Unit III — Binary Arithmetic: Addition & Subtraction

Binary arithmetic follows the same rules as decimal arithmetic but uses only two digits: 0 and 1. Carries and borrows work on powers of 2 instead of powers of 10.

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Binary Addition Rules

Key rule: 1 + 1 = 10 (binary) — write 0, carry 1.

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Binary Addition Examples

Example 1: Add (1011)₂ + (0110)₂

  1 0 1 1   (11 in decimal)
+ 0 1 1 0   (6 in decimal)
---------
  1 0 0 0 1  (17 in decimal)

Step-by-step (right to left):

• Col 0: 1+0 = 1
• Col 1: 1+1 = 10 → write 0, carry 1
• Col 2: 0+1+1(carry) = 10 → write 0, carry 1
• Col 3: 1+0+1(carry) = 10 → write 0, carry 1
• Col 4: carry 1 → write 1

Result: (10001)₂ = 17 ✓

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Example 2: Add (11001)₂ + (10111)₂

  1 1 0 0 1   (25)
+ 1 0 1 1 1   (23)
-----------
1 1 0 0 0 0   (48)

• Col 0: 1+1 = 0 carry 1
• Col 1: 0+1+1 = 0 carry 1
• Col 2: 0+1+1 = 0 carry 1
• Col 3: 1+0+1 = 0 carry 1
• Col 4: 1+1+1 = 1 carry 1
• Col 5: 0+0+1 = 1

Result: (110000)₂ = 48 ✓

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Binary Subtraction Rules

Key rule: 0 − 1 → borrow from left; treated as 10 − 1 = 1, with borrow propagating.

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Binary Subtraction Examples

Example 1: Subtract (1101)₂ − (0101)₂

  1 1 0 1   (13)
- 0 1 0 1   (5)
---------
  1 0 0 0   (8)

• Col 0: 1−1 = 0
• Col 1: 0−0 = 0
• Col 2: 1−1 = 0
• Col 3: 1−0 = 1

Result: (1000)₂ = 8 ✓

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Example 2: Subtract (10100)₂ − (01011)₂

  1 0 1 0 0   (20)
- 0 1 0 1 1   (11)
-----------
  0 1 0 0 1   (9)

Work through each column, borrowing as needed.

Result: (01001)₂ = 9 ✓

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Verification Tip

Always verify by converting binary results to decimal.

Key Takeaway: Binary addition carries when a column sums to 2 or more (1+1=10). Binary subtraction borrows from a higher bit when subtracting 1 from 0. Carry/borrow management is the most error-prone part — always work right to left and note each carry/borrow explicitly.