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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.
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 ...
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 ...
The following list includes the continued fractions of some constants and is sorted by their representations. Continued fractions with more than 20 known terms have been truncated, with an ellipsis to show that they continue. Rational numbers have two continued fractions; the version in this list is the shorter one.
For the continued fraction expansion. of a number, see simple continued fraction, of a function, see continued fraction This page was last edited on 11 ...
For example, can be expanded to the periodic continued fraction [;,,,...]. This article considers only the case of periodic regular continued fractions . In other words, the remainder of this article assumes that all the partial denominators a i ( i ≥ 1) are positive integers.
The continued fraction representation of a real number can be used instead of its decimal or binary expansion and this representation has the property that the square root of any rational number (which is not already a perfect square) has a periodic, repeating expansion, similar to how rational numbers have repeating expansions in the decimal ...
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 =! (=!