Binary Addition and Subtraction Explained with Examples (1’s & 2’s Complement Method)

Binary Addition and Subtraction are fundamental arithmetic operations used in digital electronics and computer systems. Since computers understand only binary numbers (0 and 1), all calculations are performed using binary arithmetic.

What is Binary Addition and Subtraction?

If a computer works with 5-bit numbers such as -1101, where the first bit represents the sign and the remaining bits represent magnitude, then:

• 11101 represents -1101 (1 indicates negative sign)
• 01101 represents +1101 (0 indicates positive sign)

Negative binary numbers can also be represented using:

• 1’s Complement Method
• 2’s Complement Method

These methods allow computers to perform subtraction using only the addition process, which simplifies hardware design.

Binary Addition

Binary addition is similar to decimal addition, but it uses only two digits: 0 and 1.

Binary Addition Rules

Important: Whenever the sum exceeds 1, a carry is generated.

Example: Binary Addition

   11011  (27)
+  10101  (21)
------------
  110000  (48)

Step-by-Step Explanation:

• 1 + 1 = 0, Carry 1
• 1 + 1 + 0 = 0, Carry 1
• 1 + 0 + 1 = 0, Carry 1
• 1 + 1 + 0 = 0, Carry 1
• 1 + 1 + 1 = 1, Carry 1

Final Result: 111000

Binary Subtraction

Binary subtraction can be performed using two methods:

1. Direct Subtraction Method
2. 2’s Complement Method

Method 1: Direct Binary Subtraction

Binary Subtraction Rules

Example:

   1101101
-  0011011
------------
   1010010

Subtraction is performed from right to left. Borrow is taken whenever the top digit is smaller than the bottom digit.

Final Result: 1010010

Method 2: Two’s Complement Method

This method is widely used in computer systems.

Steps to Find 2’s Complement:

1. Make both numbers equal in digits.
2. Find 1’s complement (change 0 to 1 and 1 to 0).
3. Add 1 to the 1’s complement to get 2’s complement.

Example:

Subtrahend: 0011011

Step 1: 1’s Complement

0011011 → 1100100

Step 2: Add 1

 1100100
+0000001
---------
 1100101

Step 3: Add to Minuend

 1101101
+1100101
---------
1 1010010

Discard the extra Most Significant Bit (MSB).

Final Result: 1010010

Conclusion

Binary addition and subtraction are essential concepts in digital electronics and computer architecture. While binary addition follows simple carry rules, binary subtraction can be performed either directly or by using the 2’s complement method. In modern computer systems, the 2’s complement method is preferred because it simplifies hardware design and calculations. Understanding these concepts is crucial for students of computer science, electronics, and embedded systems.