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The mathematical constant e (occasionally called Euler's number after the Swiss mathematician Leonhard Euler, or Napier's constant in honor of the Scottish mathematician John Napier who introduced logarithms) is the base of the natural logarithm function. It is approximately equal to
Alongside the number π and the imaginary unit i, e is one of the most important mathematical constants. It has a number of equivalent definitions; some of them are given below.
The three most common definitions of e are the following.
A proof of the equivalence of these definitions is available here.
Many growth or decay processes can be modeled with an exponential function. The exponential functionThe exponential function is one of the most important functions in mathematics. It is written as exp x or e x where e is the base of the natural logarithm. As a function of the real variable x the graph of e x is always positive (above the x axis) and inc is important because it is the unique function which is its own derivativeCalculus In mathematics, the derivative of a function is one of the two central concepts of calculus. The inverse of a derivative is called the antiderivative, or indefinite integral. The derivative of a function at a certain point is a measure of the rat (up to a constant factor; the most general function that is its own derivative is , for any ).
The number e is known to be both irrationalIn mathematics, an irrational number is any real number that is not a rational number, i. one that cannot be written as a fraction a ''b with a and b integers, and b not zero. It can readily be shown that the irrational numbers are precisely those numbers and transcendentalIn mathematics, a transcendental number is any irrational number that is not an algebraic number, i. it is not the solution of any polynomial equation of the form : where n ≥ 1 and the coefficients a are integers (or, equivalently, rationals), not all. It was the first number to be proved transcendental without having been specifically constructed for this purpose (c.f. Liouville numberIn number theory, a Liouville number is a real number x with the property that, for any positive integer n there exist integers p and q with q > 1 and such that :0 n''. A Liouville number can then be approximated "quite closely" by a sequence of rational); the proof was given by Charles HermiteCharles Hermite (pronounced "air meet ) ( December 24, 1822 January 14, 1901) was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Hermite polynomials, He in 1873Events The United Kingdom declares war against Ghana's King Kofi KariKari, who was involved in the trading of slaves. The war ended by July and the British established the Gold Coast Colony. January 17 Indian Wars: First Battle of the Stronghold during th. It is conjectured to be normalIn mathematics, a normal number is, roughly speaking, a real number whose digits show a random distribution with all digits being equally likely. Digits" refers to the finitely many digits before the point (the integer part) and the infinite sequence of d. It features in Euler's Formula, one of the most important identities in mathematics:
The special case with x = π is known as Euler's identity:
described by Richard Feynman as "Euler's Jewel".
The infinite continued fraction expansion of e contains an interesting pattern that can be written as follows:
