eulers-number

Euler’s Number $e$

Summary: §122 of Chapter 7. The base $a$ is at the analyst’s disposal; choose it so the constant $k$ in $a^{\omega }=1+k\omega$ equals 1. Then the defining series $a=\sum k^n/n!$ collapses to

$e=1+\frac{1}{1}+\frac{1}{1\cdot 2}+\frac{1}{1\cdot 2\cdot 3}+\frac{1}{1\cdot 2\cdot 3\cdot 4}+\cdots =2.71828182845904523536028\dots$

This is the first appearance of the symbol $e$ in mathematical history. The resulting logarithms are called natural or hyperbolic — see natural-logarithm.

Sources: chapter7 (§122)

Last updated: 2026-04-26

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The defining choice

From §116, every base $a>1$ comes paired with a constant $k$ via

$a=1+\frac{k}{1}+\frac{k^2}{1\cdot 2}+\frac{k^3}{1\cdot 2\cdot 3}+\frac{k^4}{1\cdot 2\cdot 3\cdot 4}+\cdots$

For $a=10$, $k\approx 2.30258$. For $a=2$, $k$ is some other finite value. Euler observes (source: chapter7, §122) that we are free to choose the base, and the simplest analytical choice is the one that makes $k=1$.

Substituting $k=1$ in the right side gives

$a=1+\frac{1}{1}+\frac{1}{1\cdot 2}+\frac{1}{1\cdot 2\cdot 3}+\frac{1}{1\cdot 2\cdot 3\cdot 4}+\cdots$

Summing as decimal fractions:

$n$	$1/n!$	partial sum
0	1.00000	1.00000
1	1.00000	2.00000
2	0.50000	2.50000
3	0.16667	2.66667
4	0.04167	2.70833
5	0.00833	2.71667
6	0.00139	2.71806
…	…	…

Euler reports the value to twenty-three digits:

$a=2.71828182845904523536028\dots$

(source: chapter7, §122). He denotes it $e$ — “for the sake of brevity for this number 2.718281828459… we will use the symbol $e$, which will denote the base for natural or hyperbolic logarithms.”

Why “natural” or “hyperbolic”

Euler gives only a brief etymology: the logarithms with this base are called natural because the resulting series are simplest (the factor $1/k$ in §119 disappears, and $\log (1+\omega )=\omega$ for infinitely small $\omega$ — a property unique to base $e$, see §123). They are also called hyperbolic “since the quadrature of a hyperbola can be expressed through these logarithms” — referring to the fact that the area under $y=1/x$ from $1$ to $b$ equals ${\log }_{e}b$. The hyperbolic terminology is older; the natural-logarithm terminology survives in modern use.

Why $k=1$?

The series for $a^z$ is $\sum (kz{)}^n/n!$ (§116). With $k=1$ this becomes the cleanest possible:

$e^z=1+\frac{z}{1}+\frac{z^2}{1\cdot 2}+\frac{z^3}{1\cdot 2\cdot 3}+\frac{z^4}{1\cdot 2\cdot 3\cdot 4}+\cdots$

Every other base requires the constant $k={\log }_{e}a$ to appear in every term, and inverse logarithm series similarly carry a $1/k$ in front. Picking $k=1$ is the analyst’s choice for simplicity, not a forced choice. Euler is explicit (§122) that any base could be used; $e$ is just the most convenient one.

Status as a constant

By the time Euler writes down the series $\sum 1/n!$, he has already established that:

• The series converges, since $1/n!$ decays faster than any geometric.
• Its sum is a real number, computable to arbitrary decimal precision.
• It is not a rational power of any rational base — by §105 applied in reverse, $e$ would have to be transcendental for ${\log }_{e}a$ to be transcendental for “generic” $a$. (Euler does not press this, but the argument is implicit.)

He does not prove $e$ is irrational in §122 — but in §381 Example III Euler discovers (by running the Euclidean algorithm on the decimal expansion of $(e-1)/2$) that $(e-1)/2=[0;1,6,10,14,18,\dots ]$ has partial quotients in arithmetic progression. The infinite, unbounded simple [[continued-fraction-for-e|continued fraction for $e$]] proves irrationality immediately; Euler asserts the pattern “can be confirmed by infinitesimal calculus” and proved it rigorously in 1737 via the Riccati equation. He does not prove $e$ is transcendental either; that is Hermite (1873).

Connection to $(1+1/n{)}^n$

§125 returns to the infinite-power form:

$e^z={\left(1+\frac{z}{j}\right)}^j$

with $j$ infinitely large. Setting $z=1$:

$e={\left(1+\frac{1}{j}\right)}^j.$

In modern notation $e={\lim }_{n\to \infty }(1+1/n{)}^n$ — the standard limit definition. Euler’s $\sum 1/n!$ definition and the limit definition are the binomial expansion of one another, related by the same $(j-m)/j=1$ collapse that produced the exponential series.

What changes once $e$ is on the table

The whole apparatus of Chapter 7 simplifies once the base is $e$:

Identity (general $a$)	Identity (base $e$)
$a^z=\sum (kz{)}^n/n!$	$e^z=\sum z^n/n!$
$\log (1+x)=(1/k)\sum (-1{)}^{n+1}{x}^n/n$	$\log (1+x)=\sum (-1{)}^{n+1}{x}^n/n$
$\log \frac{1+x}{1-x}=(2/k)\sum x^{2n+1}/(2n+1)$	$\log \frac{1+x}{1-x}=2\sum x^{2n+1}/(2n+1)$
$b^z=\sum (kz\log b{)}^n/n!$	$b^z=\sum (z\log b{)}^n/n!$ ($=e^{z\log b}$)

Every other base $a$ is recovered via the conversion factor $k={\log }_{e}a$ — see change-of-base and §124.

Related pages

• exponential-series
• logarithmic-series
• natural-logarithm
• infinitesimal-and-infinite-numbers
• exponential-function
• logarithm
• change-of-base
• chapter-7-on-exponentials-and-logarithms-expressed-through-series
• continued-fraction-for-e — the §381 Example III simple-CF expansion with arithmetic-progression quotients
• chapter-18-on-continued-fractions