The table below shows all possible values in a 4-bit system, from −7
to +7.
+ -
0
0000 1111 —Note that +0 and −0 return TRUE when tested
for
zero , FALSE when tested for non-zero.
1
0001 1110
2
0010 1101
3
0011 1100
4
0100 1011
5
0101 1010
6
0110 1001
7
0111 1000
Sign-Magnitude
Form
Another method of representing negative numbers is sign-magnitude.
Sign-magnitude representation also uses the most significant bit of the number
to indicate the sign. A negative number is the 7-bit binary representation of
the positive number with the most significant bit set to one. The drawbacks to
using this method for arithmetic computation are that a different set of rules
are required and that zero can have two representations (+0, 0000
0000 and -0, 1000 0000).
For example:
1.
In this form, a
negative number is the 1’s complement of the corresponding positive number
2.
The 1’s complement of a binary number is found by
changing all 1s to 0s and all 0s to 1s as shown below:
+2510
= 000110012
-2510=
111001102(1’s complement of +25)
Definition
Property
Two's
complement representation allows the use of binary arithmetic operations on
signed integers, yielding the correct 2's complement results.
Positive Numbers
Positive 2's
complement numbers are represented as the simple binary.
Negative Numbers
Negative 2's
complement numbers are represented as the binary number that when added to a
positive number of the same magnitude equals zero.
2’s
complement = (1’s complement) + 1
The most significant
(leftmost) bit indicates the sign of the integer; therefore it is sometimes called the sign
bit.
If the sign bit is zero,
then the number is greater
than or equal to zero, or positive.
If the sign bit is one,
then the number is less
than zero, or negative.
Calculation of 2's Complement
To calculate the 2's
complement of an integer, invert the binary equivalent of the number by
changing all of the ones to zeroes and all of the zeroes to ones (also called 1's complement), and then add
one.
For example,
0001 0001(binary 17)
 1110 1111(two's complement -17) |
||
NOT(0001 0001)
|
=
|
1110 1110 (Invert bits)
|
1110 1110 + 0000 0001
|
=
|
1110 1111 (Add 1)
|
FOR EXAMPLE:
Express the
decimal number -55 as an 8-bit in the sign-magnitude, 1’scomplement, and 2’s
complement forms.
SOLUTION:
8-bit number for + 5510 =
001101112
Sign-magnitude form for -5510
= 101101112
Change the
sign bit to a 1 and leave the magnitude bits as they are 1’s complement form
for -5510 = 110010002
Take the 1’s
complement of +55 by changing all 1’s to 0s and 0s to 1s
2’s complement form for -5510 =
11001000 1 's complement
+ 1
11001001 2 's complement
Two's Complement Addition
1. 47 + 23
Since
47 and 23 are not negative integers, we don't have to use two's complement for
addition.
47
= (32 + 8 + 4 + 2 + 1) = 101111
23
= (16 + 4 + 2 + 1) = 10111
1 1 1 1 1 carry
row
1 0 1 1 1 1
+ 1 0 1 1 1
1 0 0 0 1 1 0
1 0 1 1 1 1
+ 1 0 1 1 1
1 0 0 0 1 1 0
1000110
= 70
Notice that
no sign bit is added in this instance, nor is the leftmost bit to be interpreted
as a negative.
2. 72 + (-100)
72
= 01001000
100
= 01100100, and the two's complement of 01100100 = 10011100.
1 1 carry row
0 1 0 0 1 0 0 0
+ 1 0 0 1 1 1 0 0
1 1 1 0 0 1 0 0
0 1 0 0 1 0 0 0
+ 1 0 0 1 1 1 0 0
1 1 1 0 0 1 0 0
There
is no overflow, and the leftmost digit is a 1, indicating that the result is a
negative number. First use 2's complement to convert 11100100 to
00011011, then convert 00011011 to decimal representation and take the
negative. The answer is -28.
4. (-35) + (-58)
-35
= -00100011 = 11011101 (two's complement of 35)
-58
= -00111010 = 11000110 (two's complement of 58)
1 1 1 1
carry row
1 1 0 1 1 1 0 1
+ 1 1 0 0 0 1 1 0
1 1 0 1 0 0 0 1 1
1 1 0 1 1 1 0 1
+ 1 1 0 0 0 1 1 0
1 1 0 1 0 0 0 1 1
After
truncating the overflow 1, we get the negative binary number 10100011.
Converting 10100011 to decimal representation by two's complement gives us a
result of -93. This last problem illustrates that, if you had decided to
use only 7 bits to represent your numbers, you would have come up with an
incorrect, positive number. For now, when adding 2 negative number , make
sure that the "carries" don't carry the negative sign bit out -
compensate by adding an extra bit when making calculations
Two's
complement subtraction is the binary addition of the minuend to the 2's
complement of the subtrahend (adding a negative number is the same as subtracting
a positive one).
For example,
7
- 12 = (-5)
|
0000 0111
|
=
|
+7
|
|
+ 1111 0100
|
=
|
-12
|
||
1111 1011
|
=
|
-5
|
Two's
complement multiplication follows the same rules as binary multiplication.
For example,
(-4)
× 4 = (-16)
|
1111 1100
|
=
|
-4
|
|
× 0000 0100
|
=
|
+4
|
||
1111 0000
|
=
|
-16
|
Two's
complement division is repeated 2's complement subtraction. The
2's complement of the divisor is calculated, then added to the dividend. For
the next subtraction cycle, the quotient replaces the dividend. This repeats
until the quotient is too small for subtraction or is zero, then it becomes the
remainder. The final answer is the total of subtraction cycles plus the
remainder.
For example,
7
÷ 3 = 2 remainder1
|
00000111
|
=
|
+7
|
00000100
|
=
|
+4
|
||
11111101
|
=
|
-3
|
11111101
|
=
|
-3
|
|||
00000100
|
=
|
+4
|
0000 0001
|
=
|
+1
(remainder)
written by TEH WEI HAN |
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