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The ones' complement of a binary number is defined as the value obtained by inverting all the bits in the binary representation of the number swapping 0s for 1s and vice versa. The ones' complement of the number then behaves like the negative of the original number in some arithmetic operations. A ones' complement system or ones' complement arithmetic is a system in which negative numbers are represented by the inverse of the binary representations of their corresponding positive numbers.

In such a system, a number is negated converted from positive to negative or vice versa by computing its ones' complement. The ones' complement binary numeral system is characterized by the bit complement of any integer value being the arithmetic negative of the value. That is, inverting all of the bits of a number the logical complement produces the same result as subtracting the value from 0.

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. Adding two values is straightforward. Simply align the values on the least significant bit and add, propagating any carry to the bit one position left.

If the carry extends past the end of the word it is said to have "wrapped around", a condition called an " end-around carry ". When this occurs, the bit must be added back in at the right-most bit.

This phenomenon does not occur in two's complement arithmetic. Subtraction is similar, except that borrows, rather than carries, are propagated to the left. If the borrow extends past the end of the word it is said to have "wrapped around", a condition called an " end-around borrow ". When this occurs, the bit must be subtracted from the right-most bit. It is easy to demonstrate that the bit complement of a positive value is the negative magnitude of the positive value.

Negative zero is the condition where all bits in a signed word are 1. This follows the ones' complement rules that a value is negative when the left-most bit is 1, and that a negative number is the bit complement of the number's magnitude. The value also behaves as zero when computing. Negative zero is easily produced in a 1's complement adder. Simply add the positive and negative of the same magnitude. Although the math always produces the correct results, a side effect of negative zero is that software must test for negative zero.

The generation of negative zero becomes a non-issue if addition is achieved with a complementing subtractor. The first operand is passed to the subtract unmodified, the second operand is complemented, and the subtraction generates the correct result, avoiding negative zero.

If the second operand is negative zero it is inverted and the original value of the first operand is the result. The result can be only 1 of two cases. In case 2, the subtraction will generate a value that is 1 larger than operand 1 and an end-around borrow. Completing the borrow generates the same value as operand 1.

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