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Simple proportion one that is a one to one function

  1. Jun 8, 2003 #1
    is a simple proportion one that is a one to one function of some sort. I dont understand this term.
     
    Last edited by a moderator: Feb 5, 2013
  2. jcsd
  3. Jun 8, 2003 #2

    Integral

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    To say A is poportional to B is to say that there is some constant k such that A=kB. It makes no other statement or assumptions on the behavior of A and B.

    With that said I am going to bump this into the Math Forum
     
  4. Jun 8, 2003 #3
    I don't want to create a new thread. This thread may be a start for my question:

    I'm trying to determine whether there's a relationship between Phi=Golden Mean, and the natural log base=e. If there's a relationship, (other than by a simple constant - not what I'm looking for) it is almost certainly irrational.

    I'm running into a brick wall; perhaps because there is no relationship. Can anyone help?

    Gratefully,
     
  5. Jun 8, 2003 #4

    HallsofIvy

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    Of course, there's a relationship (there's a "relationship" between any two numbers!) but that's exactly what you say you are not looking for. It's pretty easy to calculate that phi= (1+ sqrt(5))/2 and don't think you'll find any simple relationship between an algebraic number like that and a transcendental number like e.
     
  6. Jun 8, 2003 #5

    Some people might have once said that about Pi!

    Although this number does not seem to be as important as Pi.
     
  7. Jun 8, 2003 #6

    HallsofIvy

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    pi is transcendental like e.

    You are not going to find a "simple" relationship between an algebraic number and a transcendental number.

    Of course, it is possible to make up some formula that changes phi into e. Given any two numbers, it's possible to make up a formula that will change one into the other. That's clearly not what r637h meant.
     
  8. Jun 9, 2003 #7
    I would like to avoid number theory (very uncomfortable):

    Phi is an algebraic number, granted, and satifies the quadratic, x^2-x-1=0. (Although that contains the irrational square root of 5.)

    But e is transcendental, and cannot satisy a quadratic or any other algebraic expression, as far as I know.

    But Phi is also an irrational number, which can satisfy a Taylor Series or be expressed as a continued fraction.

    I thought a number had to be either irrational or algebraic. Is that the flaw in my thinking?

    Or is Phi "unique", in that it is both? Surely not.

    Anyway, Pi and e can easily be related, and although the relationship may be complex, can be demonstrated in several series.

    Am I running around in circles? Is the reasoning non-sequetur?

    Help!
     
    Last edited by a moderator: Jun 9, 2003
  9. Jun 9, 2003 #8

    HallsofIvy

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    No, a number is either "rational" or "irrational".

    Both rational and irrational numbers can be "algebraic".

    An algebraic number is any number that can be found as a root of polynomial equation with integer coefficients. (If "n" is the lowest possible degree of such a polynomial, the roots are "algebraic of order n".)

    Rational numbers are precisely those numbers that are algebraic of order 1: x= a/b if and only if x satifies bx- a= 0, a polynomial equation of order 1 with integer coefficients.

    Square root of 2, on the other hand, is not rational but is algebraic of order 2.

    Phi, since it satisfies the equation x^2- x- 1= 0, is also algebraic of order 2.

    Transcendental numbers are those numbers that are not algebraic of any order. Pi and e are the best known of those, but, technically speaking, "almost all" numbers are transcendental.
     
  10. Jun 9, 2003 #9
    To Halls of Ivy:

    Thanks. I was confusing "rational" with "algebraic."

    Right: Order 2.

    Any thoughts on "constructing" a relationship of Phi with e.(other than a simple constant)?

    If Pi with e, then why not Phi with e?

    Again, Thanks.
     
  11. Jun 9, 2003 #10
    Am I meant to be aware of some result which states that there exists no 'simple' relationship between any transcendental number and a non transcental number.

    Clearly e^1 = e,

    Which gives a simple relationship between e and 1.

    I think that we require a working definition of 'simple'.
     
  12. Jun 10, 2003 #11
    As it turns out, relating phi to e is simpler than I thought:

    phi=2cos(pi/5), etc. and the identity:
    e^(i.pi)+1=0, (special case of Euler's Formula, with x=pi)

    From there, a simple matter of substitution.

    Now, the answer may be a complex number (I haven't figured it out, yet), but at least the relationship is expressed.

    Now comes a tougher job: Trying to relate fundamental (or natural) # like i,pi,e,phi, to physical # such as Planck's constant, c, gravity acceleration, Avogadro's Number, etc.

    Any thoughts?
     
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