Binary Addition & Subtraction

Overview

Binary addition and subtraction are fundamental operations in computing, often used for arithmetic calculations. Unlike decimal addition and subtraction, binary uses only two digits (0 and 1), which makes operations simpler but requires an understanding of carrying and borrowing in binary form. For handling both positive and negative numbers, we use Two's Complement representation, which allows for straightforward addition and subtraction of negative values.

Binary Addition

Binary addition follows simple rules similar to decimal addition, with extra care needed for carrying:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 10 (result is 0 with a carry of 1)
• 1 + 1 + 1 = 11 (result is 1 with a carry of 1)

  1 1 0 1   (carry)
  1 0 1 1
+ 1 1 0 1
---------
1 1 0 0 0

Binary Subtraction

Binary subtraction uses borrowing when subtracting a larger bit from a smaller one.

Borrowing Rules

• 0 - 1 = 1 (borrow 1 from the next higher bit)
• 1 - 0 = 1
• 1 - 1 = 0
• 10 - 1 = 1 (where 10 represents 2 in binary)

  1 0 1 1   (borrowed bits)
  1 1 0 1
- 1 0 0 1
---------
  0 0 0 0

Two's Complement for Negative Numbers

Two's Complement allows straightforward addition and subtraction of negative binary numbers without special handling.

Converting to Two's Complement

• For a positive number, represent it in binary.
• For a negative number, invert the bits of its positive counterpart and add 1.

Binary Addition with Two's Complement (Including Negatives)

To add a positive and a negative binary number, simply add their Two's Complement representations.

  00000101
+ 11111101
-----------
  00000010

Binary Subtraction with Two's Complement

To subtract binary numbers using Two's Complement, convert the number to be subtracted into its Two's Complement form and then add.

  00000101
+ 11111101
-----------
  00000010

Examples

    1 1 0 1
  + 0 1 1 0
  ---------
    1 0 0 1 1

  00000101
+ 11111001
----------
  11111110