# 1683

## Euler’s number

## The book of science

Tom Sharp

Tom Sharp

Jacob Bernoulli, Leonhard Euler mathematics |

- 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.

- 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.

- 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