Continued Fraction

DOWNLOAD Mathematica Notebook EXPLORE THIS TOPIC IN the MathWorld Classroom Contribute to this entry

The term "continued fraction" is used to refer to a class of expressions of whichgeneralized continued fractionof the form

 b_0+(a_1)/(b_1+(a_2)/(b_2+(a_3)/(b_3+...))) 
 =b_0+K_(n=1)^infty(a_n)/(b_n)

(and the terms may be integers, reals, complexes, or functions of these) are the most general variety (Rocket and Szüsz 1992, p. 1).

Wallis first used the term "continued fraction" in his Arithmetica infinitorum of 1653 (Havil 2003, p. 93), although other sources list the publication date as 1655 or 1656.An archaic word for a continued fraction is anthyphairetic ratio.

这个simple continued fractiona_n=1for allN, leaving

 b_0+1/(b_1+1/(b_2+1/(b_3+...)))=b_0+K_(n=1)^infty1/(b_n).

Ifb_0is an integer and the remainder of thepartial denominators b_kk>0are positive integers, the continued fraction is known as aregular continued fraction

Wolfram Web Resources

Mathematica »

The #1 tool for creating Demonstrations and anything technical.

Wolfram|Alpha »

Explore anything with the first computational knowledge engine.

Wolfram Demonstrations Project »

Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.

Computerbasedmath.org »

Join the initiative for modernizing math education.

Online Integral Calculator »

Solve integrals with Wolfram|Alpha.

Step-by-step Solutions »

Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own.

Wolfram Problem Generator »

Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.

Wolfram Education Portal »

Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more.

Wolfram Language »

Knowledge-based programming for everyone.