Many students have trouble understanding the difference between must and have to, and it's really hardly surprising.
Before we get into that, I'd like to say, first, that we can use have to and have got to without any difference in meaning. The difference is the same as that between have and have got: the former is more formal, the latter is more colloquial. Have is more common in US English, and have got is more common in UK speech.
Most of the time, the difference between must and have to (when they are used in the positive) is so subtle as to make it rather pedantic.
I've got to go to the supermarket tomorrow.
I must go to the supermarket tomorrow.
The general explanation is that have (got) to implies a third party obligation and must implies a personal obligation.
Therefore the first sentence suggests that someone else wants me to go to the supermarket while the second sentence suggests that I personally want to. Regardless of which is used, it doesn't prevent the listener from understanding the essence of what is being communicated: the action of going to the supermarket.
What is important to note, however, is that when an adverb of frequecy is present, we generally prefer to use have (got) to.
I always have to eat breakfast.
I've always got to eat breakfast.
On the other hand, in negative usage, the difference is significant.
When we want to say that something is not allowed, we use mustn't, but to say that it is not necessary, we use don't have to or haven't got to.
You mustn't smoke in here. (Smoking is prohibited)
You don't have to come in early tomorrow. (You can if you want, but it isn't necessary)
Rather than spending too much time trying to explain the subtle differences between must and have to when used positively it is far more important to help them understand the frequent errors they make with must. Here are two very common examples.
He doesn't must come tomorrow. (He mustn't/doesn't have to come tomorrow)
Remember that the modal verbs must, can, could, may, might, will, would, shall, and should are never followed by 'to' nor are they preceded by an auxiliary such as 'do'.
Note that these verbs do require a 'to':
have (got) to
need to (need functions as a normal verb in this case, although need also exists as a modal verb)
I've got to publish this article before the end of this week.
They ought to give you the contract - you're the best person for the job!
I need to have my hair cut.
Note the difference: You needn't come and get me; I'll make my own way.
For some practice, you can try Grammar Secrets.
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