Comparison of Continued Fractions
Consider x = and y = . If k is the smallest index for which ak is unequal to bk then x < y if (−1)k(ak − bk) < 0 and y < x otherwise.
If there is no such k, but one expansion is shorter than the other, say x = and y = with ai = bi for 0 ≤ i ≤ n, then x < y if n is even and y < x if n is odd.
Read more about this topic: Continued Fraction
Famous quotes containing the words comparison of, comparison and/or continued:
“We teach boys to be such men as we are. We do not teach them to aspire to be all they can. We do not give them a training as if we believed in their noble nature. We scarce educate their bodies. We do not train the eye and the hand. We exercise their understandings to the apprehension and comparison of some facts, to a skill in numbers, in words; we aim to make accountants, attorneys, engineers; but not to make able, earnest, great- hearted men.”
—Ralph Waldo Emerson (1803–1882)
“When we reflect on our past sentiments and affections, our thought is a faithful mirror, and copies its objects truly; but the colours which it employs are faint and dull, in comparison of those in which our original perceptions were clothed.”
—David Hume (1711–1776)
“The cause of Sense, is the External Body, or Object, which presseth the organ proper to each Sense, either immediately, as in the Taste and Touch; or mediately, as in Seeing, Hearing, and Smelling: which pressure, by the mediation of Nerves, and other strings, and membranes of the body, continued inwards to the Brain, and Heart, causeth there a resistance, or counter- pressure, or endeavor of the heart, to deliver it self: which endeavor because Outward, seemeth to be some matter without.”
—Thomas Hobbes (1579–1688)