The Natural Base e

Definition Interest is compounded when the interest earned in each period is added to the balance, so that later interest is computed on the larger balance. If a principal $P > 0$ earns interest at an annual rate $r > 0$, written as a decimal, and the interest is compounded once a year, then after $t$ whole years the balance is

Each year multiplies the balance by the growth factor $1 + r$.

Definition Suppose the annual rate $r$ is compounded $n$ times per year, where $n$ is a positive integer. Each period then applies the rate $\frac{r}{n}$, and after $t$ years, that is after $nt$ whole periods, the balance is

The number $r$ is called the nominal annual rate. Monthly compounding has $n = 12$, quarterly has $n = 4$ and daily is usually taken as $n = 365$.

Definition An infinite list of numbers $a_1, a_2, a_3, \dots$ converges to a real number $L$ if the distance $\lvert a_n - L \rvert$ becomes smaller than any tolerance you choose and stays smaller, as long as $n$ is large enough. More precisely, for every $\varepsilon > 0$ there is a number $N$ such that $\lvert a_n - L \rvert < \varepsilon$ for every $n > N$. The number $L$ is the limit, and we write

Watching the computed values $a_n$ for larger and larger $n$, and seeing more and more of their leading digits stop changing, is called observing numerical convergence.

Definition The number $e$ is the limit

where $n$ runs through the positive integers. The limit exists because the values $(1 + 1/n)^n$ increase with $n$ and all of them stay below $3$, and an increasing list of real numbers that stays below a fixed bound always converges. Since $e > 1$, the function $f(x) = e^x$ is an increasing exponential function with domain all real numbers and range $(0, \infty)$. It is called the natural exponential function. Books introduce $e$ in different ways, and this course defines it by the limit above. The calculus course starts instead from another property of the same number, that $(e^h - 1)/h$ approaches $1$ as $h$ approaches $0$, in the lesson on special limits, and then proves the limit above, so both routes lead to the same number.

Definition For a fixed principal $P$, rate $r > 0$ and time $t > 0$, the balances under finite compounding approach a limit as the number of periods per year grows without bound,

Continuous compounding is the model in which the balance after $t$ years is

To see where $e$ enters, put $m = n/r$. Then

and as $n$ grows so does $m$, which pushes the bracket toward $e$. The number $m$ need not be a whole number. That the bracket still approaches $e$ in that case is proved in calculus in the lesson on limits at infinity, and from there the whole power approaches $e^{rt}$.

Definition To fifteen decimal places

and $e \approx 2.71828$ to five decimal places. The number $e$ is irrational, so it is not a ratio of two integers and its decimal expansion never ends and never repeats. That is a theorem with a proof, not something any table of digits could show. Every decimal you write down for $e$ is an approximation. A much faster way to compute it than compounding is the infinite sum

where $k! = 1 \cdot 2 \cdot 3 \cdots k$.

Definition A table of computed values is numerical evidence. It can suggest that a limit exists and suggest its first digits, but it cannot prove either, because it shows only finitely many terms and each computed value may carry rounding error. A proof is an argument that covers every $n$ at once. For $e$, what is proved is that for every positive integer $n$

Together with the completeness of the real numbers, one form of which says that an increasing list with an upper bound has a limit, this guarantees that $e$ exists. Both inequalities can be proved with the binomial theorem, from the later lesson on the binomial theorem and Pascal’s triangle.