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I Is there a geometrical derivation of e

  1. Jan 12, 2017 #1
    Hi All,

    Mr. James Grime from Numberphile channel has said () that the Euler´s number e has basically nothing to do with geometry.
    I would like to know if there is any derivation of e based on geometrical arguments.

    Best Regards,

    DaTario
     
  2. jcsd
  3. Jan 12, 2017 #2

    Mark44

    Staff: Mentor

    None that I've ever seen...
     
  4. Jan 12, 2017 #3
    Thank you, Mark44.
     
  5. Jan 12, 2017 #4

    fresh_42

    Staff: Mentor

    Probably not what you're looking for, as it is no construction method (as there can't be any with compass and ruler). But you can define ##e## to be the right boundary of the area under the standard hyperbola with its left boundary ##1## that equals ##1##.
    $$ 1 = \int_1^e \frac{1}{x}dx$$
     
  6. Jan 12, 2017 #5
    Yes, fresh_42,

    This kind of geometrical association seems to be more inclined to the calculus itself.
    Like e is the value of a base ## b ## such that ##b^{i \theta}## corresponds to the complete unit circle as ## \theta ## goes from ## 0 ## to ## 2\pi ##.
     
  7. Jan 12, 2017 #6

    fresh_42

    Staff: Mentor

    How would you answer your question if it was ##\pi##? This is clearly a geometric number, but cannot be constructed either. It's the relation between diameter and circumference of a circle or the area of a circle to its squared radius. Circles and hyperbolas are closely related, so I find the definition above quite similar.
     
  8. Jan 12, 2017 #7
    Radius, length, areas (total area), curvature are kind of anatomic parts of geometrical objects. Taking a function and doing a definite integral from a to b is a method which can generate almost anynumber, don´t you agree?
     
  9. Jan 12, 2017 #8
    Take all squares having integer sides, take the inverse of these areas and sum. Finally multiply by 6 and voi la ##\pi ^2##.
     
  10. Jan 12, 2017 #9
    Besides, the position of the hyperbola in relation to the cartesian system plays an important role.

    But the axis in this case are the assimptote...
     
  11. Jan 12, 2017 #10

    fresh_42

    Staff: Mentor

    No, I don't. The integral was simply shorter to write than to make a picture of the area. You can also generate any number as the area of a circle which is also an integral.


    upload_2017-1-13_5-57-46.png
     
  12. Jan 12, 2017 #11
    In the case of circle one can speak of total area. Regarding the hyperbola, it doesn´t happen. I see some difference, but I agree that there is some geometry involved here.
     
  13. Jan 12, 2017 #12
    What is your opinion with respect to the example of the squares above?
     
  14. Jan 12, 2017 #13

    Mark44

    Staff: Mentor

    What does this have to do with e? Wasn't that what you first asked about?
     
  15. Jan 12, 2017 #14
    Sorry Mark44.
     
  16. Jan 12, 2017 #15
    The importance of my question was in determining what is a geometrically based defnition. The use of ##\pi## was an exercise,
     
  17. Jan 12, 2017 #16

    fresh_42

    Staff: Mentor

    Here you can find some more "natural" geometric identities involving ##e##
    https://en.wikipedia.org/wiki/Logarithmic_spiral

    I admit, ##e## is kind of hidden in comparison to ##\pi## and I wouldn't call it a geometric definition as you required. But at least it's not completely out of business.
     
  18. Jan 12, 2017 #17
    Ok, but the identities on this web page seem to be related to the logarithm as an operation which does not select, in principle, a prefered base.
     
  19. Jan 12, 2017 #18

    Mark44

    Staff: Mentor

    The first definition they give, for a logarithmic curve, is ##r = ae^{b\theta}##. The next equation they give is for ##\theta##, in terms of the natural log function, ##\ln##. The parametric equations are given in terms of exponential functions involving e.
     
  20. Jan 12, 2017 #19
    ok, but the parameter b tells us that any base could in principle be adopted, isn´t it?
     
  21. Jan 13, 2017 #20

    Mark44

    Staff: Mentor

    The parameter doesn't have anything to do with it, as far as I can see. You can change an exponential expression in one base to any other base.

    $$e^x = b^{x\log_b(e)}, b > 0, b \ne 1$$
    However, in the wiki article that fresh_42 linked to, the base was e.
     
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