Finite Continued Fractions
Every finite continued fraction represents a rational number, and every rational number can be represented in precisely two different ways as a finite continued fraction. These two representations agree except in their final terms. In the longer representation the final term in the continued fraction is 1; the shorter representation drops the final 1, but increases the new final term by 1. The final element in the short representation is therefore always greater than 1, if present. In symbols:
For example,
Read more about this topic: Continued Fraction
Famous quotes containing the words finite and/or continued:
“All finite things reveal infinitude:”
—Theodore Roethke (1908–1963)
“Along the journey we commonly forget its goal. Almost every vocation is chosen and entered upon as a means to a purpose but is ultimately continued as a final purpose in itself. Forgetting our objectives is the most frequent stupidity in which we indulge ourselves.”
—Friedrich Nietzsche (1844–1900)