## You are here

Homenatural log base

## Primary tabs

# natural log base

The *natural log base*, or $e$, has value

$2.718281828459045\ldots$ |

$e$ was extensively studied by Euler in the 1720’s, but it was originally discovered by John Napier.

$e$ is defined by

$\lim_{{n\rightarrow\infty}}\left(1+\frac{1}{n}\right)^{n}$ |

It is more effectively calculated, however, by using the Taylor series for $f(x)=e^{x}$ at $x=1$ to get the representation

$e=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\cdots$ |

Related:

ExampleOfTaylorPolynomialsForTheExponentialFunction, EIsTranscendental,EIsIrrationalProof, ApplicationOfCauchyCriterionForConvergence

Synonym:

Euler number, Eulerian number, Napier's constant, e

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

33B99*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections

## Attached Articles

e is irrational by mathwizard

e is not a quadratic irrational by mathcam

$e^r$ is irrational for $r\in\mathbb{Q}\setminus\{0\}$ by Cosmin

$(1 + 1/n)^n$ is an increasing sequence by rspuzio

values of $\displaystyle \sum_{i = 0}^n \frac{1}{i!}$ for $0 < n < 26$ by PrimeFan

convergence of the sequence (1+1/n)^n by kfgauss70

e is not a quadratic irrational by mathcam

$e^r$ is irrational for $r\in\mathbb{Q}\setminus\{0\}$ by Cosmin

$(1 + 1/n)^n$ is an increasing sequence by rspuzio

values of $\displaystyle \sum_{i = 0}^n \frac{1}{i!}$ for $0 < n < 26$ by PrimeFan

convergence of the sequence (1+1/n)^n by kfgauss70

## Comments

## Euler's Number

Maybe I'm thick, but I think the formula offered

should be flanked by a little explanation.

Why would you want to know about large powers

of a number just above 1 ?

The point is that if you have a list of powers

of, say, 1.00000001, there is one such number

close to any four digit decimal below ten.

If now you wish to multiply two such numbers

you take the closest powers, add the indices

and reverse lookup. This gives an *approximate*

multiplication method, and one sees that *approximate*

exponentiation and root extraction can also be carried

out. (If your result exceeds 10 you have some fancy

footwork to do, but we'll leave the reader to figure

that bit)

The only problem with this is the amount of calculation

involved and the fact that you are forever writing rather

insignificant tail digits. So ? You now divide all your

powers by n and truncate to a suitable number of digits.

This makes the number whose log was n the number whose log

is 1 - shortens your results and in no way affects their

use as approximate logs.

Of course, if you want to concatenate calculations in a

big way you have problems of accuracy and precision, but

we'll leave them for another time.

Ivan Sayer