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The odd greedy algorithm cannot terminate when given a fraction with an even denominator, because these fractions do not have finite representations with odd denominators. Therefore, in this case, it produces an infinite series expansion of its input. For instance Sylvester's sequence can be viewed as generated by the odd greedy expansion of 1/2.
An obvious necessary condition is that the starting fraction x / y have an odd denominator y, and it is conjectured but not known that this is also a sufficient condition. It is known [20] that every x / y with odd y has an expansion into distinct odd unit fractions, constructed using a different method than the greedy algorithm.
Every fraction 2 / y requires two terms in its greedy expansion if and only if y ≡ 1 (mod 2); the simplest such fraction is 2 / 3 . A fraction 3 / y requires three terms in its greedy expansion if and only if y ≡ 1 (mod 6), for then −y mod x = 2 and y(y + 2) / 3 is odd, so the fraction remaining after a ...
Fractions: A representation of a non-integer as a ratio of two integers. These include improper fractions as well as mixed numbers . Continued fraction : An expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of ...
For example, 1 / 4 , 5 / 6 , and −101 / 100 are all irreducible fractions. On the other hand, 2 / 4 is reducible since it is equal in value to 1 / 2 , and the numerator of 1 / 2 is less than the numerator of 2 / 4 . A fraction that is reducible can be reduced by dividing both the numerator ...
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.
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 TI-108 is a simple four-function calculator which uses single-step execution.. The immediate execution mode of operation (also known as single-step, algebraic entry system (AES) [7] or chain calculation mode) is commonly employed on most general-purpose calculators.
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