# Thread: TA’s Challenge Problem #7

1. ## TA’s Challenge Problem #7

“Prove” that 1 is transcendental.

“Proof”:

The Gelfond–Schneider theorem states that if $\displaystyle \alpha$ and $\displaystyle \beta$ are algebraic numbers such that $\displaystyle \alpha\ne0,\,1$ and $\displaystyle \beta\notin\mathbb Q,$ then $\displaystyle \alpha^\beta$ is a transcendental number.

Hence, taking $\displaystyle \alpha=-1$ and $\displaystyle \beta=\sqrt2,$ we have that $\displaystyle (-1)^{\sqrt2}$ is transcendental.

But $\displaystyle (-1)^{\sqrt2}=(-1)^{2\frac{\sqrt2}2}=\left((-1)^2\right)^{\frac{\sqrt2}2}=1^{\frac{\sqrt2}2}=1.$

Hence 1 is transcendental.

Explain what is wrong with the above “proof”.

2. I guess maybe a couple of things are wrong with it, but for one, the proof manipulates things so that $\displaystyle \alpha=1,$ and then attempts to apply Gel'fond-Schneider, contrary to hypothesis.

3. You are on the right track, but you need to be more specific about what things are being “manipulated”.

4. Okay, well... the correct way to evaluate $\displaystyle (-1)^{\sqrt{2}}$ would probably be to use the rule $\displaystyle z^\alpha=\exp(\alpha\log z)$, which I believe has infinitely many values when $\displaystyle \alpha$ is not a rational real number.... So the expression is multi-valued, and so we can't apply Gel'fond-Schneider?

5. Gelfond–Schneider certainly applies in this case.

But you’ve the nail on the head.
Spoiler:
$\displaystyle (-1)^{\sqrt2}$ is not a real number. It is complex-valued, and for complex multiplication, the exponent laws for real numbers break down – e.g. $\displaystyle \left[(-1)(-1)\right]^{1/2}\ne(-1)^{1/2}(-1)^{1/2}.$

6. Originally Posted by TheAbstractionist
Gelfond–Schneider certainly applies in this case.

But you’ve the nail on the head.
Spoiler:
$\displaystyle (-1)^{\sqrt2}$ is not a real number. It is complex-valued, and for complex multiplication, the exponent laws for real numbers break down – e.g. $\displaystyle \left[(-1)(-1)\right]^{1/2}\ne(-1)^{1/2}(-1)^{1/2}.$
Heh... I think you've been far too generous in crediting me with providing a solution to this. My answers were clumsy all along. What I ought to have said was that Gel'fond-Schneider cannot be applied in the way that it was being applied in the OP. I could tell that something was wrong with the way the exponent was being taken, but I wasn't sure exactly what, because I've always done these "by the book" using logs, and so to be honest, I wasn't really sure where the problem was in the OP.

Which is sort of embarrassing to me, because I am supposed to be a specialist in transcendental number theory....