Euler’s number () is a mathematical constant, approximately , that appears naturally in continuous growth, continuous compounding, differential equations, probability, and calculus.
One definition is
ight)^n.$$ Another is the infinite series $$e=\sum_{n=0}^{\infty}\frac1{n!}=1+1+\frac1{2!}+\frac1{3!}+\cdots.$$ ## Why $e$ is natural The exponential function $e^x$ is its own [[Derivative]]: $$\frac{d}{dx}e^x=e^x.$$ That makes it the cleanest model for quantities whose rate of change is proportional to their current value: $$\frac{dN}{dt}=kN\quad\Rightarrow\quad N(t)=N_0e^{kt}.$$ If $k>0$ this is growth; if $k<0$ it is decay. ## Link to logarithms The natural logarithm and exponential function are [[Inverse functions]] on compatible domains: $$\ln(e^x)=x\quad(x\in\mathbb R),$$ $$e^{\ln y}=y\quad(y>0).$$ So $e^x:\mathbb R\to(0,\infty)$ is [[Bijective function|bijective]] when the codomain is chosen as positive reals. ## Physics use Exponentials appear in radioactive decay, capacitor charging/discharging, damping, thermal relaxation, and linearised systems. The constant is not “just a number”; it is the base that makes continuous-rate equations algebraically simple. Related: [[Logarithm]], [[Exponentiation]], [[Integral]], [[Number sets]]. ## Why it keeps appearing $e$ is the natural base for processes whose rate of change is proportional to their current value. This is why it shows up in compound interest, radioactive decay, population growth, heat flow, probability distributions, and differential equations. A useful characterisation israc{d}{dx}e^x=e^x.
So $e^x$ is the function that is its own derivative. That makes it the default language for continuous change.