Addition, subtraction, 2's compliments

Addition, Subtraction, and 2’s Complement in Digital Logic

In digital systems, arithmetic operations such as addition and subtraction are essential for performing calculations. These operations are typically carried out using binary numbers, which are represented by combinations of 0s and 1s. To handle negative numbers and subtraction, a system known as 2’s complement is commonly used.

1. Binary Addition

Binary addition is similar to decimal addition but is based on the binary number system (base 2). The rules for binary addition are:

• Carry is the extra bit that is carried over to the next higher bit.
• When adding 1 + 1, the result is 0 with a carry of 1, just like in decimal addition when 9 + 1 gives 10.

Example of Binary Addition:

Let's add two binary numbers: 1011 (11 in decimal) and 1101 (13 in decimal).

  1011   (11)
+ 1101   (13)
---------
 11000   (24)

• Starting from the rightmost bit, you add column by column.
• If there's a carry, you add it to the next higher bit.
• The final result is 11000, which is 24 in decimal.

2. Binary Subtraction

Binary subtraction is done using the concept of borrowing, just like in decimal subtraction. However, binary subtraction uses the rules of binary arithmetic to handle borrowing.

The subtraction rules for binary are:

Example of Binary Subtraction:

Let's subtract 1101 (13 in decimal) from 1011 (11 in decimal).

1. 1011 - 1101:
   To subtract, we need to borrow from the next higher bit.

     1011
   - 1101
   --------
     1110  (Negative result)
   

   But binary subtraction is more efficient when we use 2's complement to represent negative numbers. Let's now look at 2’s complement to handle negative numbers.

3. 2’s Complement Representation

In binary arithmetic, the 2’s complement system is widely used to represent negative numbers. It allows the subtraction of numbers by simply adding their 2’s complement, avoiding the need for explicit subtraction.

How to Find the 2’s Complement:

To find the 2's complement of a binary number:

1. Invert all the bits (1 becomes 0 and 0 becomes 1).
2. Add 1 to the result.

Example: Finding 2’s Complement

Let's find the 2's complement of 1011 (11 in decimal).

1. Invert all bits of 1011:

   • 1011 becomes 0100.

2. Add 1 to 0100:

   • 0100 + 1 = 0101, which is 5 in decimal.

Thus, the 2's complement of 1011 is 0101. The result represents -5 in decimal.

Addition Using 2’s Complement

Subtraction can be easily done by converting the subtrahend (the number to be subtracted) to its 2’s complement and then adding it to the minuend (the number from which subtraction is being done).

Example: 2’s Complement Subtraction

Suppose you want to subtract 6 from 11 (i.e., 11 - 6).

1. Convert 6 to binary: 6 in binary is 0110.

2. Find the 2's complement of 6:

   • Invert 0110: 1001
   • Add 1: 1001 + 1 = 1010, so the 2's complement of 6 is 1010.

3. Add the 2's complement of 6 to 11:

   • 11 in binary is 1011.
   • Add the 2’s complement of 6 (1010) to 1011:

     1011
   + 1010
   ------
    10101
   

4. Discard the carry bit: The final result is 0101, which is 5 in decimal, confirming that 11 - 6 = 5.

Summary of 2’s Complement Operations

• Addition: When adding binary numbers, just follow the usual addition rules (carry, no carry).
• Subtraction: Convert the number to be subtracted into its 2's complement and then add it to the original number. If the result overflows (has a carry-out), discard it.
• Representation of Negative Numbers: In a system using 2’s complement, negative numbers are represented by finding the 2’s complement of their absolute value.

Advantages of 2’s Complement

1. Single Representation for Zero: In 2's complement, there is only one way to represent zero, which avoids the issue of having two representations for zero (positive and negative) as in sign-magnitude representation.

2. Simpler Hardware Design: Arithmetic operations like addition and subtraction can be performed using the same hardware for both positive and negative numbers, simplifying circuit design.

3. Efficient Subtraction: Subtraction is handled by addition of 2's complement, which simplifies hardware design and avoids the need for a separate subtraction circuit.

Conclusion

• Addition in binary follows simple rules, where a carry is passed to the next higher bit if the sum of two bits is 1.
• Subtraction can be tricky, but using the 2’s complement method simplifies this by turning subtraction into addition.
• 2’s complement is widely used in computer systems for representing negative numbers and performing subtraction, making the system more efficient by using the same addition logic for both positive and negative numbers.