3. Given the decimal numbers: A = 49 B = - 77 and C = 83 Using their respective 2 complement, determine: (4 Pts.) a) The difference: B - A
b) The sum: C + B
a) To find the difference B - A, first we need to find the 2's complement of A:
A = 49 (decimal) = 00110001 (binary)
2's complement of A: 11001111
Now, we can subtract B from 2's complement of A:
B = -77 (decimal) = 10110011 (2's complement)
2's complement of A - B:
11001111
+ 10110011
-----------
101000110
Since the result is greater than 2's complement of 128 (10000000), this means the result is negative. To find the decimal value of the result, we need to find the 2's complement:
2's complement of 01011110: 10100010
Therefore, B - A = -82 (decimal)
b) To find the sum C + B, first we need to find the 2's complement of B:
B = -77 (decimal) = 10110011 (2's complement)
Now, we can add C and 2's complement of B:
C = 83 (decimal) = 01010011 (binary)
2's complement of B: 01001101
C + 2's complement of B:
01010011
+ 01001101
-----------
10111100
Since the result is greater than 2's complement of 128 (10000000), this means the result is negative. To find the decimal value of the result, we need to find the 2's complement:
2's complement of 01000100: 10111100
Therefore, C + B = -94 (decimal)
To calculate the difference B - A using their respective 2's complement representation, follow these steps:
Step 1: Convert the decimal numbers to their 2's complement representation:
A = 0011 0001
B = 1011 1011
Step 2: Take the 2's complement of A:
Invert all bits: 1100 1110
Add 1: 1100 1111
Step 3: Perform binary subtraction of B - A:
1011 1011 (B)
- 1100 1111 (2's complement of A)
--------------
11010 010 (Carry over the leftmost bit)
- 1 (Ignore the carry because it is outside the 8-bit representation)
Step 4: Convert the result back to decimal:
010 = 0010 (ignoring the carry)
The result is 2.
Therefore, the difference B - A is 2.
To calculate the sum C + B using their respective 2's complement representation, follow these steps:
Step 1: Convert the decimal numbers to their 2's complement representation:
C = 0101 0011
B = 1011 1011
Step 2: Perform binary addition of C + B:
0101 0011 (C)
+ 1011 1011 (B)
---------------
10000 1110 (Carry over the leftmost bit)
Step 3: Convert the result back to decimal:
1110 = 0000 1110 (ignoring the carry)
The result is 14.
Therefore, the sum C + B is 14.