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Theory Of Continued Fractions

Introduction to theory of Continued Fractions:

A continued fraction is symbol of a real number x is one of the forms


Where ai and bi are integers for all i.

  Given a0 ≥ 0, a1 ≥ 1, a2 ≥ 1, …, an ≥ 1, define [a0, a1, a2, …, an] to be the fraction in lowest terms for

                                             a0+                                1                         

                                                        a1+                      1                          

                                                              a2+                 1                       

                                                                       a3+        1                       

                                                                                          ………

                                                                                               +    1       

                                                                                                     an

          For example, [2, 3, 4] =

          2+ (1/ (3+1/4)) = 30/13.

          In theory of continued fractions, you can learn about the characteristics of continued functions.


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Theory of Periodic continued fractions:


The least number of repeating terms is called the period of the continued fraction.

the square root of a square open integer has a periodic continued fraction of the form

                                    ______________

            `sqrt(n)` = [a0, a1, a2...a2, a1, 2a0]

Comb. Interpretation:

Explain the functions p and q such that the continued fraction

                                                     P (a0, a1, a2,….an)               pn
   [a0, a1, a2, …, an         =  ------------------------------  =  -------
                                                    q(a0.a1,a2,…..,an)                qn

            When reduced to lowest terms.        

            Let Pn = P(a0, a1, a2, …, an) count the number of ways to tile an (n+1)-board with dominoes and stackable square tiles.

Height Restrictions:

            The ith cell may be covered by a stack of up to ai square tiles.

            Nothing can be stacked on top of a domino.

Recall Pn counts the number of ways to tile an n+1 board with dominoes and stackable square tiles.

Let Qn = Q (a0, a1, a2an) count the number of ways to tile an n-board with dominoes and stackable square tiles.

Define Qn = P (a1, a2an).                               Pn                                 pn   

                                                              Then           --------     =    ------------ = [a0,a1,a2,….an]
                                                                                   Qn                            qn


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Example - theory of continued fractions:


              The beginning of the “π-board” given by [4, 8, and 16] can be tiled in 336 ways:

  •               all squares = 315 ways
  •               stack of squares, domino = 3 ways
  •               domino, stack of squares = 18 ways

              Removing the initial cell, the

              [8, 16]-board can be tiled in 109 ways:

              all squares = 109 ways

              domino = 1 way

 

Thus [4, 8, 16] = ≈ 3.1415. these are the features of theory of continued fractions.