E (Number)

The mathematical constant, e, is the base for the natural logarithm. It is sometimes also called Euler’s number after Swiss mathematician Leonhard Euler (1707–1783) and Napier’s constant after Scottish mathematician John Napier (1550–1617).

The number e, like the number pi (the ratio of the circumference to the diameter of a circle), is a useful mathematical constant. Its value correct to ten places is 2.7182818284… The number e is used in complex equations to describe a process of growth or decay. It is, therefore, utilized in such fields as biology, business, demographics, physics, and engineering.

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 √−1. From the first of these relationships, one can obtain the curious equation e^iπ + 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. (The term ln x is the symbol for the natural logarithm of x.) Therefore, ln x and e^x are 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 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. Eventually, twenty decimal places can be reached; thus, e approximately equals 2.71828182845904523536.