In the theory of continued fractions, the **fundamental recurrence formulas** relate the partial numerators and the partial denominators with the numerators and denominators of the fraction's successive convergents. Let

be a general continued fraction, where the *a*_{n} (the partial numerators) and the *b*_{n} (the partial denominators) are numbers. Denoting the successive numerators and denominators of the fraction by *A*_{n} and *B*_{n}, respectively, the fundamental recurrence formulas are given by

The continued fraction's successive convergents are then given by

These recurrence relations are due to John Wallis (1616-1703) and Leonhard Euler (1707-1783)

Read more about Fundamental Recurrence Formulas: The Determinant Formula, A Simple Example

### Other articles related to "fundamental recurrence formulas, formula":

**Fundamental Recurrence Formulas**- A Simple Example

... form that represents the golden ratio φ Applying the

**fundamental recurrence formulas**we find that the successive numerators An are {1, 2, 3, 5, 8, 13...} and the successive denominators Bn are {1 ... the partial numerators in this example are equal to one, the determinant

**formula**assures us that the absolute value of the difference between successive convergents ...

### Famous quotes containing the words formulas, fundamental and/or recurrence:

“It is sentimentalism to assume that the teaching of life can always be fitted to the child’s interests, just as it is empty formalism to force the child to parrot the *formulas* of adult society. Interests can be created and stimulated.”

—Jerome S. Bruner (20th century)

“The *fundamental* steps of expansion that will open a person, over time, to the full flowering of his or her individuality are the same for both genders. But men and women are rarely in the same place struggling with the same questions at the same age.”

—Gail Sheehy (20th century)

“Forgetfulness is necessary to remembrance. Ideas are retained by renovation of that impression which time is always wearing away, and which new images are striving to obliterate. If useless thoughts could be expelled from the mind, all the valuable parts of our knowledge would more frequently recur, and every *recurrence* would reinstate them in their former place.”

—Samuel Johnson (1709–1784)