How to use “ 难道 ” in Chinese sentences?
What is the difference between 你不知道吗？ and 你难道不知道吗？
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> Don't you know it? (I thought that you knew it.)
> Don't you know that ......
> Don't you understand?
> Ain't that the truth?
> Don't you have to work today?
> Aren't you coming?
Personally, I believe it's sometimes better to put 难道 in the beginning, though most of the time, the order doesn't matter in spoken language.
Let me start with Q2:
"你不知道吗 ?" - Don't you know it already?
"你难道不知道吗？" - How come/could you don't know it?
From the example, it is clear that 难道 is used when questioning (質問/質疑) a person with doubts or suspicions in the questioner's mind about the answer/attitude of the person. It is usually expressed in a disbelieving or ridicule manner. The occasion to use the words is when you think there is a possibility that the person didn't answer your question truthfully, or straightforward, but you are unsure if he's been honest or not. On many occasions, the answer may disappoint you - "沒聽過, 我真的不知道".
你难道不認識他嗎？ How come you don't know him?
你难道不會游泳嗎? How come you don't know about swimming?
你难道不去嗎? How come you are not going?
There is another use of 难道 - 难道是...? Similar to the above, it is questioning something with uncertainties - 假如不是他殺的, 难道是她? = If he didn't kill the person, would it be her? (In the mind of the questioner, this assumption is highly doubtful.)
Not a comprehensive answer, but:
难道 gives special emphasis to the question, like "Could it be that...?" and can be used to contrast with what was previously said, especially to expose some presumption or surprise (that could serve as an explanation for what has been said previously.
[usually, there would be some pause between the two sentences because the speaker needs some time to draw the concluding second sentence from the reaction or response of the other speaker]
"Aren't you joining us to go swimming tonight? Or perhaps could it be that you can't swim?"
(Not very elegant translation, but points out the meaning of 难道)
The difference between the two sentences you provide, should now be clear, the second one expresses the surprise over the reaction or response of the other person in a form of presumption/speculation (It's not necessary that the other person does not know [whatever it is], but we just make a guess). The first sentence is just a plain question with no speculative character.
Answering Question 1):
If you break down 难道 into it's parts, the meanings of each character (in this context) is:
难 : difficult 道 : suppose/think
And if you replace "difficult to think" or "difficult to believe" in sentences where 难道 exists, it gives an accurate representation of it's meaning, AKA expression ones disbelief, sometimes used in rhetorical questions. If we look at the following sentence:
it translates to either of these two:
"Do you really not understand?" "I find it difficult to believe that you don't understand, is this actually the case?"
The second translation would be more explicitly accurate, though the first one is easier to say while conveying almost the same meaning.
Answering Question 2):
"Do you not understand?" "You don't understand, is that correct?" (more literal translation)
"Do you really not understand?" "I find it difficult to believe you don't understand, is this really the case?" (more literal translation)