Notation for every odd integer number

Question

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.

the vertical asymptotes are the zeros of the function $\cos(x)$ — Dr. Sonnhard Graubner, Sep 04 '16 at 15:52
"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. — fleablood, Sep 04 '16 at 15:54
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. — Winther, Sep 04 '16 at 16:03
@Winther I was running out space on my AP Calculus Summer assignment, that is way all these questions are popping up. — Sigma6RPU, Sep 04 '16 at 16:05
@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. — MathematicsStudent1122, Sep 04 '16 at 19:23
@Mathematics The truth is there's a continuum between writing everything completely in symbolic logic and not even using a plus sign. Use standard notation all you like, no one will argue. Beyond that, if humans are reading it, it's better to err on the side of words. — Matt Samuel, Sep 04 '16 at 21:58
$\mathbb Z_{2k+1}$ is not a good choice since it might represent the group $\mathbb Z/(2k+1)\mathbb Z$. — , Sep 05 '16 at 20:45
And let's admit that we do not have a universal agreed notation on the set of odd integers. All the answers below are not clearer/shorter than the english one. — , Sep 05 '16 at 20:47
@tired I came up with 7 different accounts, and got enough reputation to up vote myself clearly? Good question idk where anybody wanna comment why they upvoted — Sigma6RPU, Sep 13 '16 at 01:38

Dr. Sonnhard Graubner · Accepted Answer · 2016-09-04T15:58:51.907

16

they are $$(2k+1)\cdot \frac{\pi}{2}$$ with $$k \in \mathbb{Z}$$

edited Sep 04 '16 at 15:58

answered Sep 04 '16 at 15:53

Dr. Sonnhard Graubner

95,283

2

Thanks that deserves to be a high school math hack. – Sigma6RPU Sep 04 '16 at 15:57
13

@Sigma6RPU What's a "high school math hack"? – JiK Sep 04 '16 at 16:38
@JiK Anything that makes something complex, expressed in a simple elegant form. – Sigma6RPU Sep 04 '16 at 20:07
4

@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. – rschwieb Sep 06 '16 at 19:40

score 11 · Answer 2 · answered Sep 04 '16 at 15:52

11

You can go with $2\mathbb Z +1$

answered Sep 04 '16 at 15:52

Asinomás

105,651

8

A good option. Relatedly $2\mathbb Z $ for even and $ \mathbb{Z}\setminus 2\mathbb Z $ as alternative for odd. – quid Sep 04 '16 at 15:55
@quid Isn't that quotient the set of integers mod 2? – user3146 Mar 30 '22 at 19:38
@user3146 $\backslash \neq /$, but yes that notation is probably not ideal. – Asinomás Mar 30 '22 at 19:39
@Asinomás lol good point, I'm dumb – user3146 Mar 30 '22 at 19:41
@user3146 so am I , I definitely see myself falling for that. – Asinomás Mar 30 '22 at 19:45
Easier still would be $x \epsilon 2\pi\Z $, without resorting to k, right? – user3146 Mar 30 '22 at 19:46
think it would be $\pi \mathbb Z \setminus 2\pi \mathbb Z$ – Asinomás Mar 30 '22 at 19:47
I can't get script z to work. Sorry, I forgot it was odds – user3146 Mar 30 '22 at 19:49

score 8 · Answer 3 · answered Sep 04 '16 at 16:06

8

As long as we're considering alternatives, you could always write $$k\equiv 1\pmod 2$$

answered Sep 04 '16 at 16:06

Matt Samuel

58,164

Ahmed S. Attaalla · Answer 4 · 2016-09-04T16:33:58.980

6

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.

edited Sep 04 '16 at 16:33

answered Sep 04 '16 at 15:55

Ahmed S. Attaalla

18,518

2

Is that an "Oh" or a zero? – fleablood Sep 04 '16 at 15:57
2

$0$ as the set of all numbers? That's not a good idea, I wouldn't like to write $0 \in 0$ anywhere. – Santiago Sep 04 '16 at 15:57
1

Thanks @fleablood I was wondering why it went bold. – Ahmed S. Attaalla Sep 04 '16 at 15:57
2

Never seen this, but I like it. – Daniel R. Collins Sep 04 '16 at 17:23
2

Actually $\mathbb O$ is normally reserved for the octonian, as are $\mathbb H, \mathbb C, \mathbb R$. – Sep 06 '16 at 02:10
Hmm good to know, I didn't know that @Jhon Ma, but I guess if you define it first for your purposes then we are good. I've seen people use this before. – Ahmed S. Attaalla Sep 06 '16 at 02:25
Yes, but $\mathbb O$ will be bugging me all the time I guess @@.... – Sep 06 '16 at 02:52
Don't worry, @Santiago $-$ you will never have to write $0\in \mathbb O$, because $0$ is even! – TonyK Sep 12 '16 at 19:28

score 3 · Answer 5 · answered Sep 04 '16 at 16:53

3

Alternatively, you could write

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

answered Sep 04 '16 at 16:53

andrepd

433

score 1 · Answer 6 · edited Mar 30 '22 at 19:55

1

You could do $x \epsilon \pi \mathbb{Z} / 2 \pi \mathbb{Z}$, without resorting to $k$.

edited Mar 30 '22 at 19:55

Asinomás

105,651

answered Mar 30 '22 at 19:51

user3146

685

Notation for every odd integer number

6 Answers6