8

let $\alpha$ be the unique fixed point of $\cos:\mathbb{R} \rightarrow [-1,1]$

for any $t \in \mathbb{R} \setminus\{0\}$ if $t$ is algebraic then $\cos t$ is transcendental. thus if $\alpha$ were algebraic it must also be transcendental, a contradiction, since $\cos\alpha = \alpha$. hence $\alpha$ is transcendental.

QUESTION is this argument valid?

NOTE re the stated assumption, in answer to this recent question Prahlad Vaidyanathan pointed me to this Wikipedia entry re the Lindemann-Weierstrass theorem.

Community
  • 1
David Holden
  • 18,040
  • 2
    It could use a reference for the cited fact. – GEdgar Dec 29 '13 at 23:44
  • 4
    Writing \cos, with a backslash, rather than just cos, not only prevents italicization, but also provides proper spacing in things like $a\cos b$. I deleted the manually added spacing and put in the backslash. – Michael Hardy Dec 29 '13 at 23:48
  • thank you Michael, it has been irksome adding manual spacing. I suppose this applies to other trig functions and log. I will check this. I've only learned basic mathjax by frequently consulting the excellent MSE tutorial sheet, and by looking at other peoples' equations so any fine-tuning is much appreciated – David Holden Dec 29 '13 at 23:54
  • @GEdgar point taken. I'm rather new to this, hence the question. this has been puzzling me for some while, so it is remarkable to see that the solution is so simple, (if the reasoning here is is OK). – David Holden Dec 29 '13 at 23:58
  • 1
    @DavidHolden : many functions work this way, e.g., max, min, exp, sup, inf, lim, log, ... If in doubt, try putting a backslash before it. – Stefan Smith Dec 30 '13 at 00:01
  • Yes, the reasoning is OK. – Robert Israel Dec 30 '13 at 02:03
  • thank you Robert. it is amusing that once I had learned of the Lindemann-Weierstrass type of result, I was confused for a time by the thought that it would not apply because $\alpha$ and $\cos \alpha$ are equal. it took a while to twig that it is this very fact that lines up a simple reduction ad absurdum – David Holden Dec 30 '13 at 04:18
  • Wonderful question and wonderful answer! – Bumblebee Jan 02 '15 at 08:11
  • As a side note, the solution to this equation is known as the Dottie number. – Simply Beautiful Art Sep 23 '17 at 14:20

1 Answers1

3

Sifting throuht the material of https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem one concludes that

$\alpha \ne 0$ and algebraic $\implies e^{\alpha}$ transcendental

Assume that $\cos t = t$ is algebraic. Then $it$ is also algebraic and $\ne 0$, and so is $s=e^{it}$, as the solution of the equation $\frac{s+1/s}{2} = t$, contradiction with the above for $\alpha = i t$.

Yes, the argument of the OP is valid.

orangeskid
  • 53,909