"I don't know what I'd do without you"
"I'm not know what I'd do without you"
Are these sentences correct?
What are the rules to use "not" in a sentence?
To what is not pointing in the above sentences?
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When you are negating with not, look at the finite verb in the clause, the one that carries person and tense.
If it is an auxiliary verb (the first verb in a verbal construction) or a form of be, you may simply add not:
He is here. ... The verb is a form of be. >>> He is not here.
He has arrived. ... The verb has is an auxiliary. >>> He has not arrived.
He must have arrived. ... The verb must is an auxiliary. >>> He must not have arrived.
Otherwise, you require DO-support:
He came today. ... The verb came is not an auxiliary. >>> He did not come today.
He arrived today. ... The verb arrived is not an auxiliary. >>> He did not arrive today.
He has a car. ... Here has is not an auxiliary but a lexical verb meaning 'possesses'. >>> He does not have a car.
Until the twentieth century it was common to negate lexical HAVE without DO-support, but this is disappearing. Even earlier any verb might be negated without DO-support, but that died out by the nineteenth century.
The constructions you are looking at are different and represent important structures in English.
"I do X" where X is a verb is an example of do-support. Technically, you don't need to use do-support in positive statements, e.g. "I do know". When do-support is used with positive statements then do acts to emphasise the claim. E.g.
A: Do you know your twelve times table?
B: Yes, I know it.
A: I bet you're lying to sound clever.
B: I am not! I do know it!
In the above example the do is used as emphasis by B.
In negative situations, do is used as an auxillary (or helper) verb when denying something, e.g. "I do not know." This is because *"I not know" is not grammatical.
Do-support is only used with the infinitive, i.e. the form of verb that matches its name.
So you say
In these situations not is pointing to the verb being supported by do. I.e. not is negating the verb.
Your other construction: "I am X", where X is a verb. Here am is also a helper verb. Am is a form of the verb be. Using be in this way is using it as a copular verb. That means it is joining something to something else. For example in "I am happy" am is joining the subject I to the adjective (descriptive word) happy. The same is happening in "He is cold", he is the subject and cold is the adjective it is being joined to.
When used with a verb, be joins the subject to the finite forms of the verb (finite meaning the forms that change based on who is the subject and whether you're talking about past, present, future, etc.). For example: "I am writing an answer". This uses be to create the present progressive form of write.
When creating a negative statement about your actions, then you use the same form of the verb, e.g. "I am not writing an answer".
In this example not is pointing to writing.
Back to your examples: *"I am not know" is wrong because know is the infinitive form, and am must work with a finite form. However, because of the type of verb that know is, saying *"I am not knowing" is also wrong. This is complicated to explain and I can't think of other verbs that behave like this right now.
As a rule of thumb, when making statements about what you or others do not know, use do, never use be.
"I don't know " is correct. It is the negative of "I know". Other examples:
I like it. (positive) - I do not like it. (negative)
I watch TV. (positive) - I do not watch TV. (negative)
"I'm not know" is wrong. If you try to say the negative of "I know", that is "I don't know". If you want to use "I am not", it is the negative of "I am". For example"
I am from France.(positive) - I am not from France (negative)
I am tired. (positive) - I am not tired. (negative)
I am cooking. (positive) - I am not cooking. (negative)
I get that what you are trying to write is the negation of "I know" in "I know what I'd do without you."
Using the simple present, the negation of a verb that is not be, used in a sentence that is not a question, is done by adding "don't" or "doesn't" (the latter is for the third person singular) between the subject, and the verb. In your case, the sentence becomes:
I don't know what I'd do without you.