# 8.3 Euler’s number

## 8.3.1 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

$A=I(1+r)^{t}$ (8.4)

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

$A=I\left(1+\frac{r}{n}\right)^{nt}$ (8.5)

We can now consider an absurd scenario that only a mathematician can pretend is likely to have any relevance to real life44 4 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:

$\lim_{n\to\infty}I\left(1+\frac{r}{n}\right)^{n}$ (8.6)

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

$e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}$ (8.7)

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).