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E, pi, phi

  1. Oct 18, 2003 #1
    do those constants have any relation to each other?
    does something like pi-e or pi/e has any significance?
     
  2. jcsd
  3. Oct 18, 2003 #2

    HallsofIvy

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    Well, they are real numbers! Any other relationship I suspect is more "number mysticism" than mathematics. (Phi, in any case, is an algebraic number while e and pi are not.)
     
  4. Oct 18, 2003 #3

    jcsd

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    yes, there are a few identites in maths such as

    ii = e-π/2 and -1 = eπi
     
  5. Oct 18, 2003 #4
    If it's any help these are the power series for [pi] and e:

    Code (Text):

             r=[oo]
    [pi] = 4 * [sum]  ((-1)^r) = 4 - 4 + 4 - 4 + 4
             r=1 (------)       -   -   -   - ... etc.
                 ( 2r-1 )       3   5   7   9

    And

        r=[oo]
    e = [sum]  (   1  ) = 1  + 1  + 1  + 1  + 1           = 1 + 1 + 1 + 1 + 1
        r=1 (------)   --   --   --   --   -- ... etc.           -   -   -- ... etc.
            ((r-1)!)   0!   1!   2!   3!   4!                    2   6   24
     
    [pi] can also be obtained like this:

    x * Sin (180/x) where x is a very large number and 180/x is in degrees.

    I've attached a script to calculate pi and e using the above power series', however I have not been able to calculate pi using the Sin method as JavaScript assumes that the angle is measured in radians and it does not have a built in Math.pi method to allow me to convert the angle from radians into degrees.
    Be careful if you are calculating pi to 1,000,000 iterations, I have an Athlon 1800+ and it caused my PC to hang for a couple of seconds, although I was listening to music at the time.

    If you want to view the source, generally in Windows browsers, you can go View > Source.
     

    Attached Files:

    Last edited: Oct 18, 2003
  6. Oct 18, 2003 #5

    mathman

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    e(pi)i=-1
     
  7. Oct 18, 2003 #6

    Integral

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    A few years back I took Complex Analysis from Dr. King, then Chairman of the Lehigh U Math Department. He spent a fair amount of time with this relationship. He preferred to write it

    eΠi+1=0
    This expression relates 5 of the most important numbers of mathematics, Pi, e, i, 1 and 0 using all of basic mathematical operations, exponentiation, multiplication, and addition. On top of this it is an astounding, nearly unbelievable result.

    He considered it poetry in Mathematics.
     
  8. Oct 19, 2003 #7
    i forgot about this equation.
    any significance to it?
     
  9. Oct 19, 2003 #8
    the condition for the summations in both cases is the same, ie r=infinity r=1.
     
  10. Oct 19, 2003 #9

    dcl

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    What is phi exactly?
    I though it was just another unknown like 'x' 'theta' etc etc


    The above formula can also be expressed as

    e^(i*x) = cos(x) + i*sin(x)



    also 'e' can be derived from

    (1 + (1/k))^k

    as k approaches infinity, the value of 'e' is more accurate.


    Also, if you would like a few million digits of pi, download PiFast and SuperPi and you can calculate them with relative ease :) . Alot of people use these programs to benchmark their overclocked computers and to test stability.
     
  11. Oct 19, 2003 #10
    And I put that, what do you think this is:

    Code (Text):

        r=[oo]
    e = [sum]
        r=1
     
    It's just that if I were to make a script that would run forever you'd never get an answer so what would the point of it be?

    Anyway I've re-posted the script if anyone's interested, it includes the (1 + (1/k))^k way to calculate e.

    By the way, does anyone know the formula for finding the decimal places of [pi]? I have heard of a formula that when you put in a number (say n, for the nth decimal place), you get an answer. I assume there is one for e as well, so does anyone have that?
     

    Attached Files:

  12. Oct 19, 2003 #11

    Hurkyl

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    The golden ratio, (1 + 5^(1/2)) / 2 = 1.618... is often denoted by the symbol φ.
     
  13. Oct 19, 2003 #12
    I've heard of the golden ratio, but what is it used for and why is it golden?
     
  14. Oct 19, 2003 #13

    Hurkyl

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    The ancient greeks thought that the most visually pleasing rectangles had their side lengths in the proportion

    φ : 1


    Such a rectangle, called a golden rectangle, has the property that if you cut a square out of it as follows, the new rectangle has the same proportions as the original rectangle.

    Code (Text):

    +---+--+
    |   |  |
    |   |  |
    |   |  |
    +---+--+
     

    φ, like some other constants, has a tendancy to appear in unexpected places. One of the most interesting is the fact that for n >= 0, the n-th Fibbonachi number can be written as:

    Fn = round( φ^n / sqrt(5) )

    Where "round" means round to the nearest integer.

    The exact formula, incidentally, is:

    Fn = (φ^n - (1 - φ)^n) / sqrt(5)
     
  15. Oct 19, 2003 #14
    Is that assuming that the first two starting numbers are 0 and 1? Is there a formula for finding the nth term for the Fibbonacci sequence that doesn't start with 0 and 1?

    I also thought that the sequence was one of those things that didn't have a formula, I wonder where I got that idea from.
     
  16. Oct 19, 2003 #15

    Hurkyl

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    Yes, I was using F0 = 0 and F1 = 1.


    If you want a different starting point, just substute n with n + k for some k.
     
  17. Oct 19, 2003 #16
    What about values such as 0 and 2?
     
  18. Oct 19, 2003 #17

    Hurkyl

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    The general solution to the recurrence f(n+2) = f(n) + f(n+1) is:

    f(n) = A * φ^n + B * (1 - φ)^n
     
  19. Oct 19, 2003 #18
    Hmmm, sorry about chasing you around with this but, if you put in 0 and 1, for A and B respectively, you don't get:

    Fn = (φ^n - (1 - φ)^n) / sqrt(5)
     
  20. Oct 19, 2003 #19

    Hurkyl

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    Oh, A and B aren't supposed to be terms 0 and 1; they're constants for which you need to solve.
     
  21. Oct 19, 2003 #20
    So I would need the first few terms of the sequence before I could find A and B. OK, fair enough. Maybe I'll find a pattern for the values of A and B for various starting values.

    Thanks for the help.
     
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