Why do some Japanese questions end with 〜ないか?

Question

I was watching clips of “Art of Fighting 2” (Original Title: 龍虎の拳2) in Japanese, and I noticed an interesting pattern:

When Geese Howard (ギース・ハワード) is talking to Ryou Sakazaki (リョウ・サカザキ) or Robert Garcia (ロバート・ガルシア), he asks, 「私と組んでみる気はないか?」

Unless I'm mistaken, I think he's saying, “Are/aren't you interested in teaming up with me?”

In John Crawley's (ジョン・クローリー) ending, his commanding officer meets with him. John calls him 閣下, which is usually translated as “his/her/your Excellency”, but since John is in the American military, I think it would be better translated as “Sir”. Anyway, his commanding officer ends up asking, 「もう一度軍にもどる気はないかね。」

Maybe I'm wrong, but I think he's saying, “You are/aren't interested in returning to the military once again, are/aren't you?”

These are just a few examples, but it seems to show a difference between Japanese and English. Japanese seems to end a number of questions in the negative, while English seems to end a number of questions in the positive.

Why is that?

Edit:

Okay, maybe these examples are too specific. That's okay, because I found some other examples right here.

ないか

（ 連語 ）
〔打ち消しの助動詞「ない」に終助詞「か」の付いたもの〕.
① 打ち消しの疑問の意を表す。 「彼はまだ来－」
② 勧誘の意を表す。 「映画を見に行か－」
③ 婉曲的な命令の意を表す。 「もう八時過ぎだ。早く起き－」
④ 希望の意を表す。 「だれか遊びに来－なあ」
⑤ 控えめな依頼の意を表す。 「ちょっと見せてくれ－」 〔打ち消しの疑問の意では「ないのか」の形も用いられる〕

Here's a translation:

Naika

(Phrase)

[Something with sentence-ending particle “ka” attached to negative auxiliary verb “nai”].
1. Expresses the sense of negative questions. “Has he not come yet?”
2. Expresses the sense of invitations. “Would you like to go see a movie?”
3. Expresses the sense of indirect commands. “It is past 8 o’clock now. Will you get up early?”
4. Expresses the sense of desires. “Will you come to see someone?”
5. Expresses the sense of modest requests. “Could you let me look around?” [The “nainoka” form is also used with the sense of negative questions]

I translated four out of five examples without the “not”. Are invitations, indirect commands, desires, and modest requests more polite when they are phrased negatively or positively in either language?

Also, there is a chance that my translation might have mistakes. If there are, please don't hesitate to comment!

Answer 1

Let me focus on your original question. When you suggest something, a negative question such as ～ませんか/～ないか is the default choice in Japanese. It's hard to explain why, but English speakers also use a similar construction using not (eg "Why don't you go to a movie tomorrow?"), so it should not be hard to grasp the basic meaning.

If you use a non-negative version, the question tends to sound like you are plainly asking/confirming someone's existing intention ("Do you want ～?" or "Are you going to ～?"). ない signals it's a suggestion; you're trying to change someone's mind by introducing some good idea.

• 私と組んでみる気はあるか?
  ("Do you have an intention to ...?"; doesn't really sound like a direct suggestion)
• 私と組んでみる気はないか?
  (suggestion/invitation)
• サッカーやる？
  ("Do you want to ...?"; you're simply asking what the listener is going to do)
• サッカーやらない？
  (suggestion/invitation)

That said, non-negative version still can work as a suggestion depending on the context.

Answer 2

"ないか" is mostly used to inquire about something. Its translation of course depends on the context, but you can most of the time translate it to "won't you?" or "will you?". I see that you already know its meaning but have doubt about it.