Fractals/Continued fraction: Difference between revisions
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== Basic formula == |
== Basic formula == |
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A continued fraction is an expression of the form |
A continued fraction<ref>[http://file.scirp.org/Html/4-7402078_44811.htm Continued Fractions and Dynamics by Stefano Isola]</ref> is an expression of the form |
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:<math>a_0 + \cfrac{b_1}{a_1 + \cfrac{b_2}{a_2 + \cfrac{b_3}{a_3 + {_\ddots}}}}</math> |
:<math>a_0 + \cfrac{b_1}{a_1 + \cfrac{b_2}{a_2 + \cfrac{b_3}{a_3 + {_\ddots}}}}</math> |
Revision as of 20:04, 21 May 2018
"Do not worry about your problems with mathematics, I assure you mine are far greater." Albert Einstein
Notation
Basic formula
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 |
Finite simple continued fractions
Notation :
Every finite continued fraction represents a rational number :
Infinite continued fractions
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.
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
Maxima CAS
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
- number theory
- aproximation of irrational number, see rotation number in case of Siegel disk
- continued fractions based functions over the complex plan[3][4]
See also
Help
- math.stackexchange questions tagged continued-fractions
- mathoverflow questions tagged continued-fractions
References
- ↑ 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