难道 is a word I have never liked, basically because I have felt that it is impossible to use. Also, I never ever hear anyone use it. True, it can be found in writing, but words I almost never hear anyone ever use in speech (偏， 万万，难道） I tend to shy away from since I lack sufficient 语感. Not to worry, since 难道's usage is relatively simple and for our purposes you just need to know that at the end of a 难道 "sentence" you often find a 不成 or a 吗。 Just remember that and you'll be fine.
Yes, english ways of expressing this emphasis are very different leading to me not using 难道 very much, and also, just looking at examples, where 吗 or 不成 is placed at the end, the sentences often feel grammatically icky. Example:
Nciku: 你难道没看见我正忙着吗？Which they translate as "Don't you see I'm busy at this moment?" Passable. In these type of situations I will never use 难道， I will always use intonation, 怎么, 真的 to emphasize. Furthermore, in actual use I see, 难道 is nearly always placed at the very beginning of the sentence which feels even more
Example 2: Nciku:
难道你忘了自己的诺言吗？ = trans: Can you have forgotten your promise?
This is one of the major problems in comprehension and use, in translating this idea, the translations inevitably are full of these awkward constructions that don't seem to jive with the chinese or if they do, are filled with these akward double negatives (single negatives) or no negatives where one obviously would be required.
You'll often see things like "Is it possible that you've really forgotten your promise?" Which for me at least, is very hard to assimilate into a thing in English. I'll always think "Have your really forgotten your promise?". Which lead to chinese like : 你真的忘了自己的诺言吗？
Example: 3 Dict:
难道你不懂吗： Don't you understand?
Ah, negative territory. Let's look at this:
你懂吗 = You get it? You understand?
你不懂吗 = You don't understand?
你真的(还）不懂吗 = You really （still) don't understand? (Incredulous that other party doesn't understand.)
So, the only time I can think about the phrasing: 'Don't you...?" is something like "Don't you realize how important this is (to me)?" Which in Chinese I will always say 你了解这个对我有什么意义吗？ which can easily be translated back into english as: Do you realize how important this is to me? The issue is, here the "don't" seems to be the 难道 feature in chinese, but in english I have to think this usage is either a bit more extreme or somehow doesn't mesh with actual 难道 usage in Chinese.
So back to 难道你不懂吗- ...?
So, we can see why they translate this as "Don't you understand?" since it follows the "don't you" pattern seen to be used with 难道, but somehow it feels awkward. The akwardness comes from this step from
你不懂吗 to 难道不懂吗 is very different in english from the step from 你了解这个对我有什么意义吗？ to 难道你了解这个对我有什么意义吗？
In english, the pressure on and context from the second pair is clear, but with the first pair, with this question and context being so common and pedestrian, the usage and meaning and 难道 is lost or confused.
I can feel the difference relatively quickly between "Don't you realize how important this is to me?" and "Do you realize how important this is to me?" even though in 9 out of 10 cases (maybe 10 out of 10) they can be swapped with no change in stress, meaning, emphasis, etc.
With "You don't understand?" and "Don't you understand?" I really have to think of a high pressure situation where I'm going to feel like "Don't you understand?" makes any sense. This means all your common feelings about the word understand, and all common usages associated with it, like working in a classroom, with a teacher, etc, have to be scrapped and you have to put "understand" in a different context.
"You don't understand" 你不懂吗? Is a teacher's question just like 你懂吗 - "You understand?" A teacher will never rightfully say "难道你不懂吗？“
So, when will we need 难道？
One last thing before we get to that. The "classroom" 'Don't you understand what I'm talking about here?' I don't think will use 难道。 Example:
Teacher teaching students, has tons of equations and student in reponse to question is going through some of the equations on the board to answer a question. He gets tripped up in one section and the teacher interrupts him and points to the section that is confusing. He could just as easily say "...Ok, (do) you understand what I'm talking about here? This part is relates to x as a variable of coordination...blah blah blah." as he could say "...Ok, don't you understand what I'm talking about here? This part relates to x as a variable of coordination...blah blah blah." I SUPPOSE, this "don't" adds emphasis, but my gut feeling is that english usage here is very free and 难道 is much stronger and means something when used (almost always) as opposed to these type of situations in english where a respondent would not be able to report which one the teaacher used.
Ok, so, back to when to actually used 难道 with understand. That mad scientist (man do I hate that example, but it what you get from going to college) says to his colleague after discovering a way to travel through can still use both "Do you know what this means?" and "Don't you know what this means?" to great and approximately equal effect. But, BUT, the man who's wife has been cheating on him from the beginning (If you're a cylon, you've been one from the start) and finds all the love letters she has been keeping since the two commenced their relationship, and just happens to have his friend sitting next to him when he discovers all this actually has to use "Don't you know what this means?" instead of "Do you know what this means?" Why in this case, of all cases (like the "realize" case above) do we have a difference in meanning? Well, let's talk about the emtions of the speaker and the context so we can figure out what's going on here as opposed to the mad scientist and the simple classroom "understanding" case. With the husband, when he's addressing his friend, there is absolutely no sense of discovery to be found in the other party, the other party already knows, he sees the letters, he knows of the relationship in every sense that the husband now does. The husband's use of "Don't you realize what this means?" is filled with a certain sense of reproach, disgust, and "writing off". There is nothing to be said to the other party. Maybe the friend tried to say something, or suggest something and husband cut him off "Don't you realize what this means?" The is the "full rhetorical", its implication is "you know exactly what this means." There is no question here with this "full rhetorical question". It is direct statment of fact "You know exactly what this means." If, IF, the husband happened to say "Do you realize what this means?" this meaning is incredibly different, its implication is that the other party doesn't yet understand something relating to this discovery. The husband could follow this up with some observations or related things that the friend doesn't know about, "Do you realize what this means? All those trips when I trailed her I really had something to be worried about. All those times I suspected she was flirting with those guys I was right." These last two sentences are intented to inform and update the friend. They follow the question with a response. The response is the answer to the question, it IS what the letters and discovery means.
Put this in contrast to "Don't you realize what this means? All those trips she took where I tailed her for hours. All those times I suspected she was flirting with other guys, they werel all true, I was right all along." Where the follow up comments aren't the "response" to the rhetorical question, but rather the additional commentary, simply more follow up. The friend already knows these things, he can't be informed about them. These are throw-away lines, added expository details. In fact these expository lines could replace the first "full rhetorical" with that "full rhetorical" either being assumed or it's meaning is transferred to these already known truths being restated. Or, to put it another way
"Don't you realize what this means? All those trips she took where I tailed her for hours. All those times I suspected she was flirting with other guys, they werel all true, I was right all along."
is equal to
" All those trips she took where I tailed her for hours. All those times I suspected she was flirting with other guys, they werel all true, I was right all along."
But, and seriously, thanks for reading, all you really need to know is at the end of the sentence you add a 不成 or a 吗。
I'll let you figure out the essential features of the other two situations, mad scientist, teacher. I'm tired and I don't want to write this post anymore