CPS120: Introduction to Computer Science

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CPS120 Introduction to Computer Science

• Computer Math Signed Numbers

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Representing Signed Numbers

• Remember, all numeric data is represented inside
  the computer as 1s and 0s
• Arithmetic operations, particularly subtraction
  raise the possibility that the result might be
  negative
• Any numerical convention needs to differentiate
  two basic elements of any given number, its sign
  and its magnitude
• Conventions
• Sign-magnitude
• Ten's complement
• Twos complement
• Ones complement

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Representing Negatives

• It is necessary to choose one of the bits of the
  basic unit as a sign bit
• Usually the leftmost bit
• By convention, 0 is positive and 1 is negative
• Positive values have the same representation in
  all conventions
• However, in order to interpret the content of any
  memory location correctly, it necessary to know
  the convention being used used for negative
  numbers

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Comparing the Conventions
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Representing Negative Values

• You have used the signed-magnitude representation
  of numbers since grade school. The sign
  represents the ordering, and the digits represent
  the magnitude of the number.

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Representing Negative Values (Contd)

• There is a problem with the sign-magnitude
  representation
• There are two representations of zero.
• There is plus zero and minus zero. Two
  representations of zero within a computer can
  cause unnecessary complexity.

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Sign-Magnitude

• For a basic unit of N bits, the leftmost bit is
  used exclusively to represent the sign
• The remaining (N-1) bits are used for the
  magnitude
• The range of number represented in this
  convention is 2 N1 to 2 N-1 -1

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Sign-magnitude Operations

• Addition of two numbers in sign-magnitude is
  carried out using the usual conventions of binary
  arithmetic
• If both numbers are the same sign, we add their
  magnitude and copy the same sign
• If different signs, determine which number has
  the larger magnitude and subtract the other from
  it. The sign of the result is the sign of the
  operand with the larger magnitude
• If the result is outside the bounds of 2 n1 to
  2 n-1 1, an overflow results

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Other Ways of Representing Negative Values

• If we allow only a fixed number of values, we can
  represent numbers as just integer values, where
  half of them represent negative numbers.
• For example, if the maximum number of decimal
  digits we can represent is two, we can let 1
  through 49 be the positive numbers 1 through 49
  and let 50 through 99 represent the negative
  numbers -50 through -1.

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Representing Negative Values (Contd)

• To perform addition within this scheme, you just
  add the numbers together and discard any carry.

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Representing Negative Values (Contd)

• A-BA(-B). We can subtract one number from
  another by adding the negative of the second to
  the first.

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Representing Negative Values (Contd)

• There is a formula that you can use to compute
  the negative representation
• This representation of negative numbers is called
  the tens complement.

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Representing Negative Values (Contd)
Twos Complement To make it easier to look at
long binary numbers, we make the number line
vertical.
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Representing Negative Values (Contd)

• Addition and subtraction are accomplished the
  same way as in 10s complement arithmetic
• -127 10000001
• 1 00000001
• -126 10000010
• Notice that with this representation, the
  leftmost bit in a negative number is always a 1.

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Twos Complement Convention

• A positive number is represented using a
  procedure similar to sign-magnitude
• To express a negative number
• Express the absolute value of the number in
  binary
• Change all the zeros to ones and all the ones to
  zeros (called complementing the bits)
• Add one to the number obtained in Step 2
• The range of negative numbers is one larger than
  the range of positive numbers
• Given a negative number, to find its positive
  counterpart, use steps 2 3 above

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Ones Complement

• Devised to make the addition of two numbers with
  different signs the same as two numbers with the
  same sign
• Positive numbers are represented in the usual way
• For negatives
• STEP 1 Start with the binary representation of
  the absolute value
• STEP 2 Complement all of its bits