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Math Help - Transcendental Theory, Slope of a line, and how Pi+e and -(e/Pi) is transcendental

  1. #1
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    Transcendental Theory, Slope of a line, and how Pi+e and -(e/Pi) is transcendental

    I need a place for ideas to be reviewed in some way and heard this may be a good place to start.

    I have always been interested in math and science, and would like to share some ideas I have about transcendental numbers. Pi is what got me interested, until I found out that almost all numbers share its transcendental properties, and that it is hard to prove that a certain number is transcendental, specifically the sums, products and powers of transcendental numbers such as Pi and e. It is a work in progress, and I would like to get some input on if I am stating everything correctly, and what I can do to iron out any errors.

    I appreciate any comments and professional criticism.


    Transcendental Slope


    Theorem


    The slope of a straight line may be transcendental.

    Proof

    The linear slope form of any number x may be produced by:

    m=(x/1)
    m=x

    If x is transcendental, then the slope of a line m is transcendental.

    Example

    Pi is proven to be transcendental by the Lindemann-Weierstrass Theorem

    m=(Pi/1)
    m=Pi

    Slope m is transcendental.

    Points on a linear line with Transcendental Slope

    For points lying on a linear line that has transcendental slope:


    1. No more than one algebraic point exists on a linear line with transcendental slope.
    • An algebraic point can be chosen as the origin of a line with transcendental slope.
      (An algebraic point is a point in which both its x and y coordinates are algebraic numbers.)
    • Two algebraic points on a line will determine an algebraic slope.
      (If the algebraic point is unknown, a formula is needed to determine if any algebraic point lies on a line with transcendental slope m, and what its coordinates may be.)


    2. For all points not algebraic, at least one of its coordinates must be transcendental.
    • Given points P1 and P2 where P1x ne P2x or P1y ne P2y, If P1 is algebraic, the x and/or y coordinate of P2 is transcendental.
    • Two algebraic points on a line will determine an algebraic slope, a contradiction to transcendental slope.


    3. For all points not algebraic, if one coordinate is algebraic, the other must be transcendental.
    • Given points P1 and P2 where P1x ne P2x or P1y ne P2y, If P1 is algebraic, the x and/or y coordinate of P2 is algebraic, the other must be transcendental.
    • Two algebraic points on a line will determine an algebraic slope, a contradiction to transcendental slope.

    Pi+e or -(e/Pi) is Transcendental

    Proof

    Using the linear equation of a straight line where x and y are coordinates of a point, m is the slope of the line, and b is the y-intercept:

    y=mx+b (Slope Intercept Form)

    Both Pi and e are proven to be transcendental by the Lindemann-Weierstrass Theorem.

    If m=Pi, x=1, and b=e we can solve for y:

    y=(Pi/1)*1+e
    y=(Pi+e)

    The point x=1, y=(Pi + e) lies on a line with the transcendental slope of Pi.

    From "Points of a Transcendental Slope":
    3. For all points not algebraic, if one coordinate is algebraic, the other must be transcendental.

    We imply the coordinate y=(Pi + e) is transcendental because coordinate x=1 is algebraic.

    Under Investigation

    From Points of a Transcendental Slope:
    1. No more than one algebraic point exists on a line with Transcendental Slope.

    To imply "y=(Pi + e) is transcendental because x=1 is algebraic" one of the following conditions must be satisfied:


    • There is no algebraic point that lies on this line.
    • A point on this line other than x=1, y = (Pi + e) is algebraic.


    Further Applications

    Using slope intercept form, if m = Pi, y = 0, and b=e we can solve for x:

    0= (Pi/1)*x+e
    x= -(e/Pi)

    And thus:

    The point x= -(e/Pi), y=0 also lies on a line with the transcendental slope of Pi.

    Of the two points on the line with transcendental slope, coordinates x=1 and y=0 are algebraic.

    From Points of a Transcendental Slope:

    1. No more than one algebraic point exists on a line with Transcendental Slope.<br>
    3. For all points not algebraic, if one coordinate is algebraic, the other must be transcendental.

    One or both of the following coordinates:

    x= -(e/Pi) or y= Pi+e

    is transcendental.
    Last edited by madgadjt; April 1st 2012 at 10:09 PM.
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  2. #2
    Junior Member MathVideos's Avatar
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    Re: Transcendental Theory, Slope of a line, and how Pi+e and -(e/Pi) is transcendenta

    The last proof wouldn't really be considered a proof. If I understand correctly, you are trying to prove that all transcendental slopes will have one algebraic point, at the most. Giving an example is not sufficient proof of a for all statement.

    You might want to try a generic particular approach.
    Last edited by MathVideos; April 2nd 2012 at 04:37 AM.
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