Continued Fractions

A continued fraction is a number expressed as a nested fraction, constructed from integers.

It is written in the general form: $$ a_0 + \cfrac{1}{ a_1 + \cfrac{1}{ a_2 + \cfrac{1}{ a_3 + \cdots}}} $$

or, in abbreviated form: $ [ a_0;a_1;a_2;a_3;\dots ] $

The coefficient $ a_0 $ is an integer, and the subsequent coefficients $ a_1, a_2, \ldots $ are strictly positive integers.

A continued fraction is said to be finite when the expansion stops after a finite number of steps. It then represents exactly a rational number.

A continued fraction is said to be infinite when the expansion continues indefinitely. It then represents an irrational number.

The principle of continued fraction expansion relies on Euclid's algorithm for Euclidean division, as with calculating the greatest common divisor (GCD).

Each quotient obtained corresponds to a coefficient of the continued fraction.

The rational fraction approximating pi, $ \frac{355}{113} = 3.14159292035\ldots $, follows from the fact that $$ 355 = 113 \times \fbox{3} + 16 \\ 113 = 16 \times \fbox{7} + 1 \\ 16 = 1 \times \fbox{16} + 0 $$ The successive quotients are $ 3 $, $ 7 $, and $ 16 $. The corresponding continued fraction is therefore $ [3,7,16] $

The convergents of a continued fraction are the rational fractions obtained by successively truncating its expansion.

For a continued fraction $[a_0;a_1;a_2;a_3;\dots]$, the nth convergent is $[a_0;a_1;a_2;\dots;a_n]$.

The convergents provide the best rational approximations of the number represented.

Calculate an approximate value of the root (approximation as accurate as possible) and dCode will provide the corresponding continuous fraction.

The easiest way is to use cfrac: $$ e=2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{ 1+\cfrac{1}{1+\cfrac{1}{4+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{6+\cdots}}}}}}}} $$

But the shortest way is to write $$ e = [2 ; 1, 2, 1, 1, 4, 1, 1, 6, \cdots] $$

Most known continued fractions are:

— Square Root of 2: $ \sqrt{2} = [1;2,2,2,2,2,\cdots] $

— Golden Ratio: $ \Phi = [1;1,1,1,1,1,\cdots] $