# e (number)

The number e, like the number **pi**, is a useful mathematical constant that is the basis of the system of natural **logarithms**. Its value correct to nine places is 2.718281828... The number e is used in complex equations to describe a process of growth or decay. It is therefore utilized in the **biology**, business, demographics, **physics**, and **engineering** fields.

The number e is widely used as the base in the exponential **function** y = Ce^{kx}. There are extensive tables for e^{x}, and scientific calculators usually include an e^{x} key. In **calculus**, one finds that the slope of the graph of e^{x} at any **point** is equal to e^{x} itself, and that the **integral** of e^{x }is also e^{x} plus a constant.

Exponential functions based on e are also closely related to sines, cosines, hyperbolic sines, and hyperbolic cosines: e^{ix} = cos x + isin x; and e^{x} = cosh x + sinh x. Here i is the **imaginary number** RADIC-1. From the first of these relationships one can obtain the curious equation e^{iPI} + 1 = 0, which combines five of the most important constants in **mathematics**.

The constant e appears in many other formulae in **statistics**, science, and elsewhere. It is the base for natural (as opposed to common) logarithms. That is, if e^{x} = y, then x = ln y. (ln x is the symbol for the natural logarithm of x.) ln x and e^{x} are therefore inverse functions.

The expression (1 + 1/n)^{n} approaches the number e more and more closely as n is replaced with larger and larger values. For example, when n is replaced in turn with the values 1, 10, 100, and 1000, the expression takes on the values 2, 2.59..., 2.70..., and 2.717....

Calculating a decimal **approximation** for e by means of the this definition requires one to use very large values of n, and the equations can become quite complex. A much easier way is to use the Maclaurin series for e^{x}: e^{x} = 1 + x/1! +x^{2}/2! + x^{3}/3! + x^{4}/4! +.... By letting x equal 1 in this series one gets e = 1 + 1/1 + 1/2 + 1/6 + 1/24 + 1/120 +.... The first seven terms will yield a three-place approximation; the first 12 will yield nine places.

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