# Can we make $\tan(x)$ arbitrarily close to an integer when $x\in \mathbb{Z}$?

My 7-year-old son was staring at the graph of tan() and its endlessly-repeating serpentine strokes on the number line between multiples of $\pi$ and he asked me the question in the title. More precisely, is the following true or false?

For any $\epsilon > 0$, there exists some $N \in \mathbb{Z}^+$ such that $|\tan(N)-\lfloor \tan(N) \rceil| < \epsilon$.

• Take $N = 0$. :-) Feb 27, 2011 at 16:31
• I'd bet for a much stronger property: that we can make it arbitrarily close to ANY integer. Feb 27, 2011 at 16:41
• Your 7-year-old son asked you that precise question?? Wow!
– lhf
Feb 27, 2011 at 18:10
• Edited to disqualify $N=0$ :). Also, yes, my son asked that question although he didn't formalize it (I did that) and it was part of a discussion, not something out of the blue. Feb 27, 2011 at 18:41
• Let me know when your son is ready to apply for college! :) Feb 28, 2011 at 1:59

Yes, you can. In fact, you can make $\tan(N)$ arbitrarily close to your favorite integer.

Pick your favorite integer, which is probably 17. Consider $x=\arctan 17$. Then $\tan(x)=17$ and also $\tan(x+n\pi)=17$ for all integers $n$. If you could make $x+n\pi$ very close to an integer (say $K$), then since the tangent function is continuous, $\tan(K)$ will be very close to 17.

So the problem is now to make $x+n\pi$ very close to an integer. This can be done because the multiples of $\pi$ (or any irrational number) are dense mod 1, so you can make $n\pi$ arbitrarily close to $-x$ mod 1. This takes proof but it is elementary: think about a circle of circumference 1, and start walking around the circle, around and around, marking a point each time your total distance travelled is a multiple of $\pi$. You'll see the marks start to fill in the circle, and as you go on and on, the largest interval without a mark in it keeps getting smaller. Eventually you will mark a point that's within $\epsilon$ of $-x$.

• (irony on) Usually, when I think about a circle it has circumference $2\pi$ and I always keep hitting the same two spots (irony off) ;-) Feb 27, 2011 at 17:23
• Do you mean the "largest interval without a mark"? Feb 27, 2011 at 17:54
• Oh, yes. Thanks! I edited it. (Although the smallest keeps getting smaller too ... :) Feb 27, 2011 at 18:09
• So for example $\tan(212) \approx 17.37\ldots$ but $\tan(89650 ) \approx 16.999\ldots$. Feb 27, 2011 at 23:05

Fairly random (and definitely unrelated) web surfing turned up this nice short paper of Cheng and Zheng which proves in a very constructive way that if $$f: \mathbb{R} \rightarrow \mathbb{R}$$ is any continuous function which is periodic with irrational period, then $$f(\mathbb{N})$$ is dense in $$f(\mathbb{R})$$.

The ideas are first exhibited with respect to the function $$f(x) = \sin x$$.

Your question is not literally a special case of this, since $$\tan x$$ is not continuous on all of $$\mathbb{R}$$. Nevertheless I think it is close enough for the same methods to be carried over. (For instance, $$\tan x$$ can be viewed as a continuous function with values in $$\mathbb{R} \mathbb{P}^1 \cong S^1$$.) In any case the relation here is close enough so that I thought the paper would be of interest to readers of this question.