What is Continued Fraction?

By Kardi Teknomo, PhD.

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Continued fraction is nested fraction or a sequence of fraction. The following expression is called continued fraction

It forms two series of numbers and . If the sequence all become 1, we have what is called Regular Continued Fraction , written as

The number is called the approximant or the convergence of continued fraction. The variables in the series are natural numbers, which sometimes called element or term . For infinite continued fraction, the approximant will converge to a limit convergence value . Note however that convergence to infinity is also acceptable.

To save space, we usually use the following line notation to express continued fraction

The condense notation of regular continued fraction is . Personally, I prefer this condense notation as long as it is clear that nominator is always one (i.e. regular continued fraction). When we use a continued fraction but the nominators are not necessarily one (i.e. non-regular continued fraction), in this case, we should use line notation as the most condense notation.

In this tutorial, we will focus only to regular continued fraction, which have 1 as nominators.

Example: (how to read the condense notation)

,

,

This tutorial is copyrighted.

Preferable reference for this tutorial is

Teknomo, Kardi. Continued Fraction. http://people.revoledu.com/kardi/tutorial/ContinuedFraction/index.html