Two's Complement

The two's complement of an N-bit number is defined as the complement with respect to 2^N; in other words, it is the result of subtracting the number from 2^N

Three-bit two's-complement integers

Bits	Unsigned value	Two's complement value
011	3	3
010	2	2
001	1	1
000	0	0
111	7	-1
110	6	-2
101	5	-3
100	4	-4

Eight-bit two's-complement integers

Bits	Unsigned value	Two's complement value
0111 1111	127	127
0111 1110	126	126
0000 0010	2	2
0000 0001	1	1
0000 0000	0	0
1111 1111	255	−1
1111 1110	254	−2
1000 0010	130	−126
1000 0001	129	−127
1000 0000	128	−128

The term two's complement can mean either a number format or a mathematical operator. For example, 0111 represents decimal 7 in two's-complement notation, but the two's complement of 7 in a 4-bit register is actually the "1001" bit string (the same as represents 9 = 24 − 7 in unsigned arithmetics) which is the two's complement representation of −7. The statement "convert x to two's complement" may be ambiguous, since it could describe either the process of representing x in two's-complement notation without changing its value, or the calculation of the two's complement, which is the arithmetic negative of x if two's complement representation is used.

Converting to two's complement representation

Invert all bits and add 1.

Example

1.
+ 15   ->  0000 1111
+  5   ->   0000 0101
------       --------------------
  20          0001 0100

2.

+ 15  ->   0000 1111       ->    0000 1111
-  5    -> -  0000 0101  2's ->   1111 1011 
                                              --------------------
                                             1 0000 1010  -> 10 , ignore Carry

3.
- 15   ->   -  0000 1111  2's  ->  1111 0001
+  5  ->      0000 0101         ->  0000 0101
                                                -------------------
                                                 1111 0110   ->  sign bit is 1 , take 2's complement
                  2's ->   0000 1010   -> -10 (Answer)

4.
- 15  ->   - 0000 1111   2's ->  1111 0001
-  5   ->   - 0000 0101   2's ->  1111 1011
                                               --------------------
                                             1 1110 1100  -> Ignore Carry ,Sign bit is 1 , take 2's complement
             1110 1100   -> 2's ->  0001 0100  ->  -20 (Answer)       

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