Purely Periodic and Periodic Fractions
Since all the partial numerators in a regular continued fraction are equal to unity we can adopt a shorthand notation in which the continued fraction shown above is written as
where, in the second line, a vinculum marks the repeating block. Some textbooks use the notation
where the repeating block is indicated by dots over its first and last terms.
If the initial non-repeating block is not present – that is, if
the regular continued fraction x is said to be purely periodic. For example, the regular continued fraction for the golden ratio φ – given by – is purely periodic, while the regular continued fraction for the square root of two – – is periodic, but not purely periodic.
Read more about this topic: Periodic Continued Fraction
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