# e (Euler's Number -- What is it?)
**URL:** https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458
**Category:** Factual Questions
**Created:** [December 31, 2004, 6:47pm UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458 "2004-12-31T18:47:59Z")
**Posts on this page:** 20
**Page:** 1
### Author:  [@Strinka](https://boards.straightdope.com/u/Strinka)
#### Post date: [December 31, 2004, 6:47pm UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/1 "2004-12-31T18:47:59Z")
There’s this number e that is approximately 2.718281828. e is a special number because it occurs naturally.
Where? What are some examples of e in nature?
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### Author:  [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)
#### Post date: [December 31, 2004, 6:52pm UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/2 "2004-12-31T18:52:29Z")
> [@Strinka](#):
>
> What are some examples of e in natur **e**?
Well, there’s one. 😃
Sorry, couldn’t resist.
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### Author:  [@Scruloose](https://boards.straightdope.com/u/Scruloose)
#### Post date: [December 31, 2004, 6:53pm UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/3 "2004-12-31T18:53:43Z")
Does [this](http://mathforum.org/library/drmath/view/53903.html) help any?
It doesn’t help me any. I’m sure someone will be by shortly who knows this stuff.
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### Author:  [@Spectre\_of\_Pithecanthropus](https://boards.straightdope.com/u/Spectre_of_Pithecanthropus)
#### Post date: [December 31, 2004, 7:20pm UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/4 "2004-12-31T19:20:18Z")
IA Extremely NA mathematician, but IIRC _e_ is simply the number defined so that in calculus, the integral of the function e^x is equal to itself. It’s similar to the way phi is defined to be the number whose square is a unit greater than itself, or pi is the ratio of a circle’s circumference to its diameter.
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### Author:  [@aahala](https://boards.straightdope.com/u/aahala)
#### Post date: [December 31, 2004, 7:36pm UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/5 "2004-12-31T19:36:03Z")
If you had a bunch of married couples and you divided them into men and women, then randomly assigned the women to the men, the chances of no pairs were properly reunited is 1/e or .367879.
This is probabiloity is accurate to six decimal places whether the number of pairs are ten or ten billion or any number in between.
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### Author:  [@Improv\_Geek](https://boards.straightdope.com/u/Improv_Geek)
#### Post date: [December 31, 2004, 7:44pm UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/6 "2004-12-31T19:44:46Z")
There is a book called _Why Do Buses Travel in Three’s_ and your question made me try td dig my copy out. I couldn’t find it but if the answers you get here are not up to snuff then check it out and see what it has to say.
Some links:
[http://mathforum.org/library/drmath/view/53903.html](http://mathforum.org/library/drmath/view/53903.html)
And from another forum, “Euler’s number will arise in any process in which the rate of change of a quantity is proportional to the quantity itself. Radioactive decay, Newton’s Law of Cooling, Population Growth models and continuous compounding of interest are a few examples.” – [Physics Forum](http://physicsforums.com/showthread.php?p=404499#post404499)
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### Author:  [@Bytegeist](https://boards.straightdope.com/u/Bytegeist)
#### Post date: [December 31, 2004, 7:54pm UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/7 "2004-12-31T19:54:35Z")
> [@Strinka](#):
>
> What are some examples of e in nature?
A [catenary curve](http://mathworld.wolfram.com/Catenary.html) — the curve adopted by a rope, string, chain, vine, or what have you when it’s suspended from two fixed points — follows the hyperbolic cosine function (_cosh_), which is essentially the sum of two related exponentials with base _e_.
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### Author:  [@spingears](https://boards.straightdope.com/u/spingears)
#### Post date: [December 31, 2004, 8:36pm UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/8 "2004-12-31T20:36:57Z")
> [@Strinka](#):
>
> There’s this number e that is approximately 2.718281828. e is a special number because it occurs naturally.
> Where? What are some examples of e in nature?
There was an extensive thread on the ‘natural’ number e and pi, relating the radius/diameter of a circle to it’s area and circumference. Scan through General Questions for all the answers.
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### Author:  [@spingears](https://boards.straightdope.com/u/spingears)
#### Post date: [December 31, 2004, 8:39pm UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/9 "2004-12-31T20:39:54Z")
Strinka: Sugest you read “Sticky” #3 in General Questions.
