Rational numbers

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Sequence rank

The position of a rational number in the sequence , , , , , , , , , , ..., ordered in terms of increasing numerator+denominator.

Rational number

A rational number is a number that can be expressed as a fraction where and are integers and . A rational number is said to have numerator and denominator . Numbers that are not rational are called irrational numbers. The real line consists of the union of the rational and irrational numbers. The set of rational numbers is of measure zero on the real line, so it is "small" compared to the irrationals and the continuum.The set of all rational numbers is referred to as the "rationals," and forms a field that is denoted . Here, the symbol derives from the German word Quotient, which can be translated as "ratio," and first appeared in Bourbaki's Algèbre (reprinted as Bourbaki 1998, p. 671).Any rational number is trivially also an algebraicnumber.Examples of rational numbers include , 0, 1, 1/2, 22/7, 12345/67, and so on. Farey sequences provide a way of systematically enumerating all rational numbers.The..

Devil's staircase

A plot of the map winding number resulting from mode locking as a function of for the circle map(1)with . (Since the circle map becomes mode-locked, the map winding number is independent of the initial starting argument .) At each value of , the map winding number is some rational number. The result is a monotonic increasing "staircase" for which the simplest rational numbers have the largest steps. The Devil's staircase continuously maps the interval onto , but is constant almost everywhere (i.e., except on a Cantor set).For , the measure of quasiperiodic states ( irrational) on the -axis has become zero, and the measure of mode-locked state has become 1. The dimension of the Devil's staircase .Another type of devil's staircase occurs for the sum(2)for , where is the floor function (Böhmer 1926ab; Kuipers and Niederreiter 1974, p. 10; Danilov 1974; Adams 1977; Davison 1977; Bowman 1988; Borwein and Borwein 1993; Bowman..

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