Correct way of writing this sentence [duplicate]

Possible Duplicate:
A number of questions “has been” or “have been” asked?

Which one of these sentences is correct?

The number of students are increasing.

or

The number of students is increasing.

Initially I thought the first option wass correct, but later in the solution section of a book, the answer shown is that the sentence is with "is" not "are" is correct.

Am I correct, or is the book correct?

Here only the second can be correct, because the definite article means that "the number" refers to an exact number/group of students, which is singular.

However, were you to say "a number of students", this could be interpreted as "a few students" or "a couple of students", in which case you would say "are".

Like this:

The number of students is increasing. = The total count of students is getting bigger.

Versus

A number of students are complaining about the situation. = A couple of students are complaining about the situation.