e (mathematical constant)

The number e is approximately 2.718281828 and is an important transcendental number which arose in the study of compound interest. It may be expressed as the limit of $\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}.$   The function $f(x)=e^{x}$ is called the exponential function, and the natural logarithm is its inverse function, for which e is considered the base.

Quotes

• Euler wrote... in 1748 his Introductio in Analysin Infinitorum, which was intended to serve as an introduction to pure analytical mathematics. ...He ...shewed that the trigonometrical and exponential functions were connected by the relation $\cos \theta +i\sin \theta =e^{i\theta }.$
  • Walter William Rouse Ball, A Short Account of the History of Mathematics (1908)

• The meaning of the differential equation now follows:${\frac {df(t)}{dt}}=Af(t)$expresses the claim that the rate of change in $f(t)$... is proportional at $t$ to $f(t)$ itself.
  And this makes sense. How fast a colony of bacteria will grow is contingent on the... number of bacteria on hand and the relative percentage of bacteria engaged in reproduction. ...
  Equations are... acts of specification in the dark; something answers to some condition. ...Specification in the dark corresponds to the...process by which a sentence in which a pronoun figures—He smokes—acquires the stamp of specificity when the antecedent... is dramatically or diffidently revealed—Winston Churchill, say, or a lapsed smoker seeking an errant cigarette in a bathroom.
  The differential equation describing uniform growth admits a simple but utterly general solution by means of the exponential function$f(t)=ke^{At}$.The number e is an irrational number lying on the leeward side of the margin between 2 and 3 and playing, like $\pi$, a strange and essentially inscrutable role throughout all of mathematics; exponentiation takes e to a power... in this case... specified by A and t. The constant k has an interpretation as the problem's initial value... some... (weight or mass) of bacteria. ...
  as time scrolls backward or forward in the... imagination, $ke^{At}$ provides a running account of growth or decay...
  This is in itself remarkable, the temporal control achieved by what are after all are just symbols, quite unlike anything else in language or its lore or law, but when successful, specification in the dark achieves an analysis of experience that goes beyond any specific prediction to embrace a universe of possibilities loitering discreetly behind the scenes.
  • David Berlinski, The Advent of the Algorithm: the Idea that Rules the World (2000)

• The natural exponential function is identical with its derivative. This is the source of all the properties of the exponential function and the basic reason for its importance in applications.
  • Richard Courant & Herbert Robbins, What is Mathematics? (1940)

• The number e has an established place in mathematics alongside the Archimedian number π ever since the publication in 1748 of Euler's Introductio in Analysin Infinitorum. It provides an excellent illustration of how the principle of monotone sequences can serve to define a new real number.
  • Richard Courant & Herbert Robbins, What is Mathematics? (1940)

• To Euler is due the very remarkable formula $e^{ix}=\cos {x}+i\sin {x},$ which, for $x=\pi$ becomes $e^{i\pi }+1=0,$ a relation connecting five of the most important numbers in mathematics. By purely formal processes, Euler arrived at an enormous number of curious relations, like $i^{i}=e^{-{\frac {\pi }{2}}}.$
  • Howard Eves, An Introduction to the History of Mathematics (1964)

• Euler wrote... Introductio in Analysin infinitorum, 1748, which was intended to serve as an introduction to pure analytical mathematics. ...He ...showed that the trigonometrical and exponential functions are connected by the relation $cos\theta +isin\theta =e^{i\theta }$. Here too we meet the symbol e used to denote the base of the Naperian logarithms, namely the incommensurable number 2.7182818... The use of the single symbol to denote the incommensurable number 2.7182818... seems to be due to Cotes, who denoted it by M. Newton was probably the first to employ the literal exponential notation, and Euler using the form a^z, had taken a as the base of any system of logarithms. It is probable that the choice of e for a particular base was determined by its being a vowel consecutive to a, or, still more probable because e is the initial of the word exponent.
  • Benjamin Franklin Finkel, A Mathematical Solution Book (1893) p. 449.

• The exponential function, y = e^x, is the instrument used, in one form or another, to describe the behavior of growing things. For this it is uniquely suited: it is the only function of x with a rate of change with respect to x equal to the function itself.
  • Edward Kasner, James R. Newman, Mathematics and the Imagination (1940)

• There is a familiar formula—perhaps the most compact and famous of all formulas—developed by Euler from a discovery of De Moivre: $e^{i\pi }+1=0$. ...It appeals equally to the mystic, the scientist, the philosopher, the mathematician.
  • Edward Kasner, James R. Newman, Mathematics and the Imagination (1940)

• Think of it: of the infinity of real numbers, those that are most important to mathematics—0, 1, √2, e and π—are located within less than four units on the number line. A remarkable coincidence? A mere detail in the Creator's grand design? I let the reader decide.
  • Eli Maor, e: The Story of a Number (1994)

External links

A Short Account of the History of Mathematics by Walter William Rouse Ball