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  2. Liouville number - Wikipedia

    en.wikipedia.org/wiki/Liouville_number

    The terms in the continued fraction expansion of every Liouville number are unbounded; using a counting argument, one can then show that there must be uncountably many transcendental numbers which are not Liouville. Using the explicit continued fraction expansion of e, one can show that e is an example of a transcendental number that is not ...

  3. List of representations of e - Wikipedia

    en.wikipedia.org/wiki/List_of_representations_of_e

    Since e is an irrational number (see proof that e is irrational), it cannot be represented as the quotient of two integers, but it can be represented as a continued fraction. Using calculus, e may also be represented as an infinite series, infinite product, or other types of limit of a sequence.

  4. Euler's continued fraction formula - Wikipedia

    en.wikipedia.org/wiki/Euler's_continued_fraction...

    Euler derived the formula as connecting a finite sum of products with a finite continued fraction. (+ (+ (+))) = + + + + = + + + +The identity is easily established by induction on n, and is therefore applicable in the limit: if the expression on the left is extended to represent a convergent infinite series, the expression on the right can also be extended to represent a convergent infinite ...

  5. Continued fraction - Wikipedia

    en.wikipedia.org/wiki/Continued_fraction

    Lagrange's discovery implies that the canonical continued fraction expansion of the square root of every non-square integer is periodic and that, if the period is of length p > 1, it contains a palindromic string of length p − 1. In 1813 Gauss derived from complex-valued hypergeometric functions what is now called Gauss's continued fractions ...

  6. Continued fraction expansion - Wikipedia

    en.wikipedia.org/wiki/Continued_fraction_expansion

    Upload file; Search. Search. Appearance. ... Download QR code; Print/export ... move to sidebar hide. For the continued fraction expansion. of a number, see simple ...

  7. Lochs's theorem - Wikipedia

    en.wikipedia.org/wiki/Lochs's_theorem

    The theorem states that for almost all real numbers in the interval (0,1), the number of terms m of the number's continued fraction expansion that are required to determine the first n places of the number's decimal expansion behaves asymptotically as follows:

  8. Category:Continued fractions - Wikipedia

    en.wikipedia.org/wiki/Category:Continued_fractions

    In mathematics, regular continued fractions play an important role in representing real numbers, and have a rich general theory touching on a variety of topics in number theory. Moreover, generalized continued fractions have important and interesting applications in complex analysis

  9. Proof that e is irrational - Wikipedia

    en.wikipedia.org/wiki/Proof_that_e_is_irrational

    From this contradiction we deduce that e is irrational. Now for the details. If e is a rational number, there exist positive integers a and b such that e = ⁠ a / b ⁠. Define the number =! (=!). Use the assumption that e = ⁠ a / b ⁠ to obtain =! (=!

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