Euler's Number $(e)$ - Infrequently Asked Questions

• What do I need to know to read this?

Some basic idea of differentiation. Here is a (very) informal introduction.

• So, What is $e$?

$e$ is the unique real constant such that the differential equation $\frac{dy}{dx} = ky$ has a solution of the type $y=Ce^{kx}$, which is equivalent to saying that $e = \lim_{n \to \infty}{\left ( 1+ \frac{x}{n} \right ) ^ {\frac{n}{x}}}$

• Why should I care?

Well, these functions pop up all the time. Here is a classical example:

Say, I have a culture of bacteria which is continuously replicating. It is obvious(why?) that the rate of growth of the culture is proportional to the size of the culture itself.

Let us now try to express the size of the culture ($y$) as a function of time($t$).

Using what we just pointed out, we have:

$\frac{dy}{dt} \propto y \\ \frac{dy}{dt} = ky \quad \text{where k is the rate constant}$

Compare that with the equation in the definition!

So, we have $y(t) = Ce^{kt} \quad \text{where C was the initial size}$

• I am not very fond of bacteria. Nor do I understand why you emphasized continuously over there.

Well, you must know of compound interest.

Say, my principle was $P$, rate $R$ and I compound anually.

• At the end of the first year, I have $(P + PR) = P (1+R)$
• At the end of the second year, I have $P(1+R) + R(P(1+R)) = P(1+R)^2$
• At the end of the tth year, I have $P(1+R)^t$

Okay, now what if I compound half yearly? Wouldn't it be better because I'd have more money that is compounding during the mid-year?

Yeah, that is correct.

• At the end of the first half-year, I have $(P + P\frac{R}{2} = P (1+\frac{R}{2})$
• At the end of the year, I have $P(1+\frac{R}{2}) + \frac{R}{2}(P(1+\frac{R}{2})) = P(1+\frac{R}{2})^{2}$
• At the end of the tth year, I have $P(1+\frac{R}{2})^{2t}$

Well, if I compounded $n$ times an year, I'd be getting $A = P(1+\frac{R}{n})^{n t}$

I could keep compounding continuously too, which would mean the amount is never at rest.

That is $A_{cont} = \lim_{n \to \infty} P(1+\frac{R}{n})^{n t} \\ = \lim_{n \to \infty} P(1+\frac{R}{n})^{\frac{R n t}{R}} \\ = P \lim_{n \to \infty} (1+\frac{R}{n})^{\frac{n}{R}{Rt}} \\ = P e^{Rt}$

• Pretty Cool! Could you show this graphically?

Yes! Look at the top of the note!

Sage Code:

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a = lambda n,t: (1+(1/n))^(n*(floor(n*t)/n))
contd = [plot(lambda t: a(n,t), (0,1),ymin=0,ymax=2.8) for n in xrange(1,31)]
anim = animate(contd)
anim.show()

• You said $e$ is a real number. How do I compute it's value then?

Here is one (simple) way:

Recall the second second definition.

$e = \lim_{n \to \infty} {\left ( 1+ \frac{x}{n} \right ) ^ {\frac{n}{x}}} \\ \implies e^x = \lim_{n \to \infty}{\left ( 1+ \frac{x}{n} \right ) ^ n } \\ = \lim_{n \to \infty}{\left ( 1+ \frac{1}{n} \right ) ^ n} \\ = \lim_{n \to \infty}\sum^{n}_{k=0} {n \choose k} \frac{1}{n^k} \\ = 1+1+\lim_{n \to \infty} \sum_{k=2}^n\frac{n(n-1)(n-2)\cdots(n-(k-1))}{k!\,n^k} \\ = 1+ 1+\frac{1}{2!}\left(1-\frac{1}{n}\right)+\frac{1}{3!}\left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right)+\cdots \\ = 2 + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots$

• I still do not see why that is the unique number as promised in the first definition.

Here you go:

Proposition Let $f : \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f(0) = C$ and $f'(x) = f(x)$. Then it must be the case that $f = Ce^x$.

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Proof Let $g(x) = f(x) \frac{e^{-x}}{C}$. Then

$g'(x) = -f(x) \frac{e^{-x}}{C} + f'(x) \frac{e^{-x}}{C} = (f'(x) - f(x)) \frac{e^{-x}}{C} = 0$

by assumption, so $g$ is constant. But $g(0) = 1$, so $g(x) = 1$ identically.

Hence, $1 = f(x)\frac{e^{-x}}{C} \\ \implies f(x) = Ce^x$

• Why is the logarithm base $e$ termed natural?

Because it is!

You have seen how $e$ relates to every exponential growth scenario, now it is obvious why the natural logarithm relates to time.

The growth velocity of bacteria population varies with it's size. So, the time taken to grow by an amount would vary inversely with it. Hence, we have $\frac{d}{dx}\ln x = \frac{1}{x}$.

More precisely, our bacteria culture which grew as a function of time as $y(t) = Ce^{kt}$ is of a specific size $y$ at time $t = \frac{\ln(\frac{y}{C})}{k}$

What is important is to observe that $\frac{y}{C}$ is the amount by which it grew and $k$ is the rate constant.

• What does it mean to say that $e^{i \pi} + 1 = 0$ ?

That is one of the most beautiful equations of all time!

$e^{i x} = 1 + i x + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \cdots \\ =1 + i x - \frac{x^2}{2!} -\frac{ix^3}{3!} +\frac{(x)^4}{4!} + \cdots \\ = \left(1- \frac{x^2}{2!} + \frac{x^4}{4!}+\cdots\right) + i\left( x- \frac{x^3}{3!}+\frac{x^5}{5!}\cdots\right) \\ = \cos x + i \sin x$

So, $e^{i \theta}$ is actually an unit vector on the complex plane with argument $\theta$. In general, $R e^{i \theta}$ would be a complex number of modulus $R$ and argument $\theta$.

Clearly, $e^{i \pi} = -1 \implies e^{i \pi} + 1 = 0$.

In essence, $e$ forms the basis of exponentiation in the complex plane. Here, we use it to show that $i^i$ is real!

• You still did not answer my question!

I probably do not know what your question is. Feel free to ask me in the comments.