What is e?

Less well known than it's popular friend, pi, e is nonetheless a very important number. Approximately equal to 2.718, Euler's number pops up in everything from finance to hats!

But what is it? Let's be curious and ask 'what is e?'

Natural Logarithms

If you have only encountered e once, it was likely to do with natural logarithms (ln). A logarithm gives the power to which the base must be raised to give a number. That may sound complicated, but it is actually quite simple. A base-10 logarithm (commonly referred to as just log) would work like this:

log(1000) = 3

That's because 10 raised to the power of 3 (10³) is 1000. In other words 10x10x10=1000

Natural logarithms are logarithms to the base e. That is it gives the power to which e must be raised to give the number. For example:

ln(1000) = 6.9078 (to 4 decimal places)

That's because e raised to the power of 6.9078 (e^6.9078) is 1000. In other words exe...6.9078 times = 1000

Sometimes, though we already know the power and want to find the number. In these cases we use the inverse log. For base 10 log, that is just raising 10 to the power (for example 10⁴ is 10,000). For a natural log (base e) it is raising e to the power (for example e⁴ is 54.5982 (to 4 d.p.)).

Properties of e

Like pi, e is an irrational number. That means that there are an infinite number of decimal places and it cannot be expressed as a ratio between two integers. Also like pi, it is non-repeating meaning you can find seemingly significant strings of numbers in it.

e is actually defined as a limit. A limit is what a sequence tends to. That means if you kept going for infinity what would the sequence reach. e is the limit of the following sequence:

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

Where n is an integer (whole number). The above may look complicated, but as is often the case with maths, it is actually rather simple. As n gets larger (closer and closer to infinity) the answer gets closer and closer to the value of e. So, if we put n=10, we get 2.5937 (to 4d.p.). If we now put n=100 we get 2.7048. As you can see we are getting closer and closer to e. Once we put n=2000000 we get 2.718280 which is e correct to 5 decimal places. Although impossible to type into a calculator, if n=infinity then we would get e exactly.

Finance

There are many uses of e. In finance the primary application is within compound interest. Let's say you open a bank account. In that account you deposit £1.00. Now, this bank is a very nice bank and gives you a 100% annual interest rate. That means at the end of the year you will have £2.00 in the account.

But, rather than pay you 100% at the end of the year, the bank decides actually it will give you 50% every 6 months. Does that mean you get 50p every 6 months? No. It means you get 50% every 6 months. After the first 6 months 50p is added to your account meaning you now have £1.50. Six months later another 50% is added. 50% of £1.50 is 75p, so you now have £2.25.

But what about smaller intervals. For example you might have an interest rate of 25% per quarter, or (100/12)% per month. How could we write this as a formula?

$£1\times(1+\frac{1}{n})^n$

That looks very familiar, doesn't it. It is similar to the definition of e. If we take n as the number of compounding intervals (how often the interest is compounded in a given time, for example 12 if we are doing it monthly for a year), then we will get the amount in our bank account after.

But what happens if we continuously compound our money with an infinite number of intervals? We can say that after a length of time t (in years), and an annual rate R the amount of money in our bank account with a starting sum P is given by:

$P\times e^{Rt}$

Let's run a quick example. If we start off with £100 (P=100) in the bank account. The annual interest rate is 9% (R=0.09). The bank uses continuous compounding. How much is in our account after 5 years (t=5)? The answer is £156.83.

Hat Check Problem

The problem goes like this: n guests are invited to a party. Each guest hands their hat to the butler who puts them into a named box. Trouble is, the butler doesn't know anyone's names, so just puts them into random boxes. What's the probability that he puts the hat in the correct box? What's the probability that none of the hats are put in the correct box?

The maths is quite complicated for this problem, and beyond the scope of the blog. But for those interested (and willing to learn some more maths) I recommend looking it up.

Curious Fact

The first 9 digits of pi (3(.)141596265 appears twice in the first two billion digits of e. (www.subidiom.com/pi/)

What other applications of e do you find interesting? Do you have a curious e fact? Let me know what you think in the comments below. As always you can share this post using the links to the left and you can follow It Is All Science with the links to the right.

Remember, it is all science. Let's be curious.