Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 ?

share|improve this question
1  
Is it =Pi or =1 ? –  Claude Leibovici Dec 19 '13 at 9:33
    
The OQ stands for = π. –  daniel Azuelos Dec 19 '13 at 9:35

1 Answer 1

up vote 4 down vote accepted

Yes.

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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