A calculus of the absurd

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

$$A = I(1+r)^t$$

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

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

We can now consider an absurd scenario that only a mathematician can pretend is likely to have any relevance to real life5858 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:

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

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

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

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).