Final answer:
In 2's complement binary addition, numbers are aligned by the least significant bit and added together. The answers are expressed in 8-bit binary and decimal form, noting that a leading 1 in the result signifies a negative number. Overflow in the final carry-bit is ignored at this fixed width.
Step-by-step explanation:
To add 2's complement binary numbers, we align them by the least significant bit and perform binary addition, carrying over any bits as we would in decimal addition. If the final carry-bit is a 1 in the case of signed numbers, it is typically discarded (i.e., an overflow has occurred), since we are representing these numbers with a fixed width of 8 bits. Here are the solutions to the provided problems:
- (a) To add '01' and '1011', we first express them in 8-bit 2's complement as '00000001' and '00001011'. Then we add them to get '00001100' (base 2) or '12' (base 10).
- (b) For '11' and '01010101', we express the first number as '00000011' and add them to get '01010110' (base 2) or '86' (base 10).
- (c) '0101' and '110' are expressed as '00000101' and '11111110' in 8-bit 2's complement (since it represents the negative number -2). Adding them yields '11111111' (base 2) or '-1' (base 10).
- (d) Adding '01' (00000001) to '10' (11111110) gives us '11111111' (base 2) or '-1' (base 10).
Remember that in 2's complement, a leading 1 represents a negative number and the value is obtained by inverting the bits and adding 1.