The Discovery of the Number e

The 17$^\text{th}$ century was a time of rapid change. It was the era of the scientific revolution, the proliferation of colonialism, the emergence of mass literacy, and an explosion of international trade.The European age of exploration (and exploitation) brought disparate cultures of the world in contact, conflict, and business with each other to a degree that none of the large empires of old ever approached. Increasing the scale of commerce increased the demands for capital. Money lending began to play an increasingly large role in the prosperity of individuals, business ventures, and nations.

Given the growing presence of finance in the 17$^\text{th}$ century, historians believe that the first person to calculate $e$ was most likely a banker or trader exploring the properties of compound interest.

Interest is a fee charged by a lender on a borrower for the service of providing a loan. Interest fees offset the opportunity cost to the lender of not being able to do anything else with their assets while they are controlled by the borrower. These interest fees accumulate over time, depending on the interest rate. Borrowers think of interest as the cost of having debt, while lenders think of interest as the return on their loan as an investment.

Simple interest is where the interest accumulated is always the same proportion of the initial amount borrowed or invested (this initial amount is called the principal). For instance, if you borrow $1.00 at 5% interest per year, after one year, you will owe $1.05. After two years you will owe $1.10, etc. In effect, simple interest creates an arithmetic progression where a debt or investment grows at the same rate over all periods of time. Simple interest is rare and usually appears only in short term loans.

Compound interest is where interest accumulates on both the principal and the prior interest. Each time the interest rate is applied to the total accumulated debt or investment, it is referred to as compounding. If you invest $1 at 5% annual compounding interest, after the first year you would have $1.05. Unlike simple interest, in the second year it would increase by 5% of $1.05, yielding $1.1025. The general equation for compound interest, compounded once annually, is

$$S=P(1+r)^{t},$$

where $S$ is the total accumulated debt or investment, $P$ is the principal, $r$ is the interest rate, and $t$ is the time elapsed in years. Under compound interest, debts and investments grow by a geometric progression.

Interest can be compounded multiple times within a given year or interest rate period. For instance, if you put one dollar into a bank account that returns 5% interest per year compounded biannually, then your account will grow by 2.5% twice in a given year. For multiple compounding intervals within each rate period, the formula for compound interest becomes

$$S=P \left( 1+\frac{r}{n} \right)^{nt},$$

where $n$ is the number of times the interest is compounded in a given year and $nt$ is therefore the total number of times it is compounded.

Compound Interest Table

Compounding more times in a given time period causes your debt or investment to grow more often, but at a smaller rate each time it is compounded. As you can see, it would not take much imagination for an ambitious banker to wonder how much money could be made if the interest rate was really high and it was compounded as much as possible (daily, hourly, infinitely, etc.). If more compounding intervals make investments grow more often, would more compounding result in investments growing faster?

Historians believe that a businessman or banker likely beats the mathematicians to the thought experiment: if $1 is invested at 100% interest over one year, how much more money will be made if it is compounded often as opposed to only compounding it a few times in a year?

The equation for compound interest in this special case would simplify to

$$S= \left(1+\frac{1}{n} \right)^{n}.$$

It turns out that compounding weekly barely yields any more money than compounding monthly and at higher values of $n$, it gets closer and closer to what we recognize as the number $e$. As you can see, compounding more often does yield more money up to a point, but rapidly reaches an upper bound where growing more often does not yield faster growth. Mathematicians (led by Jacob Bernoulli, in 1683) would later go on to define $e$ as the limit

$$\lim_{n \to \infty} \left( 1+1/n \right)^{n}=e.$$

With $n$ going to infinity, growth occurs continuously, at every possible instant no matter how small the time interval. Simple interest reflects an arithmetic growth progression, compound interest reflects a discrete geometric growth progression, and infinitely compounding interest reflects a continuous growth progression (exponential growth). The number $e$ is thought of as the base that represents the growth of processes or quantities that grow continuously in proportion to their current quantity. This is why $e$ appears so often in modeling the exponential growth or decay of everything from bacteria to radioactivity.

Here is a problem to try.

• Leonhard Euler