Is the solution of the equation

$$x + \arctan(x) = \pi$$

irrational ?

The equation of $x + \arctan(x) = 1$ must be transcendental because for any nonzero algebraic $x$, $arctan(x)$ is transcendental, but this argument does not work for the above equation. The continued fraction of the above solution has more than $97000$ terms (PARI), so the answer seems to be yes. But can it be proven ?

  • 1
    $\begingroup$ Is it =Pi or =1 ? $\endgroup$ – Claude Leibovici Dec 19 '13 at 9:33
  • $\begingroup$ The OQ stands for = π. $\endgroup$ – dan Dec 19 '13 at 9:35


Let $x$ be the solution to $x+\arctan(x) = \pi$, then $$\arctan(x)=\pi-x \\ \Rightarrow x=tan(\pi-x) \\ \Rightarrow x=-tan(x).$$

Thus if $x$ would be rational, also $tan(x)$ would be rational. This is impossible: You can use the statement you gave for showing that $x+arctan(x)=1$ is irrational. Here is another reference:

Prove that if $x$ is a non-zero rational number, then $\tan(x)$ is not a rational number and use this to prove that $\pi$ is not a rational number.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.