In the process of learning certain kanji, I found that 去る and 残す both
translate to the English word "leave."

I know that 出る is used for the common "leave a place"/"exit" usage, like,
「五分前家を出ていました」. I assume that 去る and 残す have different nuances to them. Is 残す
more along the lines of "leave a note" like, 「花の世話するの忘れないでくださいと言うノートを残した」 and
去る more along the lines of "leave" in the context of [forgive the dramatic
example] 「人間がきらいだから、この世界を去りたい」? Or do I have them backwards/are they
interchangeable/something entirely different?

(大学の四がき日本話の学生から)
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question
words nuances kanji
What is the use case difference between 去る and 残す?
This might shock you, but to us native speakers, 「去{さ}る」 and 「残{のこ}す」 have
completely different meanings from each other and, for the life of me, I could
not think of a situation where the two verbs can be used interchangeably.

The evil, as usual, is the translation and the bilingual dictionary that would
tell you that both mean " ** _to leave_** ".

In the only Japanese-speaking country on earth, however,

「去る」 means " ** _to (physically) go away from a place_** " and

「残す」 means " ** _to leave something/someone behind_** "

Hope you could see the huge difference here. Since the two are so different in
meaning from each other, you could easily use both in a relatively short
sentence. Let me borrow your sentence (by correcting its particle mistakes) to
illustrate my point:

> 『花{はな}の世話{せわ}をするのを忘{わす}れないでください』と言うノートを **残した** 。 "Someone **_left_** a note
> saying 'Please don't forget to take care of the flowers!'"

It is also completely natural to say:

> 『花の世話をするのを忘れないでください』と言うノートを **残し（て）去っていった** 。 "Someone **_went away,
> leaving_** a note saying 'Please don't forget to take care of the flowers!'"

Finally, your other sentence:

> 「人間{にんげん}がきらいだから、この世界{せかい}を去りたい。」 "I want to leave this world because I hate
> people."

is valid for using 「去る」. You want to do the ultimate "going away"; You want to
kill yourself.

You can never say 「この世界を残したい」 in this context because "this world" is not your
belonging in the first place. You can only 残す the things you own, the persons
you love, your accomplishments, etc. Oops, I forgot you hated everyone. ☺
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answer