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,
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“All finite things reveal infinitude:”
—Theodore Roethke (1908–1963)
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—David Hume (1711–1776)