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  2. Proof that e is irrational - Wikipedia

    en.wikipedia.org/wiki/Proof_that_e_is_irrational

    This last fact implies that e 4 is irrational. 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]

  3. Diophantine approximation - Wikipedia

    en.wikipedia.org/wiki/Diophantine_approximation

    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 ...

  4. Irrationality measure - Wikipedia

    en.wikipedia.org/wiki/Irrationality_measure

    The irrationality exponent or Liouville–Roth irrationality measure is given by setting (,) =, [1] a definition adapting the one of Liouville numbers — the irrationality exponent () is defined for real numbers to be the supremum of the set of such that < | | < is satisfied by an infinite number of coprime integer pairs (,) with >.

  5. Square root of 2 - Wikipedia

    en.wikipedia.org/wiki/Square_root_of_2

    The rational approximation of the square root of two derived from four iterations of the Babylonian method after starting with a 0 = 1 (⁠ 665,857 / 470,832 ⁠) is too large by about 1.6 × 1012; its square is ≈ 2.000 000 000 0045.

  6. Mathematical proof - Wikipedia

    en.wikipedia.org/wiki/Mathematical_proof

    Euclid's Elements was read by anyone who was considered educated in the West until the middle of the 20th century. [10] In addition to theorems of geometry, such as the Pythagorean theorem, the Elements also covers number theory, including a proof that the square root of two is irrational and a proof that there are infinitely many prime numbers.

  7. Real number - Wikipedia

    en.wikipedia.org/wiki/Real_number

    The real numbers include the rational numbers, such as the integer −5 and the fraction 4 / 3. The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) are the root of a polynomial with integer coefficients, such as the square root √2 = 1.414...; these are called algebraic numbers.

  8. Proof of impossibility - Wikipedia

    en.wikipedia.org/wiki/Proof_of_impossibility

    One of the widely used types of impossibility proof is proof by contradiction.In this type of proof, it is shown that if a proposition, such as a solution to a particular class of equations, is assumed to hold, then via deduction two mutually contradictory things can be shown to hold, such as a number being both even and odd or both negative and positive.

  9. Wadim Zudilin - Wikipedia

    en.wikipedia.org/wiki/Wadim_Zudilin

    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. [3]