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### Author:  [@tomndebb](https://boards.straightdope.com/u/tomndebb)
#### Post date: [December 31, 2004, 9:43pm UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/10 "2004-12-31T21:43:48Z")
**Strinka** , **spingear** is suggesting that a single lower-case letter “e” is not much of a title heading. It not only fails to _clearly_ identify the type of question you are asking, it is remarkably difficult for us to click on when we have been drinking.
I have taken the liberty of expanding the title.
[/Moderator Mode]
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### Author:  [@Bytegeist](https://boards.straightdope.com/u/Bytegeist)
#### Post date: [December 31, 2004, 10:48pm UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/11 "2004-12-31T22:48:40Z")
> [@tomndebb](#):
>
> **Strinka** , **spingear** is suggesting that a single lower-case letter “e” is not much of a title heading. It not only fails to _clearly_ identify the type of question you are asking, it is remarkably difficult for us to click on when we have been drinking.
So you’d frown on the new post I was about to write, entitled “i”? Or how about “•”, my new thread about multiplication notation? Or " ", a query I had on usage of the non-breaking space character in HTML?
Anyway, however terse the “e” title might have been, note that because _e[sup]x[/sup]_ is it’s own derivative and anti-derivative, it’s one of the easiest functions to do calculus with when you _have_ been drinking.
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### Author:  [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)
#### Post date: [December 31, 2004, 10:49pm UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/12 "2004-12-31T22:49:17Z")
You need Calculus to truly appreciate _e_. Then you can marvel at facts like
The function e[sup]x[/sup] is its own derivative.
e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + … and
e[sup]x[/sup] = x[sup]0[/sup]/0! + x[sup]1[/sup]/1! + x[sup]2[/sup]/2! + …
e = the limit as n approaches infinity of (1 + 1/n)[sup]n[/sup], and
e[sup]x[/sup] is the limit of (1 + x/n)[sup]n[/sup].
This last fact is why _e_ shows up in the formula for interest compounded continuously (as opposed to a certain number of times per year), as **ronincyberpunk** noted.
_e_ is used in the formula for the bell-shaped curve that shows the standard normal distribution, or Gaussian distribution, that’s all over the place in Statistics.
**spingears** , you may be thinking of this thread: [Pi – it sure gets around](http://boards.straightdope.com/sdmb/showthread.php?t=289542). There’s also [What exactly do e, Log and ln mean in math](http://boards.straightdope.com/sdmb/showthread.php?t=275704)
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### Author:  [@tomndebb](https://boards.straightdope.com/u/tomndebb)
#### Post date: [January 1, 2005, 12:29am UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/13 "2005-01-01T00:29:49Z")
> [@](#):
>
> So you’d frown on the new post I was about to write, . . .
Not at all. I would smile and chortle with merry delight–as I moved it into the unrecoverable æther.
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### Author:  [@spingears](https://boards.straightdope.com/u/spingears)
#### Post date: [January 1, 2005, 1:32am UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/14 "2005-01-01T01:32:42Z")
> [@Thudlow Boink](#):
>
> [What exactly do e, Log and ln mean in math](http://boards.straightdope.com/sdmb/showthread.php?t=275704)
That’s the one! 🙂
> [@Bytegeist](#):
>
> So you’d frown on the new post I was about to write, entitled “i”? Or how about “•”, my new thread about multiplication notation? Or " ", a query I had on usage of the non-breaking space character in HTML?
I would assume that either you hadn’t read the forum stickies or had forgotten the contents. 😃
An appreciation of/for natural numbers will require a broader exposure to mathematics than can be covered by post and links on SDMB’s as time and space do not permit.
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### Author:  [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)
#### Post date: [January 1, 2005, 1:58am UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/15 "2005-01-01T01:58:56Z")
**strinka**
Here’s a thread I started about ‘e’.
> **[a^x = x^a](https://boards.straightdope.com/sdmb/showthread.php?t=276338)**
>
> I think there was a thread similar to this a few months ago but it was the recent thread about logarithms that made me think about this. I’ve researched a great many websites to see why ‘e’ is so special. Oh sure, I get the usual textbook routine -...
**tomndebb**
You think just having ‘e’ was a poor title for a thread?
Geez, I’ve seen a lot worse.
How about those incredibly vague threads that start “Anyone know about this”?
Then clicking on the thread just leads to some inane link:
[www.some-inane-link.com](http://www.some-inane-link.com)
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### Author:  [@Strinka](https://boards.straightdope.com/u/Strinka)
#### Post date: [January 1, 2005, 3:41am UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/16 "2005-01-01T03:41:15Z")
Thank you for fixing the title **tomndebb**.
And thank you to the various people ( **wolf\_meister** , **Bytegeist** , **ronincyberpunk** , **aahala** to name a few) for the informtive answers.
I would ask why e does all this stuff but, I have a feeling that I would need to know calculus to understand the answer and that **Thudlow Boink** answered that question.
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### Author:  [@moriah](https://boards.straightdope.com/u/moriah)
#### Post date: [January 1, 2005, 7:21am UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/17 "2005-01-01T07:21:21Z")
Related anecdote:
While in High School Senior year, I was taking Calculus II. I asked the teacher, “What is _e_? Where does it come from, what is it used for?”
He couldn’t answer the question. He just didn’t know! :eek:
He couldn’t come up with a use for polar coordinates either. It was a math class from hell. Nice way to treat the advance math program, stupid school district. :rolleyes:
Needless to say, I felt compelled to retake calculus in college.
Peees.
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### Author:  [@Ludovic](https://boards.straightdope.com/u/Ludovic)
#### Post date: [January 1, 2005, 8:18am UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/18 "2005-01-01T08:18:33Z")
> [@ronincyberpunk](#):
>
> And from another forum, “Euler’s number will arise in any process in which the rate of change of a quantity is proportional to the quantity itself. Radioactive decay, Newton’s Law of Cooling, Population Growth models and continuous compounding of interest are a few examples.”
Except in cases where you want to know how much of something or other is left after a certain time period (or how long something’s been around,) and your reference materials tell you the _half_-life. Sure, you can contort the formula to use e, but if you really wanted to you could contort the formule to use π. Simpler to use
C[sub]final[/sub]=C[sub]orig/sub[sup]time/halflife[/sup]
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### Author:  [@etv78](https://boards.straightdope.com/u/etv78)
#### Post date: [July 23, 2011, 6:56am UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/19 "2011-07-23T06:56:57Z")
> [@Improv\_Geek](#):
>
> There is a book called _Why Do Buses Travel in Three’s_ and your question made me try td dig my copy out. I couldn’t find it but if the answers you get here are not up to snuff then check it out and see what it has to say.
>
> Some links:
> [http://mathforum.org/library/drmath/view/53903.html](http://mathforum.org/library/drmath/view/53903.html)
>
> And from another forum, “Euler’s number will arise in any process in which the rate of change of a quantity is proportional to the quantity itself. Radioactive decay, Newton’s Law of Cooling, Population Growth models and continuous compounding of interest are a few examples.” – [Physics Forum](http://physicsforums.com/showthread.php?p=404499#post404499)
I asked Dr. Math about Real Life applications of Quadratic Equations. He had to create ahackneyed situation to get a situation.
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### Author:  [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)
#### Post date: [July 23, 2011, 7:04am UTC](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458/20 "2011-07-23T07:04:59Z")
Despite this thread being a zombie, there’s something I’d like to say: a number of purported instances of “e” in nature are really just instances of exponential growth that someone chose to describe in terms of “e”, but which don’t really have anything to do with the particular numeric value 2.71828… in any significant way. Sure, you can choose to see “e” wherever there’s exponential growth, always rewriting b[sup]t[/sup] into e[sup]ln(b)t[/sup], but I don’t think you should think of it as “There’s this magic number 2.71828… that happens to show up in a lot of disparate contexts for mysterious reasons”. It’s just “Exponential growth shows up a lot (and it’s often conveniently described in terms of its natural logarithm)”.
Speaking of which, nor should ln(b) generally be thought of as fundamentally “The power to raise 2.71828… to to reach b”. I think it’s better (particularly when ln(b) is non-real) thought of as fundamentally “The invariant ratio between the t-derivative of b[sup]t[/sup] and b[sup]t[/sup] itself”. The constant defined by e = ln[sup]-1/sup = 2.71828… doesn’t really have anything to do with it; b = e[sup]whatever[/sup] is just a roundabout notation for describing the rate of growth of b[sup]t[/sup].
In short, it’s exponential growth and its measurement via natural logarithm which are ubiquitous; the particular number whose natural logarithm is 1 is also relevant from time to time, but not nearly as often as it is given credit for. The more general idea of exponential growth is the real star.
[Next page](https://boards.straightdope.com/t/e-eulers-number-what-is-it/282458.md?page=2)