# Fractals/Continued fraction

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"Do not worry about your problems with mathematics, I assure you mine are far greater." Albert Einstein

## Contents

# Notation[edit]

A continued fraction^{[1]} is an expression of the form

where :

- and are either integers, rational numbers, real numbers, or complex numbers.
- , etc., are called the
*coefficients*or*terms*of the continued fraction

Variants or types :

- If for all the expression is called a
*simple*continued fraction. - If the expression contains a finite number of terms, it is called a
*finite*continued fraction. - If the expression contains an infinite number of terms, it is called an
*infinite*continued fraction.^{[2]}

Thus, all of the following illustrate valid finite simple continued fractions:

Formula | Numeric | Remarks |
---|---|---|

All integers are a degenerate case | ||

Simplest possible fractional form | ||

First integer may be negative | ||

First integer may be zero |

## simple continued fractions[edit]

### Finite[edit]

Notation :

```
```

Every finite continued fraction represents a rational number :

```
```

### Infinite[edit]

Notation :

```
```

Every infinite continued fraction is irrational number :

```
```

The rational number obtained by limited number of terms in a continued fraction is called a **n-th convergent**

```
```

because sequence of rational numbers converges to irrational number

```
```

In other words irrational number is the limit of convergent sequence.

Nominator p and denominator q can be found using the relevant recursive relation:

so

Key words :

- the sequence of continued fraction convergents of irrational number
- sequence of the convergents
- continued fraction expansion
- rational aproximation of irrational number
- a best rational approximation to a real number r by rational number p/q

# How to use it in computer programs[edit]

## Maxima CAS[edit]

In Maxima CAS one have cf and float(cfdisrep())

(%i2) a:[0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] (%o2) [0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] (%i3) t:cfdisrep(a) (%o3) 1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/1)))))))))))))))))))))) (%i4) float(t) (%o4) 0.618033988957902

# Examples[edit]

- number theory
- aproximation of irrational number, see rotation number in case of Siegel disk

- continued fractions based functions over the complex plan
^{[3]}^{[4]} - " a continued fraction may be regarded as a sequence of Möbius maps" Alan F. Beardone
^{[5]}

# See also[edit]

- fractional iteration
- dynamics of continued fractions

# Help[edit]

- math.stackexchange questions tagged continued-fractions
- mathoverflow questions tagged continued-fractions
- Continued Fractions - Professor John Barrow
- Chaos in Numberland: The secret life of continued fractions By John D. Barrow

# References[edit]

- ↑ Continued Fractions and Dynamics by Stefano Isola
- ↑ Darren C. Collins, Continued Fractions, MIT Undergraduate Journal of Mathematics,[1]
- ↑ continued fractions based functions over the complex plane
- ↑ continued-fractions-with-applications by L. Lorentzen H. Waadeland
- ↑ Continued Fractions, Discrete Groups and Complex Dynamics by Alan F. Beardone. Beardone, A.F. Comput. Methods Funct. Theory (2001) 1: 535. https://doi.org/10.1007/BF03321006