Fractals/Continued fraction

"Do not worry about your problems with mathematics, I assure you mine are far greater." Albert Einstein Notation Generalized form A continued fractionContinued Fractions and Dynamics by Stefano Isola 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.Darren C. Collins, Continued Fractions, MIT Undergraduate Journal of Mathematics, Thus, all of the following illustrate valid finite simple continued fractions: +Examples of finite simple continued fractionsFormulaNumericRemarksAll integers are a degenerate caseSimplest possible fractional formFirst integer may be negativeFirst integer may be zero simple form 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 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 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 decimal number ( real or rational) to continued fraction abacus CAS Maxima CAS : cf (expr) Converts expr into a continued fraction. 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 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 number theory aproximation of irrational number, see rotation number in case of Siegel disk continued fractions based functions over the complex plancontinued fractions based functions over the complex planecontinued-fractions-with-applications by L. Lorentzen H. Waadeland " a continued fraction may be regarded as a sequence of Möbius maps" Alan F. BeardoneContinued 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 Help 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 binary expansion ( representation of real number) fractional iteration dynamics of continued fractions Dynamics of continued fractions and kneading sequences of unimodal maps by Claudio Bonanno, Carlo Carminati, Stefano Isola, Giulio Tiozzo References CALENDARS AND CONTINUOUS FRACTIONS by Anne Broise ( in fr.) Dynamics of continued fractions and kneading sequences of unimodal maps by Claudio Bonanno, Carlo Carminati, Stefano Isola, Giulio Tiozzo