The number e | Mathematico

Money lending

Let's suppose I lend you some money, $£ M_{0},$ and because I'm so generous I say you can pay me back double at the end of the year. So, after one year, you owe me

M_{0} \times (1 + 1)

Ok, that's a weird way of writing it, but it's useful because I'm going to make you an offer (because I'm so generous, you see): after each month, I'll increase your debt by a mere $\frac{1}{12}$ and that's how much you owe me. Smaller increments, more often - fair is fair. You're welcome.

Read the small print

The wording of our agreement is very important. The amount you owe is

\begin{aligned} M_{0} (1 + \frac{1}{12}) & after 1 month \\ M_{0} {(1 + \frac{1}{12})}^{2} & after 2 months \\ M_{0} {(1 + \frac{1}{12})}^{3} & after 3 months \\ ⋮ \end{aligned}

What if it still takes you a year to raise the money to pay me back? Then you owe me

M_{0} {(1 + \frac{1}{12})}^{12}

Without a calculator, you might think sure, probably fair. But this is

M_{0} \times 2.613035290 \dots

Are you still happy with this arrangement?

Towards $e$

What about if I offer to increase your debt by a tiny $\frac{1}{365}$ each day? Surely you're happier with that? Then again, at the end of the year you owe

M_{0} {(1 + \frac{1}{365})}^{365} = M_{0} \times 2.714567482 \dots

That's still better for the lender, but they might be a little disappointed that the coefficient leaps from $2$ to $2.6$ (ish) the first time but only $2.6$ to $2.7$ the second time.

But still, the tendency is to push this further: what about every hour?

M_{0} {(1 + \frac{1}{365 \times 24})}^{365 \times 24} = M_{0} \times 2.718126692 \dots

The lender is a little happier! Every minute?

M_{0} {(1 + \frac{1}{365 \times 24 \times 60})}^{365 \times 24 \times 60} = M_{0} \times 2.718279243 \dots

Every second?

M_{0} {(1 + \frac{1}{365 \times 24 \times 60^{2}})}^{365 \times 24 \times 60^{2}} = M_{0} \times 2.718281785 \dots

We're making the coefficient grow less and less each time. It's approaching a limit, and that limit is called the number $e$

The definition of $e$

We define $e$ as

lim_{n \to \infty} {(1 + \frac{1}{n})}^{n}

If you choose a large number for $n,$ say ten million, your calculator gives you

e \approx 2.718281693

whereas the true value of $e$ would be

e = 2.718281828 \dots

The digits would keep going and would have no pattern to them. In fact, $e$ is an irrational number like $π$ or $\sqrt{2} .$

The most money the lender could possibly make by increasing the frequency of the compounding is

M_{0} \times e

Differentiating $e^{x}$

We're going to use differentiation from first principles to see why

\frac{d}{d x} e^{x} = e^{x}

Let $f (x) = e^{x} .$ Then

\begin{aligned} f^{'} (x) & = lim_{h \to 0} \frac{f (x + h) - f (x)}{h} \\ = lim_{h \to 0} \frac{e^{x + h} - e^{x}}{h} \\ = lim_{h \to 0} \frac{e^{x} (e^{h} - 1)}{h} \\ = e^{x} lim_{h \to 0} \frac{e^{h} - 1}{h} \end{aligned}

Now let $k = e^{h} - 1,$ so $h = \ln (1 + k)$ and

k \to 0 \Leftrightarrow h \to 0

f^{'} (x) = e^{x} lim_{k \to 0} \frac{k}{\ln (1 + k)}

Finally, let $n = \frac{1}{k}$ so

k \to 0 \Leftrightarrow n \to \infty

Then we get

\begin{aligned} f^{'} (x) & = e^{x} lim_{n \to \infty} \frac{1}{n \ln (1 + \frac{1}{n})} \\ = e^{x} lim_{n \to \infty} \frac{1}{\ln [{(1 + \frac{1}{n})}^{n}]} \\ = e^{x} \frac{1}{\ln (e)} \\ = e^{x} \end{aligned}

That was a lot of work! But we managed to use nothing but the definition of $e$ and some laws of logarithms to prove what we wanted, which is a nice itch to scratch.

The uniqueness of $e$

Is $e^{x}$ the only type of function that differentiates to itself? Suppose $g (x)$ is another function with the property that $g^{'} (x) = g (x)$ and consider, by the quotient rule,

\begin{aligned} \frac{d}{d x} \frac{g (x)}{e^{x}} & = \frac{e^{x} g^{'} (x) - e^{x} g (x)}{e^{2 x}} \\ = \frac{e^{x} g (x) - e^{x} g (x)}{e^{2 x}} \\ = 0 \end{aligned}

which implies

\frac{g (x)}{e^{x}} = k

for some $k \in R .$ So $g (x)$ must be a multiple of $e^{x} .$ This means $e$ really is special! (Note: if we include the property that $g (0) = 1$ then $k = 1$ so $g (x) = e^{x}$ .)

Conclusions

The financial origins of the number $e$ are interesting, and can be adapted into a fun task where students are led to the value of $e$ by greedily trying to swindle more and more money from their customers.

The fact that the fundamental property of $e^{x}$ follows from this definition and first principles is pretty astonishing, but it is to be hoped for because these properties should depend only on the definitions. The working is probably a bit too technical to be enjoyable for most A Level students, but it could be demonstrated at break time if anyone ever asks.

Finally, $e^{x}$ is somehow unique amongst all functions for having this property, which should convince anyone of its importance. $π$ and $ϕ$ steal a lot of the limelight, but $e$ is by far my favourite number.

Money lending ​

Read the small print ​

Towards e ​

The definition of e ​

Differentiating ex ​

The uniqueness of e ​

Conclusions ​