無い and 無く difference


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?




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'.

