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
Famous quotes containing the words purely and/or periodic:
“It was a purely wild and primitive American sound, as much as the barking of a chickaree, and I could not understand a syllable of it.”
—Henry David Thoreau (1817–1862)
“But parents can be understanding and accept the more difficult stages as necessary times of growth for the child. Parents can appreciate the fact that these phases are not easy for the child to live through either; rapid growth times are hard on a child. Perhaps it’s a small comfort to know that the harder-to-live-with stages do alternate with the calmer times,so parents can count on getting periodic breaks.”
—Saf Lerman (20th century)