Two questions about Euler's number $e$

I am on derivatives at the moment and I just bumped into this number $e$, "Euler's number" . I am told that this number is special especially when I take the derivative of $e^x$ , because its slope of any point is 1. Also it is an irrational ($2.71828\ldots$) number that never ends, like $\pi$.

So I have two questions, I can't understand

2. Where do we humans use this number that is so useful, how did Mr Euler come up with this number?

and how come this number is a constant? where can we find this number in nature?

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The slope of $e^x$ is $1$ only at $x=0.$ More generally, the slope of $e^x$ at $x$ is $e^x.$ An exponential function, $a^x,$ has the property that its derivative is proportional to the function itself. Only when the base is $e$ is the derivative equal to the function itself. – Will Orrick May 31 '13 at 2:04
This post may be of interest. – Will Orrick May 31 '13 at 2:08
Oh Nice thank you – themhz May 31 '13 at 2:10
If I may recommend a book, check out "e: The story of a number" by Eli Maor. It's an excellent read that talks all about the history of the number e and all of its implications. You can find it here (or of course look for it at a library at school or otherwise): amazon.com/Story-Number-Princeton-Science-Library/dp/0691141347 – cemulate May 31 '13 at 2:21

You don't take the derivative of a constant. You could, but it's zero.

What you should be talking about is the exponential function, $e^x$ commonly denoted by $\exp(\cdot )$. Its derivative at any point is equal to its value, i.e. $\frac{d}{dx} e^x \mid_{x = a} = e^a$. That is to say, the slope of the function is equal to its value for all values of $x$.

As for how to arrive at it, it depends entirely on definition. There are numerous ways to define $e$, the exponential function, or the natural logarithm. One common definition is to define $$\ln x := \int\limits_1^x \frac{1}{t} \ dt$$ From here, you can define $e$ as the sole positive real such that $\ln x = 1$ and arrive at it numerically.

Another common definition is $e = \lim\limits_{n \to \infty}\left(1 + \frac{1}{n}\right)^n$, although in my opinion it is easier to derive properties from the former definition.

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aha I see. And why is it so important? Do we see it nature someware? – themhz May 31 '13 at 1:17
integral from 1 to $x$. – i707107 May 31 '13 at 1:18
Thanks, I have fixed it. – Jon Claus May 31 '13 at 1:19
ok it's in the compounded interest, I checked it out, that's what we humans deside for our interest rate, so this could have been manipulated by humans right? Is it someware that nature creates it by it's self like π happens with the circle? – themhz May 31 '13 at 1:30
just a note: I liked the first sentence quite a bit – DanZimm May 31 '13 at 1:49

Just a slight correction, as Jon Claus notes about the derivative of $e^x$:

what you may be remembering is that "$e$ is the unique real number such that the value of the derivative (slope of the tangent line) of the function $f(x) = e^x$ at the point $x = 0$ is equal to $1$.

See the Wikipedia article on Euler's number $e$ for more fascinating information:

1. The number e is the unique positive real number such that

$$\frac{d}{dt}e^t = e^t.$$

2. The number e is the unique positive real number such that

$$\frac{d}{dt} \log_e t = \frac{1}{t}.$$

The following three characterizations can be proven equivalent:

3. The number e is the limit

$$e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n$$

Similarly:

$$e = \lim_{x\to 0} \left( 1 + x \right)^{1/x}$$

4. The number e is the sum of the infinite series

$$e = \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots$$

where $n!$ is the factorial of n.

5. The number e is the unique positive real number such that

$$\int_1^e \frac{1}{t} \, dt = 1.$$

The number $e$ is of eminent importance in mathematics, alongside $0, 1, \pi, \;\text{and}\; i.$ All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity: $$e^{i\pi} + 1 = 0$$ Like the constant $π, e$ is irrational: it is not a ratio of integers; and it is transcendental: it is not a root of any non-zero polynomial with rational coefficients.

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That $e^{iπ} +1=0$ , that is like a glue of all those sacred numbers? – themhz May 31 '13 at 1:35
man this thing looks so weard, you just made me more curious now. Thank you. Can we find this $e^x$ in nature someware? – themhz May 31 '13 at 1:39
You might also suggest that differential equations governing harmonic oscillators in nature are frequently solved by exponential functions. Exponentials also naturally arise when determining statistics for particle collider events leading to our understanding of the standard model – Dan May 31 '13 at 2:10
Read the Wikipedia article, you'll see how it relates to compound interest; it's also related to the unit circle: Euler's Formula: $e^{i\theta} = \cos \theta + i\sin\theta$, see this link for more, and the exponential function is used in exponential growth and decay processes and models. – amWhy May 31 '13 at 2:11
@themhz I beleive that it kind of occurs as the ptobability of a certain event. If we have $n$ men wearing hats and they put them in a pile then collect some hat then the probability of no man collecting his own hat is approximately $e^{-1}$. This becomes exact if we let $n$ go to infinity – Ben May 31 '13 at 2:15