An Egyptian fraction is the 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 a/b; for instance the Egyptian fraction above sums to 43/48. Every positive rational number can be represented by an Egyptian fraction. Sums of this type, and similar sums also including 2/3 and 3/4 as summands, were used as a serious notation for rational numbers by the ancient Egyptians, and continued to be used by other civilizations into medieval times. In modern mathematical notation, Egyptian fractions have been superseded by vulgar fractions and decimal notation. However, Egyptian fractions continue to be an object of study in modern number theory and recreational mathematics, as well as in modern historical studies of ancient mathematics.
Other articles related to "fractions, egyptian fraction, egyptians, egyptian fractions, fraction":
... The search for expansions of rational numbers as sums of unit fractions dates to the mathematics of ancient Egypt, in which Egyptian fraction expansions of this type were ... The Egyptians produced tables such as the Rhind Mathematical Papyrus 2/n table of expansions of fractions of the form 2/n, most of which use either two or three terms ... The greedy algorithm for Egyptian fractions, first described in 1202 by Fibonacci in his book Liber Abaci, finds an expansion in which each successive term is the largest unit fraction that is no ...
... notable problems remain unsolved with regard to Egyptian fractions, despite considerable effort by mathematicians ... length of the shortest expansion for a fraction of the form 4/n ... is known to be true for all n < 1014, and for all but a vanishingly small fraction of possible values of n, but the general truth of the conjecture remains unknown ...
... The unit fractions formed by the reciprocals of the values in Sylvester's sequence generate an infinite series The partial sums of this series have a simple form, This may be proved by induction, or more directly by ... For example, the first four terms add to 1805/1806, and therefore any Egyptian fraction for a number in the open interval (1805/1806,1) requires at least five terms ... as the result of a greedy algorithm for Egyptian fractions, that at each step chooses the smallest possible denominator that makes the partial sum of the series be less than ...
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