period of $\cos(x) + x - \lfloor x \rfloor$?

Question

What is the period of $\cos(x) + x - \lfloor x \rfloor$?

This is what I have done:

$x = \lfloor x \rfloor + \{x\}$

$\cos(x)$ has period $2\pi$

$\{x\}$ has period $1$

so $\cos(x) + \{x\}$ should be periodic with the period of LCM of $2\pi$ and $1$ but the solution is stated as NOT PERIODIC. How is the function non-periodic?

Well, by definition, the LCM is a integer multiple of both $1$ and $\pi$. $\pi$ is not an integer multiple of $1$. — Ben Grossmann, May 27 '15 at 20:48

score 2 · Accepted Answer · answered May 27 '15 at 20:48

2

If the function is periodic, then integer multiples of the periods would have to match, i.e. $2\pi n = 1m$ for some integer $n,m$. However this says that $2\pi = \frac{m}{n}$. This contradicts the irrationality of $\pi$.

answered May 27 '15 at 20:48

Cameron Williams

29,432

score 0 · Answer 2 · answered May 27 '15 at 20:48

0

A period of $\sin(x)+\{x\}$ would be a common period of $\sin(x)$ and $\{x\}$. That would require finding $k,n\ne0\in\mathbb{Z}$ so that $2\pi n=k$. That can't be done since $\pi\not\in\mathbb{Q}$.

answered May 27 '15 at 20:48

robjohn

345,667

period of $\cos(x) + x - \lfloor x \rfloor$?

2 Answers2