e-

Computer or computer network: e-cash; e-zine. See Usage Note at virtual.

[From E-MAIL.]

The number e is usually defined as the limit approached by the expression as n approaches infinity. If the given expression is expanded by the binomial theorem, and if one uses the theorem that the limit of the quotient of two polynomials of equal degree as the variable tends to infinity is equal to the ratio of the coefficients of highest degree, one obtains the expansion in relation (1). It can be shown
1.
by elementary methods that e is irrational; that is, it cannot be represented as the quotient of two integers. Furthermore, e is transcendental; it does not satisfy any algebraic equation with integral coefficients.

By the method outlined above, it may be shown that the limit of as n tends to infinity is e^x, and moreover that relation (2) holds.
2.
The function e^x is of great importance in mathematical analysis and is encountered in numerous problems in applied mathematics. See also Binomial theorem; Calculus; Logarithm.

Symbol for “90° elbow.”