*Banca, trapeza, mensa*: the table of the first bankers, between 1450 and 1650. Hence the words bank and banking come from.
Banca, trapeza, mensa: the table of the first bankers, between 1450 and 1650. Hence the words bank and banking come from.Euler’s Eyes Collection

Before reaching Euler, the number $e$ passed through the banks. The vocabulary of money comes from Latin: depositum (what was put, from ponere), creditum (what was trusted, from credere, to believe) and juro (from jus, juris, the right).

Imagine R$ 100 applied to **100% per year**. If the bank pays you just once, at the end of the year, you receive R$ 200. But what if he divides the year into periods and pays interest on interest?

Periods in the year Rate per period Account At the end of the year
1 (annual) 100% $(1 + \frac{1}{1})^1 = 2$ R$ 200,00
2 (six-monthly) 50% $(1 + \frac{1}{2})^2 = 2{,}25$ R$ 225,00
4 (quarterly) 25% $(1 + \frac{1}{4})^4 \approx 2{,}4414$ R$ 244,14
12 (monthly) ≈ 8,33% $(1 + \frac{1}{12})^{12} \approx 2{,}6130$ R$ 261,30
365 (daily) ≈ 0,27% $(1 + \frac{1}{365})^{365} \approx 2{,}7146$ R$ 271,46

With $n$ periods, the capital $C_0$ becomes

$$C_n = C_0\left(1 + \frac{1}{n}\right)^n.$$
The steps are the interest payments. The more periods, the closer the steps get to the dashed curve, and the final value reaches 100 · e.

The final value grows, but not without limit: it approaches a number that is neither an integer nor a fraction.

$$\begin{aligned} \left(1 + \tfrac{1}{10}\right)^{10} &= 2{,}5937\ldots \\ \left(1 + \tfrac{1}{100}\right)^{100} &= 2{,}7048\ldots \\ \left(1 + \tfrac{1}{1000}\right)^{1000} &= 2{,}7169\ldots \\ \left(1 + \tfrac{1}{10\,000}\right)^{10\,000} &= 2{,}7181\ldots \\ \left(1 + \tfrac{1}{100\,000}\right)^{100\,000} &= 2{,}7182\ldots \end{aligned}$$
$$\lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n = e = 2{,}718281828\ldots$$
“1 over infinity, raised to infinity”: the informal way of remembering the limit that gives the number e.
“1 over infinity, raised to infinity”: the informal way of remembering the limit that gives the number e.Euler’s Eyes Collection
John Napier (1550–1617), the inventor of logarithms.
John Napier (1550–1617), the inventor of logarithms.Euler’s Eyes Collection

Jakob Bernoulli had already studied this limit in 1683, without giving it a name. Euler named it $e$ and showed that it is also the sum of a series:

$$e = 1 + \frac{1}{1} + \frac{1}{2 \cdot 1} + \frac{1}{3 \cdot 2 \cdot 1} + \frac{1}{4 \cdot 3 \cdot 2 \cdot 1} + \cdots = \sum_{k=0}^{\infty}\frac{1}{k!}$$

In Introductio (1748), Euler calculated the 23 first decimal places of $e$.

Scribatur pro numero whose logarithmus is unitas, e, qui est 2,7182817… (Write e for the number whose logarithm is the unit.)

— Leonhard Euler, manuscript of 1727–1728

The path to the letter $e$:

  1. 1614
    John Napier publishes Mirifici Logarithmorum Canonis Descriptio, the first table of logarithms.
  2. 1618
    An appendix to the English translation of Napier’s work, made by Edward Wright, contains logarithms in the base that we today call $e$.
  3. 1647
    Grégoire de Saint-Vincent shows that the area under the hyperbola grows as a logarithm.
  4. 1668
    Nicolau Mercator gives the name logarithmus naturalis (natural logarithm) in Logarithmotechnia.
  5. 1683
    Jakob Bernoulli studies the limit of compound interest.
  6. 1727
    Euler uses the letter $e$ in a manuscript, published only in 1862.
  7. 1748
    In the Introductio, Euler writes $e$ as a series and calculates 23 decimal places.
The number e is the logarithm to the base e, which we write ln.
The number e is the logarithm to the base e, which we write ln.Euler’s Eyes Collection

Why “natural”? Because it is the logarithm with the simplest derivative. Using the limit that defines $e$ (with $n = x/\Delta x$, which goes to infinity when $\Delta x$ goes to zero):

$$\begin{aligned} \frac{d}{dx}\ln x &= \lim_{\Delta x \to 0}\frac{\ln(x + \Delta x) - \ln x}{\Delta x} = \lim_{\Delta x \to 0}\frac{1}{\Delta x}\ln\left(1 + \frac{\Delta x}{x}\right) \\ &= \lim_{\Delta x \to 0}\frac{1}{x}\cdot\frac{x}{\Delta x}\ln\left(1 + \frac{\Delta x}{x}\right) = \frac{1}{x}\lim_{n \to \infty}\ln\left(1 + \frac{1}{n}\right)^{n} \\ &= \frac{1}{x}\ln e = \frac{1}{x}. \end{aligned}$$

And then comes the most famous property of $e^x$. As $\ln(e^x) = x$, deriving both sides:

$$\frac{1}{e^x}\cdot\left(e^x\right)' = 1 \quad\Longrightarrow\quad \left(e^x\right)' = e^x.$$

$e^x$ is the function that is equal to the derivative itself. This property is the key to the discovery 4, about the differential equations.

All topics by Leonhard Euler · ← The Basel problem · Euler’s formula and the “most beautiful equation” →

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