Question

Perform the following operations involving elght-bit 2 s complement numbers and indicate whether arithmetic overflow occurs. Check your answers by converting to decimal representation.

(a) 00110110+01000101
(b) 00110110−00101011
(c) 01110101−11010110

Answer 1

a) 00110110 + 01000101 = 01111011 (No overflow)

b) 00110110 - 00101011 = 00001001 (No overflow)

c) 01110101 - 11010110 = 10111111 (Overflow occurs)

Step-by-step explanation:

To perform the operations involving eight-bit 2's complement numbers, addition, and subtraction are executed as follows:

a) For 00110110 + 01000101, performing addition yields 01111011. There is no overflow as the addition of these numbers within the range of eight-bit 2's complement numbers results in a valid representation.

b) Subtracting 00101011 from 00110110 gives 00001001 for 00110110 - 00101011. Again, no overflow occurs as the result falls within the valid range of eight-bit 2's complement numbers.

c) However, for 01110101 - 11010110, subtracting 11010110 from 01110101 results in 10111111. Here, an overflow occurs because the result exceeds the maximum positive value that can be represented by an eight-bit 2's complement number. This is evident from the most significant bit being 1, indicating a negative number, which exceeds the range of an eight-bit representation.

These calculations ensure that the operations are within the constraints of eight-bit 2's complement representation, with overflow identified when the result falls outside this range.

Answer 2

(a) 10111011; No overflow occurred.

(b) 00001101; No overflow occurred.

(c) 01000011; Overflow occurred.

Step-by-step explanation:

In 2's complement addition, we add the numbers bit by bit, carrying over any overflow from the previous bit. For (a), 00110110 + 01000101, starting from the least significant bit (LSB), there's no overflow, resulting in 10111011 in binary, which is -69 in decimal. For (b), 00110110 - 00101011, there's no overflow, giving 00001101 in binary, which is 13 in decimal.

However, for (c), 01110101 - 11010110, when performing subtraction, it's important to note that subtraction is equivalent to addition of the 2's complement of the second number. In this case, the 2's complement of 11010110 is 00101010. Adding 01110101 + 00101010, an overflow occurs in the MSB (most significant bit), resulting in 01000011 in binary, which represents -61 in decimal.