## £e

Everyone's second favourite transcendental number e has various properties, but I've just been reminded of a less well known one courtesy of Reddit.

Let's appreciate two facts:

1^anything=1

(anything above 1)^(anything above 1)=something above 1 and as the two things grow the answer gets really big really quickly.

But if we took a number only a tiny bit bigger than 1 we could raise it to quite a large number and the result would stay fairly small. So for example 1.1^10=2.59

If we made the number even closer to 1 we could make the power even bigger and still keep the result small. So continuing the pattern:

1.1^10=2.59

1.01^100=2.70

1.001^1000=2.7169

1.0001^10000=2.71814

1.00001^100000=2.71828203321

As the base number tends to 1 we would expect the result to tend to 1. Similarly as the power tends to infinity then we would expect the result to tend to infinity. But by taking both limits at once it looks as if the overall thing has a limit of around 2.7, it turns out that it is e.

Here is an example of where this pattern could play out: let's say you have £1 in a bank account and you get 100% yearly interest. After 1 year you will have £2 in your account.

If instead you are paid the interest of only 10% a time but 10 times in the year then you instead earn 1.1^10=£2.59 because you are being paid interest on your interest.

At only 1% but 100 times a year then you get 1.01^100=£2.70. You'll notice that it is the same pattern as above; by chopping the amount of times you calculate interest into smaller and smaller time steps (by making it closer to continuous rather than discrete) we get closer to having £e.

Putting some mathematical formalism to it we are finding lim(n->infinity)(1+1/n)^n=e