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! +x²/2! + x³/3! + x⁴/4! +.... 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.