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The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers' existence was e, by Charles Hermite in 1873. In 1874 Georg Cantor proved that the algebraic numbers are countable and the real numbers are uncountable.
The number e is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function.It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted .
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.
In particular, e 1 = e is transcendental. (A more elementary proof that e is transcendental is outlined in the article on transcendental numbers.) Alternatively, by the second formulation of the theorem, if α is a non-zero algebraic number, then {0, α} is a set of distinct algebraic numbers, and so the set {e 0, e α} = {1, e α} is linearly ...
Later, in the 1850s, he gave a necessary condition for a number to be algebraic, and thus a sufficient condition for a number to be transcendental. [5] This transcendence criterion was not strong enough to be necessary too, and indeed it fails to detect that the number e is transcendental.
In mathematics, the exponential of pi e π, [1] also called Gelfond's constant, [2] is the real number e raised to the power π. Its decimal expansion is given by: e π = 23.140 692 632 779 269 005 72... (sequence A039661 in the OEIS) Like both e and π, this constant is both irrational and transcendental.
More generally, e q is irrational for any non-zero rational q. [13] Charles Hermite further proved that e is a transcendental number, in 1873, which means that is not a root of any polynomial with rational coefficients, as is e α for any non-zero algebraic α. [14]
A power series with coefficients in the field of algebraic numbers = =! ¯ [[]]is called an E-function [1] if it satisfies the following three conditions: . It is a solution of a non-zero linear differential equation with polynomial coefficients (this implies that all the coefficients c n belong to the same algebraic number field, K, which has finite degree over the rational numbers);