Final answer:
P represents -1775 in decimal. The minimum value of Q to not cause overflow is -273, or 0xFEF in hexadecimal. Calculating P - Q with Q as 0xBAD results in a final value of -686.
Step-by-step explanation:
The student's question involves calculations with signed integers in hexadecimal and 2's complement using a 12-bit old computer system:in decimal: To find what P (0x911) represents in the decimal system, consider that in a 12-bit 2's complement system, the highest bit is the sign bit. Since the highest bit of 0x911 (1001 0001 0001 in binary) is '1', it is a negative number. To find its decimal value, first find the 2's complement which is the inversion of all bits plus one. The result is 0110 1110 1111, which is the positive binary equivalent of 1775 in decimal. Since it was originally negative, P represents -1775 in the decimal systemMinimum value of Q to prevent overflow: To prevent overflow when adding P and Q, the minimum value of Q must be such that the sum does not exceed the range of a 12-bit signed integer. The range is from -2048 to 2047.
Since the maximum positive value that can be added to P without causing an overflow is 2047 minus the absolute value of P, which is 273, the minimum value of Q to avoid overflow is -273. In hexadecimal, Q is represented as 0xFEF, which corresponds to -273 in decimalP - Q calculation: To calculate P - Q when Q = 0xBAD, first find the 2's complement of Q to turn the subtraction into addition. 2's complement of 0xBAD (1011 1010 1101) is 0100 0101 0011 (0x453). Now, add P and the 2's complement of Q. 0x911 + 0x453 = 1001 0001 0001 + 0100 0101 0011 = 1101 0101 0010. Since the resulting binary number has a '1' as the most significant bit, it is negative in a 12-bit 2's complement system. Taking the 2's complement of the result gives us 0010 1010 1110, which translates to 686 in decimal, making the final result -686.