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In 1840, Liouville published a proof of the fact that e 2 is irrational [10] followed by a proof that e 2 is not a root of a second-degree polynomial with rational coefficients. [11] This last fact implies that e 4 is irrational. His proofs are similar to Fourier's proof of the irrationality of e.
A more recent proof by Wadim Zudilin is more reminiscent of Apéry's original proof, [6] and also has similarities to a fourth proof by Yuri Nesterenko. [7] These later proofs again derive a contradiction from the assumption that ζ ( 3 ) {\displaystyle \zeta (3)} is rational by constructing sequences that tend to zero but are bounded below by ...
The condition that ξ is irrational cannot be omitted. Moreover the constant 5 {\displaystyle {\sqrt {5}}} is the best possible; if we replace 5 {\displaystyle {\sqrt {5}}} by any number A > 5 {\displaystyle A>{\sqrt {5}}} and we let ξ = ( 1 + 5 ) / 2 {\displaystyle \xi =(1+{\sqrt {5}})/2} (the golden ratio ) then there exist only finitely ...
Irrational numbers can also be expressed as non-terminating continued fractions (which in some cases are periodic), and in many other ways. As a consequence of Cantor's proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational. [3]
Written in 1873, this proof uses the characterization of as the smallest positive number whose half is a zero of the cosine function and it actually proves that is irrational. [ 3 ] [ 4 ] As in many proofs of irrationality, it is a proof by contradiction .
However, the numbers and 2 are incommensurable because their ratio, , is an irrational number. More generally, it is immediate from the definition that if a and b are any two non-zero rational numbers, then a and b are commensurable; it is also immediate that if a is any irrational number and b is any non-zero rational number, then a and b are ...
The Thue–Siegel–Roth theorem says that, for algebraic irrational numbers, the exponent of 2 in the corollary to Dirichlet’s approximation theorem is the best we can do: such numbers cannot be approximated by any exponent greater than 2.
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 ...