Proving Transcendence: Log_e(m) Conjecture

[Char. Limit]

Homework Statement

I want to try to prove something, but I don't even know if it's right. So I thought I would come to you for help. The conjecture is below:

For two transcendental numbers u and v, if u-v is not algebraic, then u+v is transcendental.

Homework Equations

The Attempt at a Solution

Now, I know that if u-v IS algebraic, then u+v is NOT transcendental. But I don't think that's what I'm looking for...

Note: I'm using this in an attempt at a proof that all numbers of the form $log_e(m)$, where m is a natural number greater than 1, are transcendental.

 

[Dick]

u=pi, v=(-pi). u-v is transcendental, u+v isn't.

 

[Char. Limit]

Ah, I see. Thanks.

 

[micromass]

Note: I'm using this in an attempt at a proof that all numbers of the form $log_e(m)$, where m is a natural number greater than 1, are transcendental.

I would take a look at the Lindemann-Weierstrass theorem: http://en.wikipedia.org/wiki/Lindemann–Weierstrass_theorem

 

[Char. Limit]

I would take a look at the Lindemann-Weierstrass theorem: http://en.wikipedia.org/wiki/Lindemann–Weierstrass_theorem

I'll take a look at it, thanks!