#e (mathmatical constant)

[ussmathematica]

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This is what makes e so important (in the calculus of functions of real or complex variables).

[ussmathematica]

e^x is special (In CALC) because its derivative at zero is equal to 1

Functions f(x) = ax are shown for several values of a. e is the unique value of a, such that the derivative of f(x) = ax at the point x = 0 is equal to 1. The blue curve illustrates this case, ex. For comparison, functions 2x (dotted curve) and 4x (dashed curve) are shown; they are not tangent to the line of slope 1 and y-intercept 1 (red).

[ussmathematica]

the number "e"

The principal motivation for introducing the number e, particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms.[11] A general exponential function y = ax has derivative given as the limit:

The limit on the right-hand side is independent of the variable x: it depends only on the base a. When the base is e, this limit is equal to one, and so e is symbolically defined by the equation:

Consequently, the exponential function with base e is particularly suited to doing calculus. Choosing e, as opposed to some other number, as the base of the exponential function makes calculations involving the derivative much simpler.

[ussmathematica]

e (mathematical constant)

From Wikipedia, the free encyclopedia

  "Euler's number" redirects here. For γ, a constant in number theory, see Euler's constant. For other uses, see List of topics named after Leonhard Euler#Euler—numbers.

  The number e is an important mathematical constant, approximately equal to 2.71828, that is the base of the natural logarithm.[1] It is the limit of (1 + 1/n)n as n becomes large, an expression that arises in the study of compound interest, and can also be calculated as the sum of the infinite series[2]

The constant can be defined in many ways; for example, e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = ex at the point x = 0 is equal to 1.[3] The function ex so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e. The natural logarithm of a positive number k can also be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case, e is the number whose natural logarithm is 1. There are also more alternative characterizations.

Sometimes called Euler's number after the Swiss mathematician Leonhard Euler, e is not to be confused with γ—the Euler–Mascheroni constant, sometimes called simply Euler's constant. The number e is also known as Napier's constant, but Euler's choice of this symbol is said to have been retained in his honor.[4] The number e is of eminent importance in mathematics,[5] alongside 0, 1, π and i. 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. Like the constant π, e is irrational: it is not a ratio of integers; and it is transcendental: it is not a root of any non-zero polynomial with rational coefficients. The numerical value of e truncated to 50 decimal places is

2.71828182845904523536028747135266249775724709369995... (sequence A001113 in OEIS).