This expression is called the continued fraction representation of the number.

It is sometimes necessary to separate a continued fraction into its even and odd parts.

This problem was solved during the 18th century by means of continued fractions.

It is used to show the repeating terms in a periodic continued fraction.

These numbers also appear in the continued fraction convergents to 2.

His work included the development of continued fractions and a method for their representation.

The continued fraction representation of an irrational number is unique.

We get other extremely large numbers as part of the continued fraction if we continue.

If a real number r is written as a simple continued fraction:

Every rational number has an essentially unique continued fraction representation.

To illustrate the use of generalized continued fractions, consider the following example.

It can be shown that this continued fraction converges in all cases.

However, he still made large contributions to number theory, infinite series and continued fractions.

This particular expansion is known as Lambert's continued fraction and dates back to 1768.

In the following year Galois's first paper, on continued fractions, was published.

He made contributions to the theory of continued fractions.

It may be of interest to represent them using continued fractions to perform various studies, including statistical analysis.

Most irrational numbers do not have any periodic or regular behavior in their continued fraction expansion.

Rational numbers have two continued fractions; the version in this list is the shorter one.

Unfortunately, this particular continued fraction does not converge to a finite number in every case.

Another meaning for generalized continued fraction is a generalization to higher dimensions.

Continued fraction analysis reveals that if is rational, its denominator must be greater than 10.

The square root of two has the following continued fraction representation:

The continued fraction expansion of Champernowne's constant has been studied as well.

Here are two continued fractions that can be built via Euler's identity.