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➕ Binary Addition and Subtraction

➕ Arithmetic Operations in Binary

Binary addition and subtraction are fundamental operations in digital systems. Understanding how these operations work is essential for comprehending computer arithmetic and digital logic.

➕ Binary Addition​

Binary addition follows similar rules to decimal addition, but with only two digits (0 and 1):

📝 Basic Rules​

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 0 = 1
  • 1 + 1 = 0, carry 1 (equivalent to 2 in decimal, which is 10 in binary)
  • 1 + 1 + 1 = 1, carry 1 (equivalent to 3 in decimal, which is 11 in binary)

🔢 Step-by-Step Process​

  1. Add the rightmost bits
  2. If the sum is 2 or 3, record the remainder and carry the 1
  3. Continue left, including any carried digits
  4. After the leftmost addition, write any final carry as an extra bit

Example 1: Simple Addition​

  1 0 1 1    (11 in decimal)
+ 0 1 0 1    (5 in decimal)
---------
  1 1 0 0    (16 in decimal)

Working right to left:

  • 1 + 1 = 0, carry 1
  • 1 + 0 + 1 (carry) = 0, carry 1
  • 0 + 1 + 1 (carry) = 0, carry 1
  • 1 + 0 + 1 (carry) = 0, carry 1
  • Final carry: 1
  • Result: 10000 (16 in decimal)

Example 2: Addition with Multiple Carries​

  1 1 1 1    (15 in decimal)
+ 0 1 1 1    (7 in decimal)
---------
 1 0 1 1 0    (22 in decimal)

Working right to left:

  • 1 + 1 = 0, carry 1
  • 1 + 1 + 1 (carry) = 1, carry 1
  • 1 + 1 + 1 (carry) = 1, carry 1
  • 1 + 0 + 1 (carry) = 0, carry 1
  • Final carry: 1
  • Result: 10110 (22 in decimal)

➖ Binary Subtraction​

There are several methods for binary subtraction. We'll cover the direct method and the complement method.

📝 Direct Method​

Basic Rules​

  • 0 - 0 = 0
  • 1 - 0 = 1
  • 1 - 1 = 0
  • 0 - 1 = 1, borrow 1 (equivalent to 2 - 1 = 1)

Example: Direct Subtraction​

  1 0 1 1    (11 in decimal)
- 0 1 0 1    (5 in decimal)
---------
  0 1 1 0    (6 in decimal)

Working right to left:

  • 1 - 1 = 0
  • 1 - 0 = 1
  • 0 - 1 = 1, borrow 1 (making the next digit 0)
  • 0 - 0 = 0
  • Result: 0110 (6 in decimal)

🔄 Two's Complement Method​

The two's complement method is widely used in computers for representing negative numbers and performing subtraction.

To subtract B from A (A - B):

  1. Find the two's complement of B
  2. Add it to A
  3. Ignore any carry beyond the original number of bits

Finding Two's Complement​

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

Example: Subtraction Using Two's Complement​

Subtract 5 (0101) from 11 (1011):

Step 1: Find two's complement of 5 (0101)
  Invert bits: 1010
  Add 1:      +   1
              -----
              1011

Step 2: Add to 11 (1011)
  1 0 1 1    (11 in decimal)
+ 1 0 1 1    (two's complement of 5)
---------
1 0 1 1 0    

Step 3: Ignore the carry bit
  0 1 1 0    (6 in decimal)

⚠️ Special Cases and Considerations​

🔄 Overflow in Addition​

Overflow occurs when the result of an operation is too large to be represented with the available bits:

  1 1 1 1    (15 in decimal, maximum 4-bit value)
+ 0 0 0 1    (1 in decimal)
---------
1 0 0 0 0    (16 in decimal, requires 5 bits)

In fixed-width systems, the leftmost carry bit is lost, potentially causing errors.

➖ Negative Results in Subtraction​

When subtracting a larger number from a smaller one using two's complement:

  • The result will be in two's complement form
  • The leftmost bit will be 1, indicating a negative number

To find the absolute value of a negative two's complement result, take its two's complement again.

💻 Practical Applications​

Binary addition and subtraction are fundamental to:

  • 🧮 Arithmetic operations in CPUs
  • 🧠 Address calculations in memory
  • 📊 Digital signal processing
  • 📈 Implementing higher-level mathematical functions
  • 🔍 Error detection and correction codes

Understanding these operations provides insight into how computers perform calculations at the most basic level.