Fixedpoint and floatingpoint numbers

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Fixed-point and floating-point numbers

• CS370
• Fall 2003

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Representations of numbers

• Unsigned integers
• Signed integers 1s and 2s complement
  representation
• To represent
• Very Large and very Small numbers
• Real numbers in general
• Fixed-point numbers
• Floating-point numbers

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Base-10 (decimal) arithmetic

• Uses the ten numbers from 0 to 9
• Each column represents a power of 10

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Base-10 (decimal) arithmetic

• Uses the ten numbers from 0 to 9
• Each column represents a power of 10

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Standard binary representation

• Uses the two numbers from 0 to 1
• Every column represents a power of 2

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Fixed-point representation

• Uses the two numbers from 0 to 1
• Every column represents a power of 2

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Addition
Base-10
Base-2
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Range of values in a byte
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Scientific notation (1)

• One billion
• 1,000,000,000
• 1 x 109
• significand or mantissa 1
• base or radix 10
• exponent 9

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Scientific notation (2)

• 1999
• 1.999 x 103
• significand or mantissa 1999
• base or radix 10
• exponent 3
• 19.99 x 10
• 199.9 x 10

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Practice (base 10)

• 258 2.58 x 102
• Mantissa 258
• Radix 10
• Exponent 2
• 24.25 2.425 x 101
• Mantissa 2425
• Radix 10
• Exponent 1

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Base-2 scientific notation

• 2.25ten
• 10.01two
• 10.01two x 20
• 1.001two x 21 ? normalized
• Numbers are usually normalized which means that
  the leading bit is always a 1.

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8-bit floating point format (1)
   
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Improvements

• Bias the exponent
• Always subtract a fixed amount, e.g., 3
• Allows representation of negative exponents
• Implicit one
• Leading one in a Phone number such as
• 1-619-556-0231 is redundant.
• Why use a bit for the leading one?

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8-bit floating-point format (2)

• Exponent (3 bits) is biased by 3
• The leading one of significand is implicit
• Zero is represented by all zeros

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IEEE standard floating-point

• Single precision
• 32 bits
• sign 1 bit
• exponent 8 bits
• significand 23 bits
• Bias 127

• Double precision
• 64 bits
• sign 1 bit
• exponent 11 bits
• significand 52 bits
• Bias 511

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Practice( base 10)

• 13 1.3 x 101
• 1.011 x 23
• 1.25 1.25 x 100
• 1.010 x 20

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