# How does "e" (2.718) help apply to applications/implications in real life?

Jun 2, 2018

Euler's number, $e$, has few common real life applications. Instead, it appears often in growth problems, such as population models. It also appears in Physics quite often.

As for growth problems, imagine you went to a bank where you have 1 dollar, pound, or whatever type of money you have. The bank offers you 100% interest every year. This means that next year you'll have 2 dollars. What a generous bank.

Instead of 100% every year, let's say they offer you 50% every 6 months. In 6 months, you'll 1.5 dollars, and in another 6 months you'll have

1.5+50% "of " 1.5 = 2.25

This is better, actually! Let's take it further.

Now, they give you 25% interest once every 3 months. If you still have 1 dollar in the bank, now you will have

$\text{In three months: " 1+25%"of "1=1+1"/} 4 = 1.25$
"In another three months: " 1.25+25%"of "1.25 = 1+1"/"4+1"/"4*(1+1"/"4)=(1+1"/"4)(1+1"/"4)=(1+1"/"4)^color(red)2
"Yet again: " (1+1"/"4)^2 + 1"/"4*(1+1"/"4)^2=(1+1"/"4)(1+1"/"4)^2=(1+1"/"4)^color(red)3

If we repeat the process, at the end of the year you will have ${\left(1 + 1 \text{/} 4\right)}^{4}$ dollars.

We can see a pattern! If we take a general case, say you get 100"/"n% interest every $12 \text{/} n$ months and you begin with 1 dollar, at the end of the year you will have

${\left(1 + \frac{1}{n}\right)}^{n}$ dollars.

So, we saw that it was advantageous to get a smaller interest over shorter intervals of time. Let's confirm this; let $f \left(n\right)$ define how much money you get after one year with 100"/"n% interest over $12 \text{/} n$ months:

$f \left(1\right) = 2$
$f \left(2\right) = 2.25$
$f \left(3\right) \approx 2.37$
$f \left(4\right) \approx 2.44$
$f \left(5\right) \approx 2.49$

Yes, it does increase, but it seems to be slowing down, converging to a value even. But what is this value?

Well, let's say your bank does the impossible and offers you an interest with $n$ going to infinity basically every nanosecond (in fact, much much faster than that). By the end of the year, you'll have:

${\lim}_{n \to \infty} f \left(n\right) = {\lim}_{n \to \infty} {\left(1 + \frac{1}{n}\right)}^{n} = \textcolor{red}{e}$

This is one of the definitions of $e$.

But this is not exactly practical, because real life banks don't work this way. However, it does offer us a pretty good image of how $e$ impacts growth.

I will continue this in another answer.

Jun 2, 2018

Continuing...

Another application of $e$ is in population models.

Suppose you have a population with $p$ people and that this population doubles every 30 years. After 180 years, say, the population will double $180 \text{/} 30 = 6$ times.

So the number of people after 180 years, which we will denote as $P$, is

$P = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot p = {2}^{6} p$

Now, we wish to find the instantenous rate of growth of the population. If we find it, it will be helpful to maybe compare it to former rates and form a pretty good impression of what the future holds. This is where $e$ comes in handy.

The population after $t$ years is going to be $P = {2}^{t \text{/} 30} p$.
Now, the instanteneous rate of change represents how much the population will have grown in an infinitesimal amount of time.
Basically, we ask what will $P$ be after a really, REALLY small period of time, like $t = {10}^{- 100}$ seconds?

If we denote the infinitesimal interval of time to be $\text{d} t$ and the effect it has on $P$ be $\text{d} P$ (which is also an infinitesimal unit), instanteneous the rate of change will be

$\left(\text{d"P)/("d"t) = (p*log_color(red)e 2)/30* 2^(t"/"30) =(p*ln2)/30 * 2^(t"/} 30\right)$

In mathematics, we usually just write ${\log}_{e}$ as $\ln$, the natural logarithm. Also, $\text{d}$ is not a constant, but rather a symbol which declares that $\text{d} P$ and $\text{d} t$ are infinitesimals.

Of course, $e$ continues to appear in growth and decay situations, but let's change subject to Physics aswell as other curiosities.

Appearances of e in Physics

The role $e$ has in Physics is somewhat complex. As it is not really my domain, I'll just offer a brief introduction.

In statistical mechanics, the Boltzmann distribution is a probability measure that gives the probability that a system will be in a certain state in terms of that state’s energy and the temperature of the system.

This is all pretty complicated stuff, especially for a precalculus student. To simplify, let's say the system can only have 2 different states. Then, the probability that it will be in one of the two states, ${p}_{1}$ for the first state and ${p}_{2}$ for the second one respectively, are:

{(p_1 = e^(-E_1"/"kT)/(e^(-E_1"/"kT)+e^(-E_2"/"kT))),(p_2=e^(-E_2"/"kT)/(e^(-E_1"/"kT)+e^(-E_2"/"kT))) :}

Where:

$\left\{\begin{matrix}T = \text{the temperature of the system" \\ E_1 and E_2 = "the different energies of the two possible states" \\ k = "Boltzmann constant" ~~1.38065 xx 10^-23 "Joules/Kelvin}\end{matrix}\right.$

Curiosities

You can often find $e$ in many probability questions and in game theory, a branch of Mathematics.
However, for the example I'm going to give, we're going to talk about sticks, just to show how far from standard Math $e$ can appear.

Let's say we have a stick of lenght $L$. We are faced with a question; in how many equal pieces should we break the stick into such that the product of their lenghts is as big as possible?

The answer is, quite surprinsingly,$\lfloor L \text{/} e \rceiling$ pieces, where the brackets represent the self-explanatory nearest integer function.

While $e$ goes on and on and keeps appearing in places where you wouldn't expect it, I will not go over them. Instead, here's a video with a few neat facts about $e$:

Conclusion

Unusually, this answer streak does have a conclusion. I wish to say that the fact that $e$, a number which you most likely won't find out about until highschool, has so many uses and amazing properties, and so I like to look at $e$ as a representative of Mathematical beauty and Mathematics as a whole. This weird and strange number, $2.718 \ldots$, dictates (or at least plays a role in) how the world around us functions, which I believe is mesmerizing.

This just makes you love Mathematics even more, doesn't it?