# Practice Math With Us # 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.

Please express your views of this topic how to reduce fractions  by commenting on blog.

## 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 Between, if you have problem on these topics Solve Absolute Value Equations ,please browse expert math related websites for more help on cbse board paper .

## 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.