A calculus of the absurd

9.3 Euler’s number

9.3.1 Definition of \(e\)

Note: the A Level doesn’t actually require any knowledge of how \(e\) is defined.

Euler’s number is defined in a number of different ways. One way which is quite nice, is to think about compound interest. When you deposit money with a bank, it lends that money to other people, with interest (they borrow money from the bank and then pay back the money, plus a percentage fee). The bank then pays back some of this money to you (or they used to).

We can write a mathematical formula to represent the amount of money that we have after a certain amount of time. Every year, the amount of money in the bank account in question increases by \(1 + p\) (where \(r\) is the annual rate of interest, e.g. \(5\% = 0.05\) or \(0.05\% = 0.00005\)). Therefore, after \(t\) years the amount of money we have, assuming that we started with \(I\) units would be

\begin{equation} A = I(1+r)^t \end{equation}

Most banks, however, don’t apply interest once a year. Instead, they apply it monthly. If we introduce a new variable, \(n\), then we can write the amount of money we have after \(t\) years as

\begin{equation} A = I \left ( 1 + \frac {r}{n} \right ) ^{nt} \end{equation}

We can now consider an absurd scenario that only a mathematician can pretend is likely to have any relevance to real life6464 Somehow, the results of this thought experiment do have a remarkable number of real-world consequences and think about what happens when we apply our interest rate an infinite number of times over one year.

We can use a limit to represent this:

\begin{equation} \lim _{n\to \infty } I \left ( 1 + \frac {r}{n} \right ) ^{n} \end{equation}

If we try to simplify things a bit, and set all the constants (\(I\) and \(r\)) equal to \(1\), we can then write that

\begin{equation} e = \lim _{n\to \infty } \left ( 1 + \frac {1}{n} \right )^{n} \end{equation}

There’s nothing special about the letter \(e\), it’s just what this limit is called in maths (in the same way that there’s nothing special about "gravity" - it’s just a word that is commonly understood to mean that all objects are attracted to each other because they have mass).