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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

  1. What is so special about this fact that it's slope is always 1?
  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?

themhz
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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 02:04
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    This post may be of interest. – Will Orrick May 31 '13 at 02:08
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    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): http://www.amazon.com/Story-Number-Princeton-Science-Library/dp/0691141347 – cemulate May 31 '13 at 02:21

2 Answers2

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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.

Jon Claus
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    aha I see. And why is it so important? Do we see it nature someware? – themhz May 31 '13 at 01:17
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    integral from 1 to $x$. – Sungjin Kim May 31 '13 at 01:18
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    @themhz, of course, it is the base of the natural logarithm. More seriously, Google didn't turn up any cursory results but I did not spend a long time on it. One important application of it is though is that the exponential function appears in infinitely compounded interest, due to the latter definition. – Jon Claus May 31 '13 at 01:20
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    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 01:30
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    just a note: I liked the first sentence quite a bit – DanZimm May 31 '13 at 01:49
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    Well I guess this thing is in us, if it was in Mr Eulers brain. I did some research on compounded interest just now and then I Tarzanjumbed into some video about decay and exponential growth http://www.youtube.com/watch?v=8GmIJ4qXVv0 and also that this $e^x$ appears when calculating population growth. Still this is a very weared number and it trigers my curiosity for further exploration. But I still have a long road before I finish calculus study and I better focus now. Thank you again – themhz May 31 '13 at 02:07
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    Finding $e$ in nature is a bit tricky. Exponential processes occur all over the place in nature (population growth, radioactive decay, logarithmic spirals, etc.) but these do not involve the number $e$ per se. The most general exponential function can be written $ca^x,$ which can also be written $ce^{kx},$ where $k=\ln a.$ If $x$ represents time, then you have to decide what units to use to measure time, and this choice will affect the value of $k.$ Usually the value of $k$ will not be $1,$ so you won't actually get the function $e^x.$ – Will Orrick May 31 '13 at 02:22
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    One place where $e$ pops out in a natural fashion is in combinatorics and probability. Ben's commment on amWhy's answer provides an example - the problem of derangements. – Will Orrick May 31 '13 at 02:30
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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.

amWhy
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    That $e^{iπ} +1=0$ , that is like a glue of all those sacred numbers? – themhz May 31 '13 at 01:35
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    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 01:39
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    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 – Triatticus May 31 '13 at 02:10
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    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 02:11
  • @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 02:15
  • @themhz, that equations, Euler's Identity, is one of the most famous equalities for that precise reason - it "glues" together quite a bit of mathematics. Equality, addition, multiplication, exponentation; $0, 1, i, \pi, e $. – Jon Claus May 31 '13 at 03:18