General Steps for Addition/Subtraction in 2’s Complement Form
The signed 2’s complement form is the standard method for handling signed binary arithmetic, enabling seamless addition and subtraction of positive and negative numbers. By converting subtraction into addition, 2’s complement simplifies the process and ensures accurate results. Below are the general steps for performing addition or subtraction in 2’s complement form:
- Express the Operation: Rewrite the operation as an addition problem. For example, to compute \(\mathsf{A − B}\), express it as \(\mathsf{A + (−B)}\). Similarly, for \(\mathsf{−A − B}\), rewrite it as \(\mathsf{(−A) + (−B)}\).
- Perform Addition: Convert both numbers to their 2’s complement form and add them using standard binary addition.
- If there is an end-carry, discard it. The remaining digits represent the result, which may be positive or negative.
- If there is no end-carry, the result is the direct output of the addition, also potentially positive or negative.
This method leverages the 2’s complement system’s ability to handle signed numbers efficiently, making arithmetic operations consistent and reliable.
Converting Between Decimal and 2’s Complement Form
When working with decimal inputs and expecting decimal outputs, additional steps are needed to convert numbers to and from 2’s complement form. The number of bits used for representation is critical to avoid overflow and ensure accurate results.
Determining the Number of Bits
If the problem specifies the number of bits (e.g., 6 bits), use that as the basis for representation. If no bit count is provided, determine the number of bits required as follows:
- Identify the largest number (ignoring its sign) in the problem.
- Convert this number’s magnitude to binary to find the minimum number of bits (\(\mathsf{n}\)) needed to represent it.
- Add one bit for the sign and an additional bit to accommodate potential overflow, resulting in \(\mathsf{n+2}\) bits. This ensures the sum of two n-bit numbers, which may require up to \(\mathsf{n+1}\) bits, fits within the representation without overflow issues.
Converting Decimal to 2’s Complement
To convert a decimal number to its 2’s complement form:
- Positive Numbers: Convert the magnitude to binary using the allocated magnitude bits and prefix a 0 for the sign bit.
- Negative Numbers: Convert the positive magnitude to binary, take its 1’s complement (invert all bits), and add 1 to get the 2’s complement representation. Prefix a 1 for the sign bit.
Converting 2’s Complement to Decimal
After performing the addition, the result is in 2’s complement form. To convert it back to decimal:
- Check the Sign Bit: If the MSB is 0, the number is positive; convert the remaining bits directly to decimal and prefix a \(\mathsf{+}\) sign.
- Negative Numbers: If the MSB is 1, the number is negative. To convert, take the 2’s complement of the result (invert all bits and add 1), convert the resulting positive magnitude to decimal, and prefix a \(\mathsf{-}\) sign.
- Alternative Method: Use a modified binary-to-decimal conversion by assigning a weight of \(\mathsf{-2^{n-1}}\) to the MSB (instead of \(\mathsf{+2^{n-1}}\)) and standard weights (\(\mathsf{2^{n-2}, 2^{n-3},}\) etc.) to the remaining bits. Sum these values to obtain the decimal result.
This approach ensures accurate conversion between decimal and 2’s complement forms, accounting for the sign bit and preventing overflow errors.
Example 1: Perform 3-4 using 6-bit signed 2’s complement method
Example 2: Perform 12+2 using 6-bit signed 2’s complement method
Example 3: Perform -5-6 using 6-bit signed 2’s complement method
Example 4: Perform -8+15 using 6-bit signed 2’s complement method
Example 5: Perform 15-8 using 6-bit signed 2’s complement method
