One of the first things students of Japanese learn is how to put verbs into their ます-form.

For example, 行く (いく // to go) becomes 行きます (いきます // go).

Then books teach us the negative form of ます conjugations, which is ～ません.

For 行く, that's 行きません (いきません // won't go; don't go).

You can even make a full sentence with a verb like this:

行きません。
いきません。
I won't go (there).
Literally: "won't go."

Of course, you'd need to have the proper context for a sentence like that to work. But mastering context is something that we'll be doing naturally throughout our studies.

But there is another usage of the negative conjugation ～ません.

By adding ～か to the end of it, giving us ～ませんか, we can make polite invitations to do things...

JLPT N5: ～ませんか (won't you~?; would you like to~?)

An example:

A:
今夜、映画に行きませんか。
こんや、　えいが　に　いきませんか。
Would you like to go to the movies tonight?
Literally: “tonight, + movie + に + won’t go + か.”

B:
いいですね。行きましょう。
いい　です　ね。　いきましょう。
That sounds nice. Yes, let's go.
Literally: “good + です + ね. + let’s go.”

See that ～ましょう in sentence B? We just looked at ～ましょう in this lesson: [NDL #469] - JLPT N5: ～ましょう.

Also, you could argue that ～ましょう ("let's VERB") and ～ませんか ("won't you VERB [with me]?") are almost the same... only using ～ませんか makes you sound more considerate of the fact that the listener may not want to participate in whatever activity you're inviting him or her to join in.

I suppose I could have given that first sentence a more literal translation of "Won't you go to the movies with me tonight?" But that sounds a little too stiff to me. The polite, "Would you like to go to the movies (with me) tonight?" seemed better.

All in all, it's a pretty simple way to invite someone to do something, though, yeah?

The trouble is that this language will probably be too polite for 90% of your conversations in Japanese in real life. I pretty much never use it. But then, I rarely find myself in a place to invite people to do things in formal situations (for example, I do not work at a Japanese company, so I rarely find myself inviting coworkers to go to lunch, the movies, etc.)

But it is useful. Especially because in the early stages of learning the language, you'll probably be using polite language quite a bit--because you haven't had time to make friends yet!

Here's another example:

A:
一緒に歌を習いませんか。
いっしょに　うた　を　ならいませんか。
Would you like to learn singing [take singing lessons] together?
Literally: “together + song / singing + を + won’t learn + か.”

B:
うーん...歌はちょっと.....。
うーん...　うた　は　ちょっと.....。
Umm… Singing’s not really my thing…
Literally: “umm… + song / singing + は + a little...”

This ちょっと in sentence B is a soft way of declining a request or invitation to do something. In the past, we had a lesson about the many uses of ちょっと: [NDL #232] - Hey! Stop it! That lesson also include an example of ～ませんか！

One more polite example, and then we'll be adding a bit more variety to this lesson.

A:
明日、一緒に買い物に行きませんか。
あした、　いっしょに　かいもの　に　いきませんか。
Would you like to go shopping with me tomorrow?
Literally: “tomorrow, + together + shopping + に + won’t go + か.”

B:
ええ、いいですね。
ええ、　いい　です　ね。
Yeah, that sounds nice.
Literally: “yeah, + good + です + ね.”

Since you're so cool, and you (will soon) have so many Japanese friends, you probably want to learn some informal language.

Well, just use the plain negative form (i.e. ～ない and not ～ません) and add か to the end of it, like this:

A:
うちで一緒に宿題をしないか。
うち　で　いっしょに　しゅくだい　を　しないか。
Want to do homework together at my house?
Literally: “home + で + together + homework + を + won’t do + か.”

B:
うん。いいよ。
うん。　いい　よ。
Yeah. Sounds good.
Literally: “yeah. + good + よ.”

Another informal example:

A:
へえ。レモンビールか。飲んでみないか。
へえ。　レモンビール　か。　のんで　みないか。
Lemon beer? Want to try it?
Literally: “whoa / oh / cool. + lemon beer + か. + drink (and) + won’t see + か.”

B:
ええ、飲んでみましょう。
ええ、　のんで　みましょう。
Yeah, let's try it.
Literally: “yeah, + drink (and) + let’s see.”

Did you notice that Person A is using informal language, but Person B is using polite language? This could, for instance, be because Person A is Person B's 先輩 (せんぱい // senior [at work or school]) or 上司 (じょうし // boss).

Note that adding the auxiliary verb ～みる to the end of a verb in て-form translates to "try VERB-ing" or "try to VERB." That's why Person A's sentence translated to "Want to try it?" and not "Won't you drink and see it?" ^_^

Check out this lesson for more: [NDL #57] - Try Doing VS Try to Do.

In informal language, it is common for female speakers to drop the か from the end of their sentences, using a rising intonation (like in English) to show that the sentence is a question.

That is, the following sentence has 作ってみない？ (つくってみない？) instead of 作ってみないか (つくってみないか), both of which mean "want to try making (something)?".

A:
明日、一緒にパンを作ってみない？
あした、　いっしょに　パン　を　つくって　みない？
Want to try making bread together tomorrow?
Literally: “tomorrow, + together + bread + を + make (and) + won’t see?”

B:
いいね。作ってみよう。
いい　ね。　つくって　みよう。
Sounds fun. Let's make some.
Literally: “good + ね. + make (and) + let’s see.”
Note: 作ってみよう is the informal form of 作ってみましょう.

Lastly--and this may seem obvious--you cannot use ～ませんか when asking the listener what they are going to do. That is, we use ～ませんか for invitations but not for interrogations.

That's why this sentence is whack:

× どこに行きませんか。
× どこ　に　いきませんか。
× Where won’t you go?
× Literally: “where + に + won’t go + か?”

That's all for this lesson.

Have fun inviting everybody to do everything. Or, if you're shy, you can have fun feeling guilty that you are not inviting everybody to do everything.

Bonus Sentences:

バスケ部の先輩に告白された。
バスケぶ　の　せんぱい　に　こくはく　された。
My senpai in Basketball Club told me that he likes me.
Literally: "basketball club + の + senpai (=senior member) + に + confession (of feelings) + was done."

げ、上司から電話だ。
げ、　じょうし　から　でんわ　だ。
Ugh, my boss is calling me.
Literally: "ugh, + boss + from + phone + だ."