Binary Arithmetic

1
Binary Arithmetic
2
Addition

• Add 2 digits ? sum carry
• 7 5
• 2 7
• ------ ------
  
• 0 9 1 2
• carry sum
  carry sum

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Binary Addition

• Adding Binary Digits
• a b sum a b carry
• ------------------ -----------------
• 0 0 0 0 0 0
• 0 1 1 0 1 0
• 1 0 1 1 0 0
  
• 1 1 0 1 1 1
• Sum Digit1 XOR Digit2
• Carry Digit1 AND Digit2

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Binary Addition
Half Adder
Full Adder
Multiple bit adder
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Example

• Decimal Addition
• 5 6 11
• Binary Addition
• 00101 5
• 00110 6
• --------------
• 01011 11

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Binary Subtraction

• Decimal Subtraction
• 20 3 20 (-3)
• Binary Subtraction
• 20 100100
• -3 ? Twos Complement of Binary
  representation of 3

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Twos Complement

• Ones Complement
• Convert 0 to 1, and 1 to 0
• What is Ones complement of 00011001?
• It is 11100110
• Twos Complement
• Add 1 to the ones complement
• What is the Twos complement of 00011001?
• It is 11100111

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Twos Complement

• (-1) (1111)
• (-2) (1110)
• (-3) (1101)
• .
• .

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• In a n bits register ? unsigned numbers in the
  range of .
• In a n bits register ? signed numbers in the
  range of .

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Binary Subtraction

• Decimal Subtraction
• 20 3 20 (-3) 17
• Binary Subtraction
• 20 10100
• (-3) 01101
• -------------------------
• 10001 17

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Unsigned Binary Addition Example
4 bit numbers can represent numbers from 0 to 15
The result 18 is not in the range 0,15, And is
too big. Consequently, Overflow happens here. So
the carry bit from adding the two most
significant bits represents a results that
overflows.
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Signed Binary Addition Example
signed 4 bit numbers (2's complement) can
represent numbers between -8 and 7
The extra carry from the most significant bit has
no meaning
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Signed Binary Addition Example
signed 4 bit numbers (2's complement) can
represent numbers between -8 and 7
if two numbers with the same sign (either
positive or negative) are added and the result
has the opposite sign, an overflow has occurred.
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Midterm Examples

• 3. Do the following signed binary arithmetic.
  Use 2's complement when appropriate. V bit is
  Overflow and C bit is Carry
• a) X 7, Y 2, Z X Y
• b) X 7, Y 2, Z X - Y
• c) X 2, Y 7, Z X - Y