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What would be the differences if I were to say 間違いない and 間違いなく? I ask that because the 無い suffix is usually explained as equivalent to the English "there is no" or "does not exist".

I've seen 無くtranslated as "without", but I haven't gotten confirmation on that. It's also taught that attaching く to adjectives makes them adverbs, Fast, quick, early = 早い and then 早く = quickly.

Where on earth do 無い and 無く fit into this?

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早く is an adjective in Japanese, not an adverb. A very common misconception among Japanese-learners. – l'électeur Dec 6 '13 at 13:06
@TokyoNagoya Really? You can say 早く行く (modifying a verb), but not 早く電車 (modifying a noun), as far as I understand... – senshin Dec 6 '13 at 13:26
In Japanese school grammar, the く form of 形容詞 (adjectives) is the 連用形, which is sometimes translated "ad-verbial form", just as 連体形 is sometimes translated "ad-nominal form". These translations work okay for adjectives, where they usually do modify later 用言 ("verbals", meaning verbs and both types of adjective) and 体言 ("nominals") respectively. But the 形容詞の連用形 is not the same thing as 副詞 (adverbs), even if both commonly function to modify 用言, which I imagine is what Tokyo Nagoya is pointing out. – snailplane Dec 6 '13 at 14:51
Consider in English "chicken soup" and "tasty soup". In each phrase, the first word functions to modify the following noun, so we say it has attributive function. But "chicken" is an attributive noun, while the "tasty" is an attributive adjective. Despite having the same function, they're different parts of speech. – snailplane Dec 6 '13 at 15:02
@user4096 That's very different. The く form of adjectives is unrelated; the なく here is not the one you've linked to. The -(a)ku in that なく long ago attached to certain forms of verbs, replacing the final vowel with a and forming a noun. For example, it attached to 「ず」の連体形「ぬ」, replacing the vowel u, giving n-aku, a noun. Likewise for 「恐る」の連体形「恐るる」, giving 恐るらく, a noun, which gave 恐るらく+は → 恐らくは → 恐らく, an adverb. That -(a)ku appears in relatively few words in modern Japanese; you'll find it in 老いらく and 曰く, for example. In contrast, the く form of adjectives is very common. – snailplane Dec 6 '13 at 20:17

To summarise / add to the discussion:

The difference between -い and -く is not one of part of speech, it's of grammatical form. Adjectives that end in -い do one of two things:

-Come at the end of a sentence, as the main verb: 車が速い。

-Modify an immediately-following noun: 速い車(が~、を~)

Adjectives that end in -く also do one of two things:

-Indicate a transition between two main clauses in the same sentence (often like English 'and'): 車が速く、自転車が遅い。

-Act as an adverb, modifying a verb: 車が速く走る。

Though, if you think about it, these are effectively the same thing - it becomes a bit more apparent if you vary the conjunction in English (using things like 'while' and so on as well as 'and'). So X無く can indeed be used to mean 'without X', but it means more 'while X is not present / does not exist'. (English fairly distinctly separates adverbs and adjectives, but on a very low level, Japanese treats adverbs as separate clauses containing an adjective - you could in theory translate 車が速く走る as 'the car is fast and goes' or 'the car goes and is fast'.)

To answer your main question directly, the difference between 間違いない and 間違いなく is one of whether or not they can stand as a sentence on their own. 間違いない can, and means something like 'there is no mistake'. 間違いなく can't really (outside of incomplete sentences), and means more 'without a mistake'.

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