In English, in order to express negation, one forms a negative sentence by adding the word "not" to the positive sentence. Often, the auxiliary verb "do" is required to express negation.
For example: WE SING SONGS (positive sentence), WE DO NOT SING SONGS (negative sentence) - requires auxiliary verb "do"
For example: SHE IS PRETTY (positive sentence), SHE IS NOT PRETTY (negative sentence) - does not require auxiliary verb "do"
Depending on the tense of the sentence, "do" must be conjugated to match the tense of the main verb in the positive sentence. For example:
I dance - present tense, positive sentence
I do not dance - present tense, negative sentence...
I danced - past tense, positive sentence
I did not dance - past tense, positive sentence - in this case, the auxiliary verb "do" is conjugated into the past tense, which is "did"
Here are some examples of negation in the six main tenses of English:
I dance, I do not dance - present tense
I danced, I did not dance - past tense
I will dance, I will not dance - future tense
I have danced, I have not danced - present perfect tense
I had danced, I had not danced - past perfect tense
I will have danced, I will not have danced - future perfect
(note that in the last 3 tenses, a form of "do" is not used because there is already an auxiliary verb. To form the perfect tenses in English, we use "have, has or had" as an auxiliary verb).
There are also these forms:
I am dancing, I am not dancing (present progressive)
I was dancing, I was not dancing (past progressive)
I will be dancing, I will not be dancing (future progressive)
I have been dancing, I have not been dancing (present perfect progressive)
I had been dancing, I had not been dancing (past perfect progressive)
I will have been dancing, I will not have been dancing (future perfect progressive)
Complicated? Ha ha!
English also uses other words to express the idea of negation - words such as "no" or "neither" or "nor" or "never" or"seldom" or "hardly", etc. These are called "negative adverbs" and can also be used to create the idea of negation. For example:
She is no friend of mine.
We are neither hot nor cold.
He never answers.
He seldom visits
She hardly speaks
I rarely dance.