Representing Real Numbers

1
Representing Real Numbers

• Scientific notation
• Precision
• Overflow and Underflow
• Conversions

2
Scientific Notation

• Why Scientific Notation?
• What is the difference between
• 9,000,000
• 9.0 x 106
• 9.0000 x 106

3
Real Numbers vs. Integers

• Real numbers have stuff after the decimal (radix)
  point
• Scientific notation already has a radix point
• How do we represent a radix point in binary?

4
Standard Representation I

• float (32 bits)
• 1 sign, 8 exponent, 23 - mantissa
• accuracy ? seven decimal digits
• range ? 10-45 to 1038

5
Standard Representation II

• double (64 bits)
• 1 sign, 11 exponent, 52 - mantissa
• accuracy ? fifteen decimal digits
• range ? 10-300 to 10300

6
Precision

• To three decimal digits, what is v2?
• To three decimal digits, what is v2 v2?
• To a finite-precision computer, does v2 v2
  2?

7
Underflow and Overflow
8
Conversions I

• float to int to float
• float a 4.5
• int b (int) a
• float c (float) b

9
Conversions II

• int to float to int
• int a 123456789
• float b (float) a
• int c (int) b

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Summary

• Be aware of format
• Be aware of precision
• Be aware of limits