We form negative declarative clauses with not after be (she is not talking), after modal verbs (they must not go) and after auxiliary verbs do and have (we did not like it; they have not eaten). Cambridge Dictionary
And I found the following sentence in Oxford Dictionary:
perhaps not surprisingly, he was cautious about committing himself.
But perhaps is not a helping verb. So, Can we use "not" without helping verb?
Update:
The Negation Rule: In English, in order to claim that something is not true, you form a negative sentence by adding the word not after the first auxiliary verb in the positive sentence. If there is no auxiliary verb in the positive sentence, as in the Present Simple and Past Simple tenses, then you add one (in both these cases, the auxiliary verb do). White Smoke
We make negatives by putting not after the first part of the verb. British Council
Another example of using not without an auxiliary verb:
I can think of a hundred reasons not to come.
is the above negative clauses? if yes, What is the rule to use not without auxiliary verb? if no, what is the part of speech of not in the above sentence?
Best Answer
If not is negating a finite verb, i.e. one that is connected to and inflected by a subject, then not should come after a helping verb, meaning verb to be, modals and auxiliary verbs be, do and have, for example:
This is what is normally called negative clause
However, if the verb being negated is nonfinite, which means infinitives with and without to, participles, except for perfect tenses, and -ing gerunds, except for progressive tenses, then not normally comes before the verb it negates:
Likewise, it is possible to negate words and phrases other than verbs, normally by placing not right before the word or phrase one wishes to negate. Your first example falls into this category.