Explain the addition of 1010 to 1101 using full adder step by step.

Q.) Explain the addition of 1010 to 1101 using full adder step by step.

 Subject: Digital Systems

A full adder is a digital circuit that performs addition of numbers. In a computer, it is used in the Arithmetic Logic Unit (ALU) to calculate the sum of two binary numbers and a carry from the previous stage.

The full adder has three inputs: A, B, and C_in (carry input), and two outputs: S (sum) and C_out (carry output). The operation of the full adder can be defined with the following truth table:

A B C_in S C_out
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1

The sum (S) can be calculated as:

S = A XOR B XOR C_in

The carry out (C_out) can be calculated as:

C_out = (A AND B) OR (C_in AND (A XOR B))

Now, let's add the binary numbers 1010 and 1101 using a full adder:

  1. Step 1: Add the least significant bits (0 from 1010 and 1 from 1101) and an initial carry in of 0.

    S = 0 XOR 1 XOR 0 = 1

    C_out = (0 AND 1) OR (0 AND (0 XOR 1)) = 0

  2. Step 2: Add the next bits (1 from 1010 and 0 from 1101) and the carry from the previous step.

    S = 1 XOR 0 XOR 0 = 1

    C_out = (1 AND 0) OR (0 AND (1 XOR 0)) = 0

  3. Step 3: Add the next bits (0 from 1010 and 1 from 1101) and the carry from the previous step.

    S = 0 XOR 1 XOR 0 = 1

    C_out = (0 AND 1) OR (0 AND (0 XOR 1)) = 0

  4. Step 4: Add the most significant bits (1 from 1010 and 1 from 1101) and the carry from the previous step.

    S = 1 XOR 1 XOR 0 = 0

    C_out = (1 AND 1) OR (0 AND (1 XOR 1)) = 1

So, the sum of 1010 and 1101 is 0111 and there is a carry out of 1. Therefore, the final result is 10111.