Jacob Bernoulli, Leonhard Euler mathematics |

## Euler’s number

The first irrational constant was *π,*
known as Archimedes’ constant.
The second was *√2,*
known as Pythagoras’ constant.
The third was *e*,
known as Euler’s number.
Jacob Bernoulli was the first
to find Euler’s number
as the result of compounding interest
when the interval approaches zero,
known as continuous compounding.
Leonhard Euler identified *e*
as a base for logarithms
because the number 1
is common to all counting systems,
because the logarithm base *e* of 1 is 1,
and because the derivative
of logarithm base *e* of 1 is 1.

## Transcendental constant

For something characterized as transcendental,
it’s odd that *e* pops up so frequently.
as the result of continuous compounding
as the probability of always losing
in integral expressions for bell curves
in expressions for the rate of radioactive decay
in the formula for distances on a Mercator map
as the area under the hyperbola 1/x
between 1 and a is the natural log of a.

## Approximation

Euler’s number is *approximately* equal to
2.718281828459045235360287471352662497757247093699959
but it is *exactly* equal to
the limit as n approaches infinity
of 1 plus 1 over n to the power of n,
but go ahead and try to write that out.
Without being subject
to the law of diminishing returns,
you may get as close as you like.

$\underset{\mathit{n\to \infty}}{lim}{\left(1+\frac{1}{n}\right)}^{n}=\sum _{n=0}^{\infty}\frac{1}{\mathit{n}!}=e$

Key to the mathematical symbols:

napproaches infinityn)^{n}nto the power ofnn= 0ngoes from zeron!nfactorial, that is, the inverse ofn(n- 1)(n- 2) . . . 1eSee also in

The book of science:Proof of the Pythagorean theorem—PythagorasGraph theory—Leonhard EulerEuler characteristic—Leonhard EulerPi—Johann Heinrich Lambert, Ferdinand von LindemannReadings in wikipedia:

e(mathematical constant)”Other readings:

e,” MacTutor Math History Archive