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A simple fraction (also known as a common fraction or vulgar fraction, where vulgar is Latin for "common") is a rational number written as a / b or , where a and b are both integers. [9] As with other fractions, the denominator (b) cannot be zero. Examples include 1 2 , − 8 5 , −8 5 , and 8 −5 .
It is sometimes necessary to separate a continued fraction into its even and odd parts. For example, if the continued fraction diverges by oscillation between two distinct limit points p and q, then the sequence {x 0, x 2, x 4, ...} must converge to one of these, and {x 1, x 3, x 5, ...} must converge to the other.
The real numbers whose continued fraction eventually repeats are precisely the quadratic irrationals. [5] For example, the repeating continued fraction [1;1,1,1,...] is the golden ratio, and the repeating continued fraction [1;2,2,2,...] is the square root of 2. In contrast, the decimal representations of quadratic irrationals are apparently ...
In the U.S. Constitution, the Three-fifths Compromise is part of Article 1, Section 2, Clause 3: . Representatives and direct Taxes shall be apportioned among the several States which may be included within this Union, according to their respective Numbers, which shall be determined by adding to the whole Number of free Persons, including those bound to Service for a Term of Years, and ...
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. [1]
The Rogers–Ramanujan continued fraction is a continued fraction discovered by Rogers (1894) and independently by Srinivasa Ramanujan, and closely related to the Rogers–Ramanujan identities. It can be evaluated explicitly for a broad class of values of its argument. Domain coloring representation of the convergent of the function , where is ...
In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, [a] which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size. With the restricted definition, each Farey sequence starts with the value 0, denoted ...
An Egyptian fraction is a finite sum of distinct unit fractions, such as That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other. The value of an expression of this type is a positive rational number ; for instance the Egyptian fraction ...
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