# Continued Fractions

**Continued Fractions**are a way of representing real numbers as a sequence of positive integers. Let's say we have a real number

$\alpha_1$

. We can form a sequence of positive integers from $\alpha_1$

in this way:$\alpha_1 = a_1 + \frac{1}{\alpha_2}, \text{where } a_1=\left \lfloor{x}\right \rfloor$

For example, let's say

$\alpha_1=1.5$

. We can say that$\alpha_1 = 1 + \frac{1}{2}, \text{with } \alpha_2 = 2$

The trick here is that if

$\alpha_2\not\in \mathbb{Z}$

, we can continue this exact process with $\alpha_2$

and keep the **continued fraction**going.$\alpha_1 = a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \frac{1}{\ddots}}}$

Let's take another example, that of

$\frac{17}{11}$

:

$\frac{17}{11} = 1 + \frac{6}{11} = 1 + \frac{1}{\frac{11}{6}} = 1 + \frac{1}{1 + \frac{5}{6}} = 1 + \frac{1}{1 + \frac{1}{\frac{6}{5}}} = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{5}}}$

The list of continued fractions is represented as a list of the coefficients

$a_i$

, in this case$\frac{17}{11} = \left[ 1; 1, 1, 5 \right]$

The

$k$

th convergent of a continued fraction is the approximation of the fraction we gain by truncating the continued fraction and using only the first $k$

terms of the sequence, for example the 2nd convergence of $\frac{17}{11}$

is $1 + \frac{1}{1} = \frac{2}{1} = 2$

while the 3rd would be $1 + \frac{1}{1 + \frac{1}{1}} = 1 + \frac{1}{2} = \frac{3}{2}$

.One of the obvious applications of these convergents is as

**rational approximations to irrational numbers**.- As a sequence, they have a limit
- This limit is$\alpha_1$, the real number you are attempting to approximate

- They are alternately greater than and less than$\alpha_1$

In Sage, we can define a continue fraction very easily:

f = continued_fraction(17/11)

We can then print off the list of coefficients:

print(f)

>>> [1; 1, 1, 5]

And the convergents are even calculated for us:

print(f.convergents())

>>> [1, 2, 3/2, 17/11]

Last modified 2yr ago