Question

Assume that a mad scientist has created a computer that has 9 bit registers. The most significant bit is the sign bit. He wants to execute the following operation using 9 bit register.

-256-2

Use 2's complement method (in binary) to find the result of the above operation in binary system. Show the computations in the answer.

Answer 1

Binary:     011111110
Decimal:    254
overflow happened.
 
Explanation:
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Number: -256
Let's convert this to two's complement binary
-256
This is negative. so, follow these steps to convert this into a 2's complement binary
Step 1:
Divide 256 successively by 2 until the quotient is 0
   > 256/2 = 128, remainder is 0
   > 128/2 = 64, remainder is 0
   > 64/2 = 32, remainder is 0
   > 32/2 = 16, remainder is 0
   > 16/2 = 8, remainder is 0
   > 8/2 = 4, remainder is 0
   > 4/2 = 2, remainder is 0
   > 2/2 = 1, remainder is 0
   > 1/2 = 0, remainder is 1
Read remainders from the bottom to top as 100000000
So, 256 of decimal is 100000000 in binary
So, 256 in normal binary is 100000000
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   100000000 is flipped to 011111111
Step 3:. Add 1 to above result
011111111 + 1 = 100000000
so, -256 in 2's complement binary is 100000000

Number: -2
Let's convert this to two's complement binary
-2
This is negative. so, follow these steps to convert this into a 2's complement binary
Step 1:
Divide 2 successively by 2 until the quotient is 0
   > 2/2 = 1, remainder is 0
   > 1/2 = 0, remainder is 1
Read remainders from the bottom to top as 10
So, 2 of decimal is 10 in binary
So, 2 in normal binary is 000000010
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   000000010 is flipped to 111111101
Step 3:. Add 1 to above result
111111101 + 1 = 111111110
so, -2 in 2's complement binary is 111111110

Adding 100000000 and 111111110 in binary
    100000000
    111111110
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 (1)011111110
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Sum produces a carry of 1. We can ignore that carry.
So, sum of these numbers in binary is 011111110

Verification:
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sum = 011111110
since left most bit is 0, this number is positive
so, we can directly convert this into a decimal value
Converting 11111110 to decimal
11111110
=> 1x2^7+1x2^6+1x2^5+1x2^4+1x2^3+1x2^2+1x2^1+0x2^0
=> 1x128+1x64+1x32+1x16+1x8+1x4+1x2+0x1
=> 128+64+32+16+8+4+2+0
=> 254
Answer: 254
-256+-2 must be -258
This is not correct since we can verify that -256+-2 not equals 254
So, there was an overflow.