8.3 Euler’s number
8.3.1 Definition of
Note: the A Level doesn’t actually require any knowledge of how 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 (where is the annual rate of interest, e.g. or ). Therefore, after years the amount of money we have, assuming that we started with units would be
Most banks, however, don’t apply interest once a year. Instead, they apply it monthly. If we introduce a new variable, , then we can write the amount of money we have after years as
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:
If we try to simplify things a bit, and set all the constants ( and ) equal to , we can then write that
There’s nothing special about the letter , 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).