I've been learning about conjugating i- and na-adjectives recently and I've stumbled onto something odd. I've seen two distinct ways of conjugating negative i-adjectives; the two ways are to replace い with either 1) くない (for present negative) or くなかった (for past negative) then add です at the end, or 2) くありません (for present negative) or くありませんでした (for past negative). I'll use 楽しい as an example:
Present Negative: 楽しくないです。 OR 楽しくありません。==> It is not fun.
Past Negative: 楽しくなかったです。 OR 楽しくありませんでした。==> It was not fun.
Are these two algorithms for conjugating negative i-adjectives equivalent? Is one better than the other? What are any differences between them? Is one more formal than the other? Is there anything I should know about the relative usage of these two algorithms? Any input will be appreciated.