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My textbook ( "Elementary Korean", Tuttle) features the question (where N is a noun)
이것은 N 입니까?
along with the following two answers:

네, (그것은) N 입니다
as the translation of "Yes, that is N", as well as

아니요, (그것은)N 이/가 아닙니다.
as the translation for "No, that is not N".

Why do I need the subject particle when negating the sentence, but not when just stating it? It's probably not a big deal but it really bothers me not to know why they chose to use the subject particle only in the second sentence.

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You have some misconceptions:

  1. 이/가 before the 아닙니다 is NOT the subject particle. Technically, it's a complement case marker(보격 조사).

  2. You can drop the complement case marker. For instance you can say "그거 먹는 거 아니야" that's not for eating.

  3. 입니다/입니까(original 이다) is a predicate case marker(서술격 조사). So you don't put a space between whatever word and 입니다. It's the only marker that can be inflected.

Complement case markers have the same form as the subject case marker, but it's technically different, although many textbooks don't make a distinction.

It's called a complement case marker when it's used with 되다/아니다.

내가 선생님 되었다. -> Complement case marker "I became a teacher"
나는 선생님 아니다. -> Complement case marker "I am not a teacher"
선생님 달린다. -> Subject case marker "The teacher runs"

The predicate case marker '이다' is exceptional in that it is the only marker that is inflected, like other verbs/adjectives. But come to think of it, it's only natural to be inflected, because the predicate case marker makes an adjective and adjectives are inflected.

나는 선생님이다. 나는 선생님입니다. 나는 선생님이야. 나는 선생님이에요. 나는 선생님입니까? 내가 선생님인가?

So you don't put a subject marker on 선생님 because it's not the subject(it's a predicate). Instead you put the predicate case marker.

To make it clear, in the sentence "그것은 N이/가 아닙니다", "그것은" is the subject, and N is the complement. So you can say "그것 A가 아닙니다" without changing the meaning. Here, the 이 is the subject marker, and 가 is the complement marker. The concept of the complement marker was introduced because there can be only one subject per sentence(or clause).

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  • Are there any other verbs that are used with the complement case marker? – busukxuan Sep 20 '16 at 2:02
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    @busukxuan No, only 되다 and 아니다. Sentences like 내가 펜이 있다 is interpreted as a sentence with two clauses, "내가 (펜이 있다)". 펜이 있다 is nested in the whole sentence. 펜이 is the subject of the predicate 있다, and 내가 is the subject of the predicate clause (펜이 있다). – MujjinGun Sep 20 '16 at 4:41
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Adding to the great answer, we, as foreigners, will interpret the concept like this:

There are some words that MEANS a verb, but ACTS as an adjective. Examples are 아니다, 되다, 있다(to have), 없다(to not have), 그립다,...

For these words, NO object particle(을/를) can be attached to the object. So when expressing sentences like "I have a pen", we use 이/가 as a complement paricle instead of the object particle.

Example: (제가) 펜이 있다.

되다 is a bit different, as using the complement particle is the only special case. It still acts as a verb in other situations, such as the plain form: 된다.

아닌다, 그립는다 are all wrong forms.

없는다 or 있는다 is correct only when it means (to be at a place).

Another special case: when using 있다 and 없다 to describe a noun, we use 있는 and 없는 instead of 있은, 없은 as it was supposed to be.

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    펜이 is not a complement, it's the subject of the predicate 있다. Analogously, "the pen" is the subject of the sentence "the pen exists". 제가 is the subject of the whole sentence whose predicate is the clause "펜이 있다". "Complement" only exists in a sentence with 되다/아니다, according to schoolbook grammar. The concept of a "complement" is heavily debated though. – MujjinGun Sep 20 '16 at 4:45
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나는 학생이다 (positive copula)
나는 학생이 아니다 (negative copula)

Some grammarians/linguists will tell you that 이/가 is always omitted in the positive copula.

나는 학생이 이다 (you will never see this)
나는 학생이다
나는 학생이다

Particle 이/가 was originally just 이. Since the sentence has the same sound twice (이+이), it simply gets contracted to 이.

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