Binary Real Numbers

1
Binary Real Numbers
2
Introduction

• Computers must be able to represent real numbers
  (numbers w/ fractions)
• Two different ways
• Fixed-point
• Floating-point
• NOTE Everything in binary uses powers of two

3
Decimal Review

• Digits to the right of the decimal point
  correspond to negative powers of 10

4
Binary Fractions
5
Fixed Point Notation

• Multiply each 1 by the corresponding power of 2
• Add up the resulting powers of 2
• Example
• 11.012 2 1 ¼ 3.2510
• 00111.0102 4 2 1 ¼ 7.2510

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Floating-Point Notation

• Floating-point notation is essentially the
  computers way of storing a number that has been
  normalized
• 3 different parts of any number
• Mantissa normalized number
• Exponent power to which the base is raised
• Sign of both mantissa and exponent
• Decimal Example
• 12.5 0.125 x 102 ? normalized!

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Normalization Steps

• Beginning with a fixed point number
• Normalize the number such that the radix point
  (decimal point) is all the way to the left
  (produces the mantissa)
• Multiply the resulting number by the base raised
  to an exponent

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Floating-Point Example

• What is 12.5 in floating-point representation?
• Convert 12.5 to binary fixed point
• 12.510 1100.12
• Normalize the number by moving the radix point,
  producing the mantissa
• 1100.12 0.11001 24
• Fill in the bits for each of the three parts of
  any real number
• Sign (2 bits)
• Mantissa ( bits varies)
• Exponent ( bits varies)
• NOTE 2s complement may be applied to the
  mantissa or the exponent if either are negative

9
Placing the Bits

• Assume you have the following
• 1 bit for the mantissa sign
• 8 bits for the mantissa
• 1 bit for the exponent sign
• 6 bits for the exponent
• SM M M M M M M M M S E E E E E E E
• Example
• 0.11001 24 ? 0 00011001 0 000100

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Another Example

• Convert -12.510 to binary
• Convert 12.5 to fixed point ? 01100.1
• Normalize ? 0.11001 24
• Convert exponent base to binary 4 ? 0100
• 2s complement the mantissa by flipping bits and
  adding 1 011001 ? 100111
• Final number ? 1 00111 0100

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Upper Lower Bounds

• Assume you have the following
• 1 bit for the mantissa sign
• 8 bits for the mantissa
• 1 bit for the exponent sign
• 6 bits for the exponent
• What is the upper bound for the floating-point
  number?
• What is the lower bound for the floating-point
  number?
• What happens if we convert a floating-point
  number to an integer?

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Integers vs. Floating-point

• integers
• smaller range than floating-point
• all numbers within the range are 100 accurate
• floating-point
• large range of numbers
• not all numbers within the range can be
  represented accurately
• Example 2.9999999999999 repeating

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Possible Errors

• truncation error
• round off errors using floating-point numbers
  because not all real numbers can be represented
  accurately
• overflow error
• attempting to represent a number that is greater
  than the upper bound for the given number of bits
• underflow error
• attempting to represent a number that is less
  than the lower bound for the given number of bits