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His proofs are similar to Fourier's proof of the irrationality of e. In 1891, Hurwitz explained how it is possible to prove along the same line of ideas that e is not a root of a third-degree polynomial with rational coefficients, which implies that e 3 is irrational. [12] More generally, e q is irrational for any non-zero rational q. [13]
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Here is a proof by contradiction that log 2 3 is irrational (log 2 3 ≈ 1.58 > 0). Assume log 2 3 is rational. For some positive integers m and n , we have
Rational numbers have irrationality exponent 1, while (as a consequence of Dirichlet's approximation theorem) every irrational number has irrationality exponent at least 2. On the other hand, an application of Borel-Cantelli lemma shows that almost all numbers, including all algebraic irrational numbers , have an irrationality exponent exactly ...
He has reproved Apéry's theorem that ζ(3) is irrational, and expanded it. Zudilin proved that at least one of the four numbers ζ(5), ζ(7), ζ(9), or ζ(11) is irrational. [ 2 ] For that accomplishment, he won the Distinguished Award of the Hardy - Ramanujan Society in 2001.
In mathematics, an irrational number is any real number that is not a rational number, i.e., one that cannot be written as a fraction a / b with a and b integers and b not zero. This is also known as being incommensurable, or without common measure. The irrational numbers are precisely those numbers whose expansion in any given base (decimal ...
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Thus the accuracy of the approximation is bad relative to irrational numbers (see next sections). It may be remarked that the preceding proof uses a variant of the pigeonhole principle: a non-negative integer that is not 0 is not smaller than 1. This apparently trivial remark is used in almost every proof of lower bounds for Diophantine ...