What happens to the carry-out bit when subtracting two n-bit numbers using 2's complement and the result is positive?

Explanation:

Correct Option - The carry-out bit is ignored

When subtracting two n-bit numbers using 2's complement and the result is positive, the carry-out bit does not have a significant role in determining the outcome of the subtraction. The carry-out bit is essentially ignored in this scenario. This is due to the way 2's complement arithmetic handles overflow and underflow conditions.

Understanding 2's Complement Representation:

2's complement representation is a method used to encode signed integers in binary. In this representation, positive numbers are represented as usual in binary, while negative numbers are represented by taking the binary representation of the absolute value, inverting all the bits (creating the 1's complement), and then adding one to the least significant bit.

For example:

• +5 in 4-bit binary: 0101
• -5 in 4-bit 2's complement: 1011 (1's complement of 0101 is 1010, add 1 to get 1011)

Subtraction Using 2's Complement:

When performing subtraction using 2's complement, we add the 2's complement of the number to be subtracted. For instance, to compute A - B, we actually calculate A + (-B). The process involves the following steps:

• Compute the 2's complement of B (which gives -B).
• Add this value to A.
• Check the result for any overflow or underflow conditions.

Carry-out Bit:

The carry-out bit is a bit that is carried out of the most significant bit position when performing binary addition or subtraction. In the context of 2's complement subtraction, the carry-out bit can indicate whether an overflow has occurred, but it does not affect the correctness of the result itself. If the result of the subtraction is positive, it means that the result fits within the range of representable values for the given bit-width, and thus, the carry-out bit is ignored.

Analysis of Other Options:

Option 1: The carry-out bit indicates an overflow

This option is incorrect because the carry-out bit alone does not indicate an overflow in the context of 2's complement arithmetic. Overflow is determined by examining the sign bits of the operands and the result, not just the carry-out bit. For example, if adding two positive numbers results in a negative number, or adding two negative numbers results in a positive number, an overflow has occurred.

Option 2: The carry-out bit is added to the result

This option is incorrect because the carry-out bit is not added to the result in 2's complement arithmetic. The carry-out bit is discarded, and the result is taken as-is unless overflow has occurred, which needs to be handled separately.

Option 3: The carry-out bit is inverted

This option is incorrect because inverting the carry-out bit has no meaningful impact on the result of the 2's complement subtraction. The carry-out bit does not affect the final result and is simply ignored if the result is within the valid range.

Option 4: The carry-out bit is ignored

This is the correct option, as explained above. The carry-out bit does not affect the result of the subtraction if the result is positive and within the representable range of the bit-width.

Important Information:

Understanding the carry-out bit and its role in binary arithmetic is crucial for anyone working with digital systems and computer architecture. The carry-out bit can indicate different conditions depending on the context, such as overflow or borrow in subtraction, but in the case of 2's complement arithmetic, it is ignored if the result is positive and valid.

Additionally, understanding the principles of 2's complement representation and arithmetic is foundational for designing and analyzing algorithms that involve signed integer calculations. This knowledge is essential for fields like embedded systems, computer engineering, and digital signal processing, where precise manipulation of binary data is required.