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
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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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]
Explain the functions p and q such that the continued fractionP (a0, a1, a2,….an) pn
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.
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, a2… an) count the number of ways to tile an n-board with dominoes and stackable square tiles.
Define Qn = P (a1, a2… an). Pn pnThen -------- = ------------ = [a0,a1,a2,….an]
The beginning of the “π-board” given by [4, 8, and 16] can be tiled in 336 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.