Binary Subtraction Using First Complement

This solution details the process of subtracting the binary number (1010)_2 from (1101)_2 using the first complement (also known as one's complement) method. This technique is fundamental in understanding digital arithmetic.

First Complement Method Explained

The first complement method is a way computers perform subtraction. Instead of direct subtraction, it uses addition with the complement of the subtrahend. The process involves these key steps: finding the first complement of the number being subtracted (subtrahend), adding this complement to the other number (minuend), and then interpreting the result based on whether a carry-out occurs.

Binary Subtraction Steps

1. Step 1: Identify Minuend and Subtrahend
   • Minuend: The number from which another is to be subtracted. Here, it is (1101)_2.
   • Subtrahend: The number to be subtracted. Here, it is (1010)_2.
2. Step 2: Find the First Complement of the Subtrahend

   The first complement (one's complement) is found by changing every 0 to a 1 and every 1 to a 0 in the binary number. For the subtrahend (1010)_2:

   First Complement of (1010)_2 = (0101)_2.

3. Step 3: Add Minuend and First Complement

   Next, add the minuend (1101)_2 to the first complement of the subtrahend, which is (0101)_2. Ensure both numbers have the same number of bits for addition; in this case, both are 4 bits.

   The addition is performed as follows:

   \[ \begin{array}{@{}c@{\,}c@{}c@{}c@{}c} & 1 & 1 & 0 & 1_2 \quad \text{(Minuend)} \\ + & 0 & 1 & 0 & 1_2 \quad \text{(1's Complement of Subtrahend)} \\ \hline & 10 & 0 & 1 & 0_2 \\ \hline \end{array} \]

   The sum obtained is \( (10010)_2 \).

4. Step 4: Interpret the Result with Carry

   When performing subtraction using the first complement method, if the addition results in a carry-out (an extra bit to the left of the most significant bit), it signifies a positive result. In our sum \( (10010)_2 \), the leftmost '1' is the carry-out.

   To get the final answer, this carry-out bit is added to the least significant bit (the rightmost bit) of the sum. This is often called the 'end-around carry'.

   Adding the carry:

   \[ \begin{array}{@{}c@{\,}c@{}c@{}c@{}c} & 0 & 0 & 1 & 0_2 \quad \text{(Lower 4 bits of sum)} \\ + & & & & 1_2 \quad \text{(Carry)} \\ \hline & 0 & 0 & 1 & 1_2 \\ \hline \end{array} \]

   The final result after adding the carry is \( (0011)_2 \).

Result of Binary Subtraction

The subtraction of (1010)_2 from (1101)_2 using the first complement method results in (0011)_2.