E (mathematics)

e is a constant real number equal to 2.71828 18284 59045 23536.... Irrational and transcendental, e is the base of the natural logarithms. The inverse function, the exponential function, $\scriptstyle f(x)=e^{x}$, is the only function equal to its derivative, i.e. $\scriptstyle f^{'}(x)=f(x)$. In fact, for any differentiable function u, $\scriptstyle {\frac {d}{dx}}(e^{u})=e^{u}{\frac {du}{dx}}$ and $\scriptstyle \int e^{u}du=e^{u}+C$. For this reason, the exponential function plays a central role in analysis.

e is sometimes called "Euler's number" in honor of the Swiss mathematician Leonhard Euler who studied it and has shown its mathematical importance. Equally, it is sometimes called "Napier's constant" in honor of the Scottish mathematician John Napier who introduced logarithms.

Properties

In 1737, Leonhard Euler proved that e is an irrational number, i.e. it cannot be expressed as a fraction, only as an infinite continued fraction. In 1873, Charles Hermite proved it was a transcendental number, i.e. it is not solution of any polynomial having a finite number of rational coefficients.

e is the base of the natural logarithms. Its inverse, the exponential function, $\scriptstyle f(x)=e^{x}$, is the only function equal to its derivative, i.e. $\scriptstyle f^{'}(x)=f(x)$. This function plays a central role in analysis since, for any differentiable function u, $\scriptstyle {\frac {d}{dx}}(e^{u})=e^{u}{\frac {du}{dx}}$ and $\scriptstyle \int e^{u}du=e^{u}+C$. The solutions of many differential equations are based on those properties.

History

There is no precise date for the discovery of this number. In 1624, Henry Briggs, one of the first to publish a logarithm table, gives its logarithm, but does not formally identify e. In 1661, Christiaan Huygens remarks the match between the area under the hyperbola and logarithmic functions. In 1683, Jakob Bernoulli studies the limit of $\scriptstyle (1+{\frac {1}{n}})^{n}$, but nobody links that limit to natural logarithms. Finally, in a letter sent to Huyghens, Gottfried Leibniz sets e as the base of natural logarithm (even if he names it b).

Definitions

There are many ways to define e. The most common are probably

$e=\lim _{n\rightarrow \infty }(1+{\frac {1}{n}})^{n}$

and

$e={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+{\frac {1}{5!}}+\cdots$