"I am a child."
Translation:Ich bin ein Kind.
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I'm not sure what you mean by "counter", but the form "einen" is only used with masculine nouns.
Masculine: ein (nominative) -> einen (accusative)
Feminine: eine (nominative & accusative)
Neuter: ein (nominative & accusative)
The accusative case is used for most direct objects (e.g. A man reads A BOOK/ A dog bites A CHILD). The nominative case is used for the subject (e.g. A MAN reads a book/ A DOG bites a child) and after the verbs "sein" (to be) and "werden" (to become) (e.g. The man is A TEACHER).
Here, "a child" is used after the verb "to be", so we have to use the nominative case. "Kind" is a neuter noun, so the article is "ein".
"Bin" (am) is used with "ich" (I): ich bin = I am
"Bist" (are) is used with "du" (you): du bist = you are
Note that German has several different words for "you" (du, ihr, Sie), but the verb form "bist" is only used with "du". "Du" (you) is used when talking to a close friend, relative, child, God, fellow student or someone on the Internet. The ending -st (bist) is generally used with "du", also with other verbs. "Du" corresponds to the archaic English pronoun "thou" that you can find in texts by Shakespeare, e.g. "thou canst" (modern English: you can, German: du kannst).
Yes, as in English, you have to add the personal pronouns. However, again as in English, it's sometimes possible to use incomplete sentences in colloquial speech (What did I do this morning? Got up, had breakfast, brushed my teeth and rushed to work.). But this is not "proper" formal grammar.