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The Rhind Mathematical Papyrus, [1] [2] an ancient Egyptian mathematical work, includes a mathematical table for converting rational numbers of the form 2/n into Egyptian fractions (sums of distinct unit fractions), the form the Egyptians used to write fractional numbers. The text describes the representation of 50 rational numbers.
By considering the complete quotients of periodic continued fractions, Euler was able to prove that if x is a regular periodic continued fraction, then x is a quadratic irrational number. The proof is straightforward. From the fraction itself, one can construct the quadratic equation with integral coefficients that x must satisfy.
Continued fractions can also be applied to problems in number theory, and are especially useful in the study of Diophantine equations. In the late eighteenth century Lagrange used continued fractions to construct the general solution of Pell's equation, thus answering a question that had fascinated mathematicians for more than a thousand years. [9]
A unit fraction is a common fraction with a numerator of 1 (e.g., 1 / 7 ). Unit fractions can also be expressed using negative exponents, as in 2 −1, which represents 1/2, and 2 −2, which represents 1/(2 2) or 1/4. A dyadic fraction is a common fraction in which the denominator is a power of two, e.g. 1 / 8 = 1 / 2 3 .
The game does not actually teach the player about fractions; [8] the player's score is given in fractions, but no knowledge of them is necessary to play. Its sequel, Frog Fractions 2, announced by a Kickstarter in 2014, was released in December 2016 after players successfully completed a multi-segmented alternative reality game assembled by ...
The convergents of the continued fraction for φ are ratios of successive Fibonacci numbers: φ n = F n+1 / F n is the n-th convergent, and the (n + 1)-st convergent can be found from the recurrence relation φ n+1 = 1 + 1 / φ n. [31] The matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1.
The continued fractions on the right hand side will converge uniformly on any closed and bounded set that contains no poles of this function. [7] In the case , the radius of convergence of the series is 1 and the fraction on the left hand side is a meromorphic function within this circle. The continued fractions on the right hand side will ...
[6] [7] [8] Oliver Heaviside introduced the practical use of fractional differential operators in electrical transmission line analysis circa 1890. [9] The theory and applications of fractional calculus expanded greatly over the 19th and 20th centuries, and numerous contributors have given different definitions for fractional derivatives and ...