Editing Two's complement (section)

===From the ones' complement===
To get the two's complement of a negative binary number, all [[bit]]s are inverted, or "flipped", by using the [[bitwise NOT]] operation; the value of 1 is then added to the resulting value, ignoring the overflow which occurs when taking the two's complement of 0.

For example, using 1 byte (=8 bits), the decimal number 5 is represented by
{{block indent|0000 0101<sub>2</sub>}}
The most significant bit (the leftmost bit in this case) is 0, so the pattern represents a non-negative value. To convert to −5 in two's-complement notation, first, all bits are inverted, that is: 0 becomes 1 and 1 becomes 0:
{{block indent|1111 1010<sub>2</sub>}}
At this point, the representation is the [[ones' complement]] of the decimal value −5. To obtain the two's complement, 1 is added to the result, giving:
{{block indent|1111 1011<sub>2</sub>}}
The result is a signed binary number representing the decimal value −5 in two's-complement form. The most significant bit is 1, signifying that the value represented is negative.

Alternatively, instead of adding 1 after inverting a positive binary number, 1 can be subtracted from the number ''before'' it is inverted.  The two methods can easily be shown to be equivalent.  The inversion (ones' complement) of <math>x</math> equals <math>(2^N - 1) - x</math>, so the sum of the inversion and 1 equals <math>(2^N - 1) - x + 1 = </math><math>2^N - x - 1 + 1 = </math><math>2^N - x</math>, which equals the two's complement of <math>x</math> as expected.  The inversion of <math>x - 1</math> equals <math>(2^N - 1) - (x - 1) = </math><math>(2^N - 1) - x + 1 = </math><math>2^N - x</math>, identical to the previous equation.  Essentially, the subtraction inherent in the inversion operation changes the −1 added to <math>x</math> before the inversion into +1 added after the inversion.  This alternate subtract-and-invert algorithm to form a two's complement can sometimes be advantageous in computer programming or hardware design, for example where the subtraction of 1 can be obtained for free by incorporating it into an earlier operation.<ref>... e.g. by reducing an added constant by 1, increasing a subtracted constant by 1, or setting the carry/borrow flag before a subtract-with-borrow operation.  For example, to compute <math>-(m + 4)</math>, instead of adding 4 to <math>m</math>, inverting the result, and then adding 1, one can merely add 3 (= 4 − 1) to <math>m</math> and then invert the result.  (Of course, it is also an option, using the invert-and-add scheme, to invert <math>m</math> first and then subtract 3 [equivalent to adding −3 = −4 + 1].)</ref>

The two's complement of a negative number is the corresponding positive value, except in the special case of the [[most negative number]]. For example, inverting the bits of −5 (above) gives:
{{block indent|0000 0100<sub>2</sub>}}
And adding one gives the final value:
{{block indent|0000 0101<sub>2</sub>}}
The two's complement of the most negative number representable (e.g. a one as the most-significant bit and all other bits zero) is itself. Hence, there is an 'extra' negative number for which two's complement does not give the negation, see {{section link||Most negative number}} below.

The case for the most negative number is one of only two special cases.  The other special case is for zero, the two's complement of which is zero: inverting gives all ones, and adding one changes the ones back to zeros (since the overflow is ignored).  Mathematically, in the two's complement system of signed integers (which represents the negative of each number as its two's complement), this is obviously correct: the negative of 0 is in fact 0 (<math>-0 = 0</math>).  This zero case also makes sense by the definition of two's complements: by that definition, the two's complement of zero would be <math>2^N - 0 = 2^N</math>, but in <math>N</math> bits, all values are taken modulo <math>2^N</math>, and <math>2^N</math>{{math|mod }}<math>2^N = 0</math>.  In other words, the two's complement of 0 in <math>N</math> bits is (by definition) a single 1 bit followed by <math>N</math> zeros, but the 1 gets truncated, leaving 0.<ref>Zero is the one value which when added to its two's complement, using machine (modular) binary arithmetic, does not sum to <math>2^N</math>, but observe that in arithmetic modulo <math>2^N</math>, <math>0</math> is congruent to <math>2^N</math>.</ref>

In summary, the two's complement of any number, either positive, negative, or zero, can be computed in the same ways.  In two's complement signed integer representation, the two's complement of any integer is equal to -1 times that integer, except for the most negative integer representable in the given number of bits <math>N</math>, i.e. the integer <math>-2^{N-1}</math>, the two's complement of which is itself (still negative).