Translation:That is not right.
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The only rule i know with "nga", is that is used when you are talking about what you want, to make it sound "softer" or less demanding I guess. I.e. 「ビールはほしいですが」"biiru wa hoshii desu nga". But ive only heard this on a Pimsleur CD and it may be outdated. What quick and little research I did just stated "nga" is an old way of pronouncing が. Or that you only add the "n" if the "g" sound falls between two vowel sounds.
In my own experience, I've noticed that female speakers tend to nasalize が, at least in certain situations. A good example is おねがいします. I've heard multiple female speakers pronounce it as 'o-ne-nga-i-shi-masu' whereas I've always heard male speakers pronounce it as 'o-ne-ga-i-shi-masu'.
I figured it was a difference in male and female pronunciations. I don't know for sure though.
This is because you aren't changing a word to a negative form. As someone says previously, the word is more literally translated to "different." If you use ません you would be translating the word to mean "not different." Thus, ちがいますmeans "That's different" or "that's not correct" while ちがいません would mean "that isn't different" and likely "that's not incorrect."
It is kinda similar. 間違う (まちがう) explicitly means "to make a mistake", and 違う (ちがう) means "to differ" in a broader sense. Although I believe the latter is used more when someone is wrong, because it is less direct (and hence less confrontational or aggressive, and more respectful and polite). Actually, I think in 違う the subject is usually the fact, and in 間違う the subject is the person, which makes 違う even more desirable by not being personal. But I may be wrong. I don't have that much experience with Japanese yet
In English, it may often be more diplomatic to say "I disagree" than to say "you're wrong". (The latter would be called for, though, if there is risk involved in not being direct, e.g., in a disagreement between doctors about treating a patient.)
Is it wrong to translate ちがいます。as "I disagree."?
ちがう has the general feel that something is wrong (like an answer for example) and feels very strong (a sentiment not captured by "I disagree"). To express disagreement as a softer rejection would be more like "そうではないと思います". There is a verb to agree, but I wouldn't use it to express this kind of thought...
But that's all my personal thoughts, take it with a grain of salt if you will. Can't really back it up now without any internet resources for "reasons"...