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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 number e is one of only a few transcendental numbers for which the exact ...
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 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 ...
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]
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.
In 1858, Hermite showed that equations of the fifth degree could be solved by elliptic functions. In 1873, he proved that e, the base of the natural system of logarithms, is transcendental. [2] Techniques similar to those used in Hermite's proof of e 's transcendence were used by Ferdinand von Lindemann in 1882 to show that π is transcendental ...
Algebraic number: Any number that is the root of a non-zero polynomial with rational coefficients. Transcendental number: Any real or complex number that is not algebraic. Examples include e and π. Trigonometric number: Any number that is the sine or cosine of a rational multiple of π.