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Why e?

  1. Apr 18, 2003 #1
    Their must be over a million definitions involving the constant "e". What I would like is a description of the natural logarithm in natural terms, not just saying "e is where

    e
    [inte] dx/x=1, etc."
    1

    In other words, why choose this function to define e, and how does it most fundamentally relate to other uses of e?
     
  2. jcsd
  3. Apr 18, 2003 #2
    That's a good question. I'm sure I could answer it where it not so late, I'll think into it.
     
  4. Apr 18, 2003 #3
    e shows up everywhere is nature. Essentially it describe any phenomena that includes continuous build-up with certain rate. i.e. interests of your bank account, spiral pattern of shell of certain sea creatures. You can easily see from the definition of exponential function.

    e = lim (1 + x) ^(1/x)

    where limit takes x to zero.

    Instanton
     
  5. Apr 18, 2003 #4
    Yes, but that doesn't really explain the question.


    One thing that I remember is that for the graph of ex, the function is its own slope at any point if that helps some insight.
     
  6. Apr 18, 2003 #5
    Then, I don't know what the question is. e is just a trancedental number. Significance, or usefulness comes from the fact that it shows up everywhere in nature. Case for a pi is similar.

    Instanton
     
  7. Apr 18, 2003 #6
    actually the first function we define is not e
    it's natural log
    integrate from 1 to x for dt/t = ln(x)-ln(1)= ln(x)
    because after we define natural logarithm funtion
    we fine the base for the ln function
    so we defined the exponatial e
    hope this can help
     
  8. Apr 18, 2003 #7
    The general idea is we very often have cases where

    rate of change of X proportional to X

    eg X'(t) = c*X(t). For example this very often occurs when X is a number of objects which are independent of one another: reproducing bacteria, decaying atoms. The most mathematically natural case is where c=1; though physically, there is nothing special about that unless the units of X and t are comparable.

    It turns out the solution to this diffeq has the form of an exponential, X(t)=e^t, with e itself.
     
  9. Apr 21, 2003 #8
    Summation the reciprocal of k factorial from 1 to infinitely large is also a representation of e , but it comes from the Talor's theorem and the derivative of e^x.
     
  10. Apr 22, 2003 #9

    dg

    User Avatar

    The first definition I was given for e is as a limit for n->oo of (1+1/n)^n.

    In this form it occurs also in other scenarios like the calculation of continuous interest in finance...

    As far as considering it as a natural base for logarithms... you realize how natural it is only once you start calculating derivatives I guess...

    Another possibility is to say that e is the basis of logarithm when
    lim (log(1+x))/x=1
    x->0
    that is the logarithm that goes to zero as linearly as x does.
     
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