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What's the difference between 不可能 and 不能? Both mean 'impossible', but '不能' also means 'incapable'. Aren't they synonyms? Feel free to provide example sentences.

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As a simple na-adjective, you should use 不可能 over 不能 in nearly all cases. (But just in case, you may need ありえない instead of 不可能. 不可能 means "one is unable to do something", whereas あり得ない means "something cannot happen". 「30分でアメリカに来い? 不可能だ!」「昨日アメリカで僕を見た? ありえない!」 Both mean "That's impossible" in English, but in Japanese they are different.)

不能 is used mainly as part of longer fixed compounds meaning "un-X-able": 理解不能 (incomprehensible), 回復不能 (unrecoverable), 修理不能 (unfixable), etc. In other words, you should think of this as a suffix rather than a standalone adjective. You can use 不可能 in these cases, too, but 不能 is shorter and preferred.

不能 is used on its own also in the following rare situations, but intermediate learners may not have to remember these:

  • "impotent (sexually)": Outdated. People usually prefer "ED" or 勃起不全 today.
  • "incapable", "has no ability": Rare and outdated. 無能 is almost always preferred today.
  • "unsolvable", "has no solution" (math equation): Rare math jargon. 解なし is more common in obligatory education.
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  • Could you please also elaborate on the difference between 「ありえない」and 「不可能」? I upvoted your answer earlier today. And when I read it the first time, I thought "yeah absolutely, those two are different." But just now I heard 「前提条件がありえない!」in a show. I would've expected 「前提条件が不可能」. Let me know if I should post this in a separate question. Thanks!
    – Eddie Kal
    Dec 17 '20 at 22:31
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    @EddieKal Both make sense, but mean different things. 前提条件が不可能 means the prerequisite itself may be reasonable but the speaker is unable to achive it (eg 100mを10秒で走れたらオリンピックに出られます; アイドルになりたいならイケメンである必要がある). 前提条件がありえない means the precondition is unreasonable, or something no one can achieve, or something that is simply always false (eg 私とデートしたければ1000億円持ってきなさい; 1ドルが1円なら僕はアメリカではお金持ちだ).
    – naruto
    Dec 17 '20 at 23:24

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