# Fractals/Continued fraction

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

# Notation

[edit | edit source]## Generalized form

[edit | edit source]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 form

[edit | edit source]It is generally assumed that the numerator b of all of the fractions is 1. Such form is called a **simple** or **regular continued fraction**, or said to be in **canonical form**.

If real number is a fraction ( x < 1), then is zero and the notation is simplified:

```
```

### Finite

[edit | edit source]Notation :

```
```

Every finite continued fraction represents a rational number :

```
```

If positive real fraction x is rational number, there are **exactly two different** continued fraction expansions:

```
```

where

- Usually the first, shorter form is chosen as the canonical representation
- second form is one longer then the first

### Infinite

[edit | edit source]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 | edit source]- decimal number ( real or rational) to continued fraction
- abacus CAS
- Maxima CAS : cf (expr) Converts expr into a continued fraction.

## Maxima CAS

[edit | edit source]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

To compute n-th convergent:

(%i10) a; (%o10) [0, 3, 2, 1000, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] (%i11) a3: listn(a,3); (%o11) listn([0, 3, 2, 1000, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], 3) (%i12) a3: firstn(a,3); (%o12) [0, 3, 2] (%i13) cf3:cfdisrep(a3); 1 (%o13) ----- 1 3 + - 2 (%i14) r3:ratsimp(cf3); 2 (%o14) - 7 (%i15)

# Examples

[edit | edit source]- 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]}

# Help

[edit | edit source]- 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

# See also

[edit | edit source]- binary expansion ( representation of real number)
- fractional iteration
- dynamics of continued fractions

# References

[edit | edit source]- ↑ 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