# The Natural Constant $e$ Several ways exist for defining the natural constant $e \approx 2.71828$. ## Defining the Natural Constant Based on [[The Exponential Function]] $e$ can be defined as **the value of the exponential function for 1**. $e = \exp (1)$ Approximating $e$ then becomes nothing else than evaluating the exponential function: $e = \exp (1) = \sum_{n=0}^\infty \frac {1} {n!}$. ## Defining the Natural Constant Based on the Derivative of Exponentiation We can define $e$ to be the base of exponentiation whose [[Derivative]] at 0 is 1. $ \frac d {dx} [e^x](0) = 1 $ This definition can be used to show that the derivative of $e^x$ is $e^x$. A variation of the above definition would then be that **$e$ is the base of exponentiation such that the exponentiation's derivative is exponentiation itself**! $ \begin{gather} \frac d {dx} [e^x](0) = \lim_{h \to 0} \frac {e^{0+h} - e^0} h = \lim_{h \to 0} \frac {e^{h} - 1} h = 1 \text{ (per definition)} \end{gather} $ $ \begin{align} \frac d {dx} e^x &= \lim_{h \to 0} \frac {e^{x +h} - e^x} h \\ &= \lim_{h \to 0} \frac {e^x e^h - e^x} h \\ &= \lim_{h \to 0} \frac {e^x \cdot (e^h - 1)} h \\ &= \lim_{h \to 0} e^x \cdot \frac {e^h - 1} h \\ &= e^x \cdot \lim_{h \to 0} \frac {e^h - 1} h \\ &= e^x \end{align} $ ## The Compounding Interest Definition of the Natural Constant The natural constant can also be defined as a solution to the problem of the continuous compounding interest. $\lim_{n \to \infty} \left(1+\frac 1 n \right)^n = e$ ^m74vgi Or more generally: $\lim_{n \to \infty} \left(1+\frac a n \right)^{bn} = e^{ab}$ ^rahqyz