Fractals/Continued fraction

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



Generalized form


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:

Examples of 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


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:




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:



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



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:




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);
(%o13)                               -----
                                     3 + -
(%i14) r3:ratsimp(cf3);
(%o14)                                 -


  • number theory
  • 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



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