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Binary Machine Numbers

1. Representing Real Numbers:

   A \(\ \ \ \ \ \ \ \ \ \ \ \ \\) (binary digit) representation is used for a real number.

   • The first bit is
   • Followed by:
   • and a \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\), called the
   • The base for the exponent is

2. Floating-Point Number Form:
3. Smallest Normalized positive Number:
   • When:
   • Equivalent to:
4. Largest Normalized positive Number:
   
   • When:
   • Equivalent to:
5. UnderFlow:
   • When numbers occurring in calculations have
6. OverFlow:
   • When numbers occurring in calculations have
7. Representing the Zero:
   
   • There are \(\ \ \ \ \\) Representations of the number zero:

Decimal Machine Numbers

1. What?
2. (k-digit) Decimal Machine Numbers:
3. Normalized Form:
4. Floating-Point Form of a Decimal Machine Number:
   
   • The floating-point form of y, denoted \(f_l(y)\), is obtained by:
5. Termination:
   There are two common ways of performing this termination:
   1. \(\ \ \ \ \ \ \ \ \ \\):
      

      This produces the floating-point form:

   2. \(\ \ \ \ \ \ \ \ \ \\): \(\ \\) which
      

      This produces the floating-point form:

      For rounding, when \(d_{k+1} \geq 5\), we
      When \(d_{k+1} < 5\), we
      If we round down, then \(\delta_i =\)
      However, if we round up,

6. Approximation Errors:
   

   • The Absolute Error: \(\ \ \ \ \ \ \\).

   • The Relative Error: \(\ \ \ \ \ \ \\).

7. Significant Digits:
   
8. Error in using Floating-Point Repr.:
   

   • Chopping:

     The Relative Error =

     The Machine Repr. [for k decimial digits] =

     \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\).

     \(\implies\)

     Bound \(\ \ \ \implies \ \ \ \ \ \ \ \ \ \ \ \\).

   • Rounding:

     In a similar manner, a bound for the relative error when using k-digit rounding arithmetic is

     Bound \(\ \ \ \implies \ \ \ \ \ \ \ \ \ \ \ \ \ \\).

9. Distribution of Numbers:
   The number of decimal machine numbers in \(\ \ \ \ \ \ \ \ \ \ \\) is \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) for

Finite-Digit Arithmetic

1. Values:

   \[x = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\]

   \[y = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\]

2. Operations:
   
3. Error-producing Calculations:
   
   • First:
   • Second:
4. Avoiding Round-Off Error:
   
   • First:
   • Second:

     \(\implies \ \ \ \ \ \ \ \ \ \ \ \ \ \ x_1 = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ \ \ \ \ \ \ \ \ \ \ \ \ \ x_2 = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)

Nested Arithmetic

1. What?

   Remember that chopping (or rounding) is performed:

   
   

   • \[\ \ \ \ \ \ \\]

   
   

   Polynomials should always be expressed \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\) , becasue, \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)

2. Why?