Where Euler’s number comes from

Euler’s number is one of the best-known mathematical constants, along with the number pi, π. Let’s see how we came across this number and what it means.

In the seventeenth century the Swiss mathematician Jacob Bernoulli was interested in compound interest. Imagine we have one pound and put it in a bank that offers us 100% interest a year. That would be sheer madness on the bank’s part, but what wouldn’t it do for the good of maths? How much money will we have in the account after exactly one year? Well, we will have two pounds.

How would the situation change if the bank offered us 50% interest, paid every six months? Would that be better or worse for us?

After the first six months we would have half as much again. In maths we write “half as much again” as $x\cdot (1+\frac{1}{2})$. We substitute our deposit, one pound, for x and work out

$$1\cdot (1+\frac{1}{2})=1.5$$

After six months we have £1.50 in the account. After the second six months our balance would again grow by half. We use the same formula, only we substitute our new balance for x, that is 1.5:

$$1.5\cdot (1+\frac{1}{2})=2.25$$

After a year we have £2.25 in the account. So paying interest more often at a lower rate works in our favour.

Compounding every month

What if the bank offered to pay interest every month, but the rate dropped to one twelfth of the original rate? Phew, that’s quite a lot of calculating. What will our balance be after the first month? We will have “one twelfth more”, which we write as $x\cdot (1+\frac{1}{12})$. So after the first month we will have:

$$1\cdot (1+\frac{1}{12})=1.0833333\dots$$

After the second month we will have

$$1.0833333\cdot (1+\frac{1}{12})=1.17361111\dots$$

But notice that instead of the number 1.0833333 we can write $1\cdot (1+\frac{1}{12})$, because we know these numbers are equal:

$$1\cdot (1+\frac{1}{12})\cdot (1+\frac{1}{12})=1.17361111\dots$$

After the third month we will have in the account:

$$1\cdot (1+\frac{1}{12})\cdot (1+\frac{1}{12})\cdot (1+\frac{1}{12})=1.27141203$$

As you can see, our calculation follows a pattern. The bracket is repeated as many times as the number of the month we are working out. We can simplify the previous expression by writing one bracket instead of three identical ones and raising it to the power 3:

$$1\cdot {(1+\frac{1}{12})}^3=1.27141203$$

Multiplying by 1 at the start is unnecessary too, so we remove it:

$${(1+\frac{1}{12})}^3=1.27141203$$

To work out the balance after twelve months, we just use the number 12 as the index instead of 3:

$${(1+\frac{1}{12})}^{12}=2.6130352$$

After twelve months we have about £2.61 in the account. That’s not bad! And it’s much better than when we had 100% interest once a year.

Continuous compounding

We don’t have to stop at months. How much would we have if we paid interest every week, at a rate of $\frac{1}{52}$ of the original rate? Since we already know the formula, we just substitute:

$${(1+\frac{1}{52})}^{52}=2.6925969$$

Daily interest?

$${(1+\frac{1}{365})}^{365}=2.7145674$$

Hourly interest? (A year has 8,760 hours.)

$${(1+\frac{1}{8760})}^{8760}=2.718126$$

We can see that each time we take a lower rate and pay it more often, we end up with a little more money in the account. But the gains are no longer so large. We could carry on like this for ever. In fact, that’s not a bad idea – we could pay interest every minute, every second, every millisecond, every nanosecond… until we reached the point of paying interest all the time. We would have a tiny continuous rate, applied to the balance at every instant. How much money would we have in the account after a year in that case?

Back to Euler’s number

That is the question Jacob Bernoulli tried to work out. Unfortunately for him, he didn’t manage it, and it was only in the 18th century that the renowned mathematician Leonhard Euler did. He worked out that after a year we would have in the account

$$2.71828182845904523536028747135\dots$$

pounds. You can see that this is not much more than when we added interest every hour. And because this balance, this number, turned out to be very important in many parts of maths, it is still called Euler’s number today, and we write it with the letter e. Euler’s number is an irrational number, that is, a number with an infinite non-recurring decimal expansion. Just like π, for example. So you will never see all of it. Even so, we can define Euler’s number exactly using the limit of a function:

$$e=\underset{n\to \mathrm{\infty }}{lim}{(1+\frac{1}{n})}^n$$

Sources and links

• A great episode of the Numberphile channel about Euler’s number.
• How to work out the interest on a mortgage