Wrong section >.< Could someone move this over to number theory??
First post, I need a place for ideas to be reviewed in some way and heard this may be a good place.
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 am not posting here to be flamed, as the general internet as its ways. If you don't like the idea, or you disagree with it, then please read through it, and comment on what my mistakes are mathematically or if I am implying something that may not be true. I ask that you disprove it instead of saying how silly, crazy, or unorthodox it sounds.
I appreciate any comments and professional criticism.
Transcendental Slope
Theorem
The slope of a line may be transcendental.
Proof
The 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 line with Transcendental Slope
For points lying on a line that has transcendental slope:
1. No more than one algebraic point exists on a 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.