What are the practical uses of $e$?

Solution 1:

$e$ is not useful. It just happens to be the value at $x = 1$ of the exponential function $$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$

which is incredibly useful because it is its own derivative. This fundamental property helps us solve differential equations, which are a very powerful language for understanding the universe.

The boring example is modeling something like the growth of bacteria. (As Raymond Manzoni's comment says, a more interesting example involving exponential growth is carbon dating.) A much more interesting example (to me, anyway) occurs if you allow $x$ to be something more general than a real number: allow it to be complex and you get Euler's formula $$e^{ix} = \cos x + i \sin x$$

which tells you that the exponential can represent rotations in the plane. It also tells you that $\cos x$ and $\sin x$ can be used to describe simple harmonic motion, a basic and fundamental example of a differential equation describing real-world phenomena (such as circuits) that we know how to solve exactly thanks to the exponential function.

To be more specific, the current $I(t)$ in an LC circuit satisfies the differential equation

$$\frac{d^2 I}{dt^2} + \frac{1}{LC} I = 0$$

where $L$ is the inductance and $C$ is the capacitance. The complex exponential can be used to deduce that the solutions to this differential equation look like

$$I(t) = A \cos \omega t + B \sin \omega t$$

where $\omega^2 = \frac{1}{LC}$. This is a very precise description: the current is periodic and we can explicitly compute its period. Add in a resistor and you get a similar differential equation whose solutions can now be damped, and again we can explicitly compute how quickly the current is damped.

Exponentials are also closely related to the theory of the Fourier transform (which also helps us solve differential equations and much more besides), the normal distribution (which we use to understand statistics), approximating factorials (which we often need to do in computer science)...

Solution 2:

$$\frac{1}{e}=\lim_{n\rightarrow\infty}\left(1-\frac{1}{n}\right)^n$$

So, it appears in a large amount of games, for example, consider a game with $n$ players, where each player has $\frac{1}{n}$ (independent) chances of win. When $n$ growth, the probability that there is no winner is (very quickly) close to $\frac{1}{e}$.

It stays true, even for some cases where it is not independent. Play the game with $n$ balls (with numbers from $1$ to $n$, one on each ball), and $n$ players (each player has a different number). Then each player draws a ball in the bag (he can't see which ball he draws, and he keeps it). Once again, the chance of no one draws his own number is close to $\frac{1}{e}$ (for enough large $n$, a class of 15 children is enough)...

If, in such experiment, you make a redraw until there is no winner, the average number of draws needed is $e$.

Solution 3:

If complex numbers are forbidden (removing much of the fun for the people here...) some practical applications :

• natural exponential decay of radioactive components, concept of half-life (suggestive subject for children!). This is the solution of the simplest differential equation : $x'=-\lambda x$ giving $x=C e^{-\lambda t}$ (many practical application in Natural sciences in the link) exponential decrease of gaz density with altitude and so on...

• simple ecologic system : predator-prey model considering a system of two coupled linear O.D.E system ($x$ is the number of predator and $y$ the number of preys) :
  $x'=(b-py)x$
  $y'=(rx-d)y$
  (well solutions may have $\ln$ too or more complicated (chaotic) things but...)

• use logarithmic scale to explain exponentials growth :

  • seismology : one graduation higher means x times more energy
  • decibels : sound level

• explaining orders of magnitude $e^{\ln(10)x}=10^x$ up to the scale of the universe (there is a nice video on the internet with powers of $10$)