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Math Help - Please help me understand e (math constant)

  1. #1
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    Please help me understand e (math constant)

    I've read through the wiki on e (it's proof of irrationality, and it's definition), and Calc BC.. and I understand all of that... but I can't help but feel like I don't intuitively grasp the number... if that makes any sense.

    Can some one who thinks they have an intuitive grasp of e try to explain it to me, intuitively? (please don't show me any more math expressions... i've seen them already --unless it's in addition to the explanation, i guess)
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    Super Member wingless's Avatar
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    Ok, the history of e starts with a limit, \lim_{n\to \infty} \left ( 1+\frac{1}{n} \right )^n. Mathematicians (Bernoulli, Euler, Leibniz) tried to calculate the limit but couldn't express the number in an elementary way. So this was a new constant, today we call it e which is e \approx 2.71828.


    e^x, the exponential function is the derivative of itself. So we use e while solving the differential equation, y' = y.


    You can also see that we need e while integrating \frac{1}{x}. It's known that \int x^n~dx = \frac{x^{n+1}}{n+1},~n \neq -1. But the integral of 1/x is the logarithm function to the base e, which is the inverse of the exponential function.
    Last edited by wingless; August 9th 2008 at 12:35 AM.
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    MHF Contributor arbolis's Avatar
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    , the exponential function is the derivative of itself. So we use e while solving the differential equation, .
    You're right. But note also that the constant function 0 is also the derivative of itself. And so is the function f(x)=ke^x with k \in \mathbb{R}.
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  4. #4
    Super Member wingless's Avatar
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    Quote Originally Posted by arbolis View Post
    You're right. But note also that the constant function 0 is also the derivative of itself. And so is the function f(x)=ke^x with k \in \mathbb{R}.
    I didn't say it's the only function ;p
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  5. #5
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    but in particular i don't understand how it appears in the laws of thermodynamics and in other formulas concerning particles and heat. i remember reading about it, and there was a formula named by some guy, whose name starts with a B, but i can't remember.. i wish i could.
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    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Skerven View Post
    but in particular i don't understand how it appears in the laws of thermodynamics and in other formulas concerning particles and heat. i remember reading about it, and there was a formula named by some guy, whose name starts with a B, but i can't remember.. i wish i could.
    Are you talking about Stefan's Law? (I think the guy your referring to is Boltzmann...)

    P=\sigma AeT^4??

    --Chris
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  7. #7
    Super Member wingless's Avatar
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    Quote Originally Posted by Skerven View Post
    but in particular i don't understand how it appears in the laws of thermodynamics and in other formulas concerning particles and heat. i remember reading about it, and there was a formula named by some guy, whose name starts with a B, but i can't remember.. i wish i could.
    Oh, it's about expontential growing. I think that guy you say is Bernoulli.

    Quote Originally Posted by Wikipedia
    The compound-interest problem

    Jacob Bernoulli discovered this constant by studying a question about compound interest.
    One simple example is an account that starts with $1.00 and pays 100% interest per year. If the interest is credited once, at the end of the year, the value is $2.00; but if the interest is computed and added twice in the year, the $1 is multiplied by 1.5 twice, yielding $1.001.5 = $2.25. Compounding quarterly yields $1.001.254 = $2.4414, and compounding monthly yields $1.00(1.0833)12 = $2.613035.

    Bernoulli noticed that this sequence approaches a limit (the force of interest) for more and smaller compounding intervals. Compounding weekly yields $2.692597, while compounding daily yields $2.714567, just two cents more. Using n as the number of compounding intervals, with interest of 1⁄n in each interval, the limit for large n is the number that came to be known as e; with continuous compounding, the account value will reach $2.7182818. More generally, an account that starts at $1, and yields (1+R) dollars at simple interest, will yield eR dollars with continuous compounding.

    For more information,

    E (mathematical constant - Wikipedia, the free encyclopedia)
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  8. #8
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    Quote Originally Posted by Skerven View Post
    I don't understand how it appears in the laws of thermodynamics and in other formulas concerning particles and heat.
    Simple. It doesn't. 'e' is not mystical. You are referring to mathematical models of natural phenomena. When you split atoms apart, there won't be little 'e's in there. It's just a model.

    Look up brightness as a function of distance from a light source (the Inverse Square Law). It's a useful model, but it has a problem. Domain restrictions are necessary or it blows up. Since light sources don't normally explode, or get so bright that there is no darkness remaining on Earth, the logical conclusion is to abandon the model when you get too close to the light source.

    All models are like this. They have limitations. Better to study the limitations than worship the mystical properties. The governing principle, I think, was said well by Box and/or Draper

    1) Essentially, all models are wrong, but some are useful.
    2) Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful.
    Last edited by TKHunny; August 9th 2008 at 12:43 PM.
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  9. #9
    Grand Panjandrum
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    Quote Originally Posted by Skerven View Post
    I've read through the wiki on e (it's proof of irrationality, and it's definition), and Calc BC.. and I understand all of that... but I can't help but feel like I don't intuitively grasp the number... if that makes any sense.

    Can some one who thinks they have an intuitive grasp of e try to explain it to me, intuitively? (please don't show me any more math expressions... i've seen them already --unless it's in addition to the explanation, i guess)

    There is nothing to intuitivly understand outside of the collection of properties that flow from its definition, and its definition pops up because it is usefull.

    RonL
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