Transcendental numbers

Source: Wikipedia. Pages: 23. Chapters: E, Transcendental number, Chaitin's constant, Liouville number, Lindemann¿Weierstrass theorem, Baker's theorem, Schanuel's conjecture, Schneider¿Lang theorem, List of representations of e, Gelfond¿Schneider theorem, Four exponentials conjecture, Gelfond¿Schneider constant, Gelfond's constant, Universal parabolic constant, Cahen's constant, Six exponentials theorem, Gauss's constant, Prouhet¿Thue¿Morse constant, Hilbert number, Hypertranscendental number. Excerpt: The mathematical constant is the unique real number such that the value of the derivative (slope of the tangent line) of the function () = at the point = 0 is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base . The number is also commonly defined as the base of the natural logarithm (using an integral to define the latter), as the limit of a certain sequence, or as the sum of a certain series (see the alternative characterizations, below). The number is sometimes called Euler's number after the Swiss mathematician Leonhard Euler. ( is not to be confused with ¿the Euler¿Mascheroni constant, sometimes called simply Euler's constant.) It is also sometimes known as Napier's constant, although the symbol is in honor of Euler. The number is of eminent importance in mathematics, alongside 0, 1, and . All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity. The number is irrational; it is not a ratio of integers. Furthermore, it is transcendental; it is not a root of any non-zero polynomial with rational coefficients. The numerical value of truncated to 50 decimal places is (sequence A001113 in OEIS). The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant. It is assumed that the table was written by William Oughtred. The discovery of the constant itself is credited to Jacob Bernoulli, who attempted to find the value of the following expression (which is in fact ): The first known use of the constant, represented by the letter , was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler started to use the letter for the constant in 1727 or 1728, and the first use of in a