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

Concept: Computers perform subtraction using the concept of complements. Instead of designing separate hardware for subtraction, digital systems convert subtraction into addition by using the 2's complement method. For two binary numbers \(X\) and \(Y\), the subtraction \[ X - Y \] can be performed as \[ X + (\text{2's complement of }Y) \] This technique simplifies arithmetic circuitry and is widely used in processors and digital computers.

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: \[ \boxed{ X-Y = X + (\text{2's complement of }Y) } \] Therefore, \[ \boxed{\text{Option (A)}} \] is the correct answer.