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In the first of these equations the ratio tends toward A n / B n as z tends toward zero. In the second, the ratio tends toward A n / B n as z tends to infinity. This leads us to our first geometric interpretation. If the continued fraction converges, the successive convergents A n / B n are eventually arbitrarily close ...
The simplified equation is not entirely equivalent to the original. For when we substitute y = 0 and z = 0 in the last equation, both sides simplify to 0, so we get 0 = 0 , a mathematical truth. But the same substitution applied to the original equation results in x /6 + 0/0 = 1 , which is mathematically meaningless .
Mixture of polymers and solvent on a lattice. Flory–Huggins solution theory is a lattice model of the thermodynamics of polymer solutions which takes account of the great dissimilarity in molecular sizes in adapting the usual expression for the entropy of mixing. The result is an equation for the Gibbs free energy change for mixing a polymer ...
Outside the circle, the continued fraction represents the analytic continuation of the function to the complex plane with the positive real axis, from +1 to the point at infinity removed. In most cases +1 is a branch point and the line from +1 to positive infinity is a branch cut for this function.
The Drake equation is a probabilistic argument used to estimate the number of active, communicative extraterrestrial civilizations in the Milky Way Galaxy. [1][2][3] The equation was formulated in 1961 by Frank Drake, not for purposes of quantifying the number of civilizations, but as a way to stimulate scientific dialogue at the first ...
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 ...
Cross-multiplication. In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable. The method is also occasionally known as the "cross your heart" method because lines resembling ...
Conversely the period of the repeating decimal of a fraction c / d will be (at most) the smallest number n such that 10 n − 1 is divisible by d. For example, the fraction 2 / 7 has d = 7, and the smallest k that makes 10 k − 1 divisible by 7 is k = 6, because 999999 = 7 × 142857. The period of the fraction 2 / 7 is ...