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I have this equation:

$$f(x)=\tan(x)$$

I found the vertical asymptotes to be:

$$x=\frac{\pi}{2}k$$

What is the proper notation for that k is equal to every odd number integer(negative,positive, and zero)?

$$k\in\mathbb{Z}$$ is for every integer, but is there such a symbol for every odd number integer?

Natural numbers are positive, and sometimes zero counting numbers, my question is about integers not natural numbers.

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    $\begingroup$ the vertical asymptotes are the zeros of the function $\cos(x)$ $\endgroup$ – Dr. Sonnhard Graubner Sep 4 '16 at 15:52
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    $\begingroup$ "the vertical asymptotes are the zeros of the function cos(x)"... which are $\pi/2(2k + 1)$... I don't at all see the point of this comment. $\endgroup$ – fleablood Sep 4 '16 at 15:54
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    $\begingroup$ Sidenote; where writing math you don't have to use fancy symbols for everything. There is nothing wrong (and imo it should be encouraged) to use words instead wherever possible. $\endgroup$ – Winther Sep 4 '16 at 16:03
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    $\begingroup$ You could simply write $k\rm~odd$, also $\endgroup$ – Akiva Weinberger Sep 4 '16 at 16:25
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    $\begingroup$ @Winther One of the most remarkable things in the history of mathematics is that we have stopped using words for everything. Having convenient notation is very important. Writing has its advantages (I prefer "for all" to $\forall$, for example), but, nevertheless, in my opinion we do need simple notation for the set of odd and even integers. $\mathbb{Z}_{2k + 1}$ is my proposal. Ahmed's idea is great as well. $\endgroup$ – MathematicsStudent1122 Sep 4 '16 at 19:23
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they are $$(2k+1)\cdot \frac{\pi}{2}$$ with $$k \in \mathbb{Z}$$

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    $\begingroup$ Thanks that deserves to be a high school math hack. $\endgroup$ – Sigma6RPU Sep 4 '16 at 15:57
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    $\begingroup$ @Sigma6RPU What's a "high school math hack"? $\endgroup$ – JiK Sep 4 '16 at 16:38
  • $\begingroup$ @JiK Anything that makes something complex, expressed in a simple elegant form. $\endgroup$ – Sigma6RPU Sep 4 '16 at 20:07
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    $\begingroup$ @Sigma6RPU I would venture to say that every serious mathematician in every field of mathematics strives to express complex things in simple, elegant forms. That being the case, it does not seem very fitting to describe the natural, commonplace behavior of every mathematician as "[high school math hacking.]" I would have thought any hacking that a high school student would do would be associated with something negative. Like expressing something simple in a complicated way or something. But if it strikes you as positive, maybe you agree and it's just a difference in interpretation. $\endgroup$ – rschwieb Sep 6 '16 at 19:40
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You can go with $2\mathbb Z +1$

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    $\begingroup$ A good option. Relatedly $2\mathbb Z $ for even and $ \mathbb{Z}\setminus 2\mathbb Z $ as alternative for odd. $\endgroup$ – quid Sep 4 '16 at 15:55
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As long as we're considering alternatives, you could always write $$k\equiv 1\pmod 2$$

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Usually people write:

$$\frac{\pi}{2}(2k+1), k \in \mathbb{Z}$$

Sometimes people would use $\mathbb{O}$ for the set of all odd integers, but because it is not so standard they will tell you ahead of time:

$$\mathbb{O}=\{ 2n+1 : n \in \mathbb{Z}\}$$

So then, after defining $\mathbb{O}$, you would say:

$$\frac{\pi}{2}k, k \in \mathbb{O}$$

Get used the $\in$, it simply means "is a member of" some set.

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    $\begingroup$ Is that an "Oh" or a zero? $\endgroup$ – fleablood Sep 4 '16 at 15:57
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    $\begingroup$ $0$ as the set of all numbers? That's not a good idea, I wouldn't like to write $0 \in 0$ anywhere. $\endgroup$ – Santiago Sep 4 '16 at 15:57
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    $\begingroup$ Thanks @fleablood I was wondering why it went bold. $\endgroup$ – Ahmed S. Attaalla Sep 4 '16 at 15:57
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    $\begingroup$ Never seen this, but I like it. $\endgroup$ – Daniel R. Collins Sep 4 '16 at 17:23
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    $\begingroup$ Actually $\mathbb O$ is normally reserved for the octonian, as are $\mathbb H, \mathbb C, \mathbb R$. $\endgroup$ – user99914 Sep 6 '16 at 2:10
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Alternatively, you could write

$$x = \frac{\pi}{2}k \quad , \quad k = \pm1, \pm3, \pm5 \dots$$

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