Question

Which of the following definition is true for 2's complement subtract operation?

• A. To subtract two numbers X and Y, form the 2's complement of Y and then add it to X.
• B. To subtract two numbers X and Y, form the 2's complement of X and then add it with Y.
• C. To subtract two numbers X and Y, form the 2's complement of Y and then subtract it from X.
• D. To subtract two numbers X and Y, form the 2's complement of X and then subtract Y from it.

Hint:

For binary subtraction: \[ X-Y \] always remember: \[ \text{Take 2's complement of the subtrahend }(Y) \] and then \[ \text{Add it to }X. \] This converts subtraction into addition, making digital hardware simpler and faster.

Answer 1

The Correct Option is A

Step 1: Understand what 2's complement means.
The 2's complement of a binary number is obtained in two steps:
• Find the 1's complement (change 0 to 1 and 1 to 0).
• Add 1 to the resulting binary number. For example, \[ 0101 \] 1's complement: \[ 1010 \] Adding 1: \[ 1011 \] Thus, \[ \text{2's complement of }0101 = 1011 \]

Step 2: Understand subtraction using 2's complement.
To compute \[ X - Y \] we do not directly subtract \(Y\) from \(X\). Instead, we:
• Find the 2's complement of \(Y\).
• Add it to \(X\). Mathematically, \[ X - Y = X + (\text{2's complement of }Y) \]

Step 3: Verify using a numerical example.
Suppose: \[ X = 9,\qquad Y = 5 \] Binary representations: \[ 9=(1001)_2 \] \[ 5=(0101)_2 \] The 2's complement of \(0101\) is: \[ 1011 \] Now add it to \(1001\): \[ \begin{array}{r} 1001 +1011 \hline 10100 \end{array} \] Discarding the overflow carry: \[ 0100 \] which equals \[ 4 \] and \[ 9-5=4 \] Hence the method works correctly.

Step 4: Analyze the given options.

• Option (A): Forms the 2's complement of \(Y\) and adds it to \(X\). \checkmark
• Option (B): Forms the 2's complement of \(X\). \(\times\)
• Option (C): Uses subtraction after taking 2's complement. \(\times\)
• Option (D): Forms the 2's complement of \(X\). \(\times\) Only option (A) matches the standard 2's complement subtraction rule.

Step 5: Write the final conclusion.
The correct definition of 2's complement subtraction is: \[ { X-Y = X + (\text{2's complement of }Y) } \] Therefore, \[ {\text{Option (A)}} \] is the correct answer.