Twos Complement

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Twos Complement

• "Counting in binary is just like counting in
  decimal if you are all thumbs."    
•  Glaser and Way.

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Example done in class

• Convert -1310 to 2s complement
• First thing to notice
• it is a negative number, so, left-most bit has
  to be a 1 in the answer
• Second thing to notice
• it cannot be represented with just 4 bits as
  this will lead to an overflow

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-1310 to 2s complement

• Convert 1310 to binary using division algorithm
• 11012 in 4-bit notation
• Before doing anything, notice that 4-bit causes
  overflow in 2s complement notation for negative
  numbers smaller than -8 and positive numbers
  greater than 7. So, represent 1310 in 8-bit
  notation.
• 0000 11012
• 3. Now, convert this 8-bit binary (00001101) to
  2s complement

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-1310 to 2s complement continued

• Convert 0000 11012 to 1s complement
• 1111 00102
• Add 1 to 1s complement to get 2s complement
• 1111 00102
• 0000 00012
• 1111 00112
• 2s complement of -1310 is therefore

1111 00112
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Another example convert -1710 to 2s complement

• First notice that it is a negative number, so,
  final answer should have the left-most bit as 1
• Convert 1710 to binary using division algorithm
• 0001 00012
• 1s complement the above binary string
• 1110 1110
• Add 1 to the above 1s complement
• 1110 1111
• Therefore the 2s complement of -1710 is

1110 1111
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Summary Decimal to 2s Complement

• Choose the appropriate number of bits for the
  bit-representation in binary to avoid overflow
• For a positive decimal number, to convert to 2s
  complement, express it in appropriate binary bits
  and stop.
• For a negative decimal number, to convert to
  2complement,
• First convert the binary to 1s complement
• Then add 1 to this 1s complement
• The result is the 2s complement of the given
  negative number

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Another Example

• Convert 1310 to 2s complement
• First thing to notice
• it is a positive number, so, left-most bit has
  to be a 0 in the answer
• Second thing to notice
• it cannot be represented with just 4 bits as
  this will lead to an overflow

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1310 to 2s complement

• Convert 1310 to binary using division algorithm
• 11012 in 4-bit notation
• As 4-bit notation causes overflow for positive
  numbers bigger than 7, represent 1310 in 8-bit
  notation.
• 0000 11012
• Stop here as this is a positive number.
• Twos Complement of 1310 is

0000 11012
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Decoding 2s Complement

• First check the left-most bit
• If left-most bit is 0, it represents a positive
  number
• If left-most bit is 1, it represents a negative
  number
• Now use the following method for decoding a
  negative number in 2s complement
• Copy the given 2s complement bits from right to
  left until a 1 is copied
• Then reverse all the remaining bits to the left
  all the way
• Now convert this binary string to decimal using
  multiplication algorithm

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Example Decoding 2s complement

• Decode 1111 0011 in 2s complement to its decimal
  equivalent
• First notice that left-most bit is a 1
• Copy the string till a 1 is copied, and reverse
  the other bits to the left of the 1 just copied
• 1111 0011
• Copy up to first 1 1
• Reverse the rest of the bits 0000 1101

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Decoding negative number continued

• 3. Convert this string (0000 1101) to decimal
  using multiplication algorithm
• 0x27 0x26 0x25 0x24 1x23 1x22
  0x21 1x20
• 0 0 0 0 8 4 0 1 1310
• 4. We are not quite done yet. Whats missing?
  Notice that the left-most bit was a 1 in the
  original string representing 2s complement, so,
  this is a negative number. The answer therefore
  is

-1310
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Decoding 2s complement positive numbers

• Decode twos complement string 01111 to its
  decimal equivalent
• First notice that left-most bit is a 0
• this string represents a positive number
• Decode using multiplication algorithm
• 0x24 1x23 1x22 1x21 1x20
• 0 8 4 2 1 1510
• The answer therefore is

1510
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Addition in 2s Complement

• To add values represented in 2s complement
• Use the same algorithm as for binary addition
• Note that the resulting bit-pattern should be the
  same length as the problem, so, truncate the
  left-end bit appropriately
• Check for overflow when adding two positive
  numbers or two negative numbers, by checking if
  the sign-bit of the result indicates correct
  arithmetic

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Example Addition in 2s complement

• Add the following numbers in 2s complement
• 0101 0110
• First, notice that both these numbers are
  positive, so the answer should be positive
• Second, be aware that as this is addition of two
  positive numbers, there could be an overflow
  error
• Perform binary addition algorithm as usual
• 0101 510
• 0110 610
• 1011 -510
• Notice that the result is the 2s complement of
  -510 !!
• This indicates an overflow, because, 56 11,
  which is out of the range of 4-bit notation of
  2s complement.

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Example Addition in 2s complement

• Add the following numbers in 2s complement
• 1010 1010
• Again, notice that it is an addition of two
  negative numbers, so the result must be a
  negative number, and there is a possibility for
  overflow
• Perform the usual binary addition
• 1010 -6
• 1010 -6
• 10100 4
• The result, after truncation to 4-bits,
  represents 410 !!
• This indicates overflow as (-6) (-6) -12
  which cannot be represented in 4-bit 2s
  complement notation

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Example Addition in 2s complement

• Here is another example with overflow error add
  01110001 in 2s complement
• Again, notice that they are both positive
  numbers, so, the result must be positive and,
  there could be an overflow error.
• Perform the usual binary addition
• 0111 7
• 0001 1
• 1000 8
• Therefore the answer is 1000 in 4-bit 2s
  complement notation and there is an overflow
  error.

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Addition in 2s complement NO Overflow

• Here is an example with no overflow add 0100
  0011 in 2s complement
• First, notice that it is an addition of two
  positive numbers, so the result must be a
  positive number, and there could be an overflow
• Perform usual binary addition
• 0100 0011 0111
• The result 0111 represents 710 which can be
  represented in 4-bit 2s complement notation